\documentclass[twoside,11pt]{article}
%\input epsf %% at top of file

\newif\ifpdf %
\ifx\pdfoutput\undefined %
    \pdffalse %
\else %
    \pdftrue %
\fi %

\usepackage[dvipsnames,usenames]{color}

\usepackage{jair}

\usepackage{epsf}
\usepackage{epsfig}
\usepackage{rotating}

\usepackage{latexsym,amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{pifont} %% for \ding
\usepackage[only bigsqcap]{stmaryrd} %% for \bigsqcap
\usepackage{hhline}

\ifpdf
  \pdfoutput=1
  \usepackage[%
    pdftex,%
    colorlinks,%
    linkcolor=blue,%
%    hyperindex,%
    plainpages=false,%
    bookmarks,
    bookmarksopen,%
    bookmarksnumbered,%
%    verbose
  ]{hyperref}

  \hypersetup{%
    pdftitle = {Combining Spatial and Temporal Logics: Expressiveness vs. Complexity}, %
%    pdfsubject = {}, %
    pdfkeywords = {spatial logics, temporal logics, combining logic, spatio-temporal logics, computational complexity, undecidability}, %
    pdfauthor = {David Gabelaia, Roman Kontchakov, Agi Kurucz, Frank Wolter and Michael Zakharyaschev},
%    pdfcreator = {\LaTeX\ with package \flqq hyperref\frqq},
%    pdfproducer = {pdfeTeX-0.\the\pdftexversion\pdftexrevision},
    pdfstartview = {FitH}
  }

%  \usepackage{thumbpdf}
\else
%  \usepackage{hyperref}
\fi


\usepackage{theapa}

\jairheading{23}{2005}{167-243}{07/04}{02/05}%
\ShortHeadings{Combining Spatial and Temporal Logics: Expressiveness vs. Complexity}%
{Gabelaia, Kontchakov, Kurucz, Wolter, \& Zakharyaschev}%
\firstpageno{167}%


\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{example}[theorem]{Example}
%\newtheorem{question}[theorem]{Question}
\newtheorem{claim}[theorem]{Claim}
%\newtheorem{sclaim}{\it Claim}[theorem]                %%% needed?
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}


\renewcommand{\qedsymbol}{\ding{113}}
\renewcommand{\proofname}{\bf Proof}
\newcommand{\proofof}[1]{\bf Proof of #1}


\newcommand{\PTL}{\mathcal{PTL}}
\newcommand{\QTL}{\mathcal{QTL}}

\newcommand{\FSA}{{\bf FSA}}
\newcommand{\FCA}{{\bf FCA}}

\newcommand{\SF}{{\mathcal{S}\mathrm{4}}}
\newcommand{\SFU}{\SF_u}
\newcommand{\RC}{\mathcal{RC}}
\newcommand{\RCm}{\RC^-}
\newcommand{\RCp}{\RC^{\textit{max}}}
\newcommand{\BRCCE}{\mathcal{BRCC}\textrm{-}8}
\newcommand{\RCCE}{\mathcal{RCC}\textrm{-}8}


\newcommand{\IB}{{\bf I}}
\newcommand{\CB}{{\bf C}}

\newcommand{\agicirc}{\raisebox{0.1em}{$\scriptstyle\bigcirc$}}

\newcommand{\Ubox}{\setlength{\unitlength}{1pt}\begin{picture}(8,10)
\put(0,0){$\Box$} \put(2.3,2.25){\small${}_{{}^\forall}$}
\end{picture}
}
\newcommand{\Udiamond}{\setlength{\unitlength}{1pt}\begin{picture}(8,10)
\put(0,0){$\Diamond$} \put(2.5,2.2){\small${}_{{}^\exists}$}
\end{picture}
}

%\newcommand{\Dvs}{\raisebox{-0.18em}{\RightDiamond}}
%\newcommand{\Dvs}{\Diamond}
%\newcommand{\Bvs}{\rlap{$\Box$}{\rule{0.35em}{0em}\rule[0.1em]{0.3em}{0.5em}}}

\newcommand{\Dvs}{{\bf S}}


\newcommand{\until}{\mathbin{\mathcal{U}}}
\newcommand{\nxt}{\mathord{\agicirc}}
\newcommand{\since}{\mathbin{\mathcal{S}}}

\newcommand{\nb}[1]{$|$\marginpar{\tiny #1}}

\newcommand{\auf}{\left\langle}
\newcommand{\zu}{\right\rangle}
\newcommand{\cc}{\rm}

\newcommand{\textv}[1]{\textsl{#1}}
\newcommand{\textp}[1]{\textsf{#1}}

\newcommand{\reg}[1]{\bigl\lceil{#1}\bigr\rceil}
\newcommand{\imp}{\sqsubset}
\newcommand{\equi}{\equiv}

\newcommand{\EQ}{\textsf{EQ}}
\newcommand{\PT}{\textsf{P}}
\newcommand{\NE}{\textsf{E}\,}

\newcommand{\cmplx}[2]{\hspace*{-1em}\scriptsize\begin{tabular}{c}\small #1\rule[-4pt]{0pt}{16pt}\\#2\end{tabular}\hspace*{-1em}}

\newcommand{\tlabel}[1]{\marginpar{\tiny #1}\label{#1}}
\renewcommand{\tlabel}[1]{\label{#1}}

\newcommand{\jump}[1]{\mathbf{J}_{2,d}\,{#1}}

\newcommand{\varset}{\Omega}
\newcommand{\aform}{\textit{AtFm}_{\varset}}
\newcommand{\aformn}{\textit{AtFm}_{\varset_0}}
\newcommand{\aformf}{\textit{AtFm}_{\varset_\varphi}}
\newcommand{\aformfp}{\textit{AtFm}_{\varset_\varphi'}}

%\usepackage{gkwzsym}

\date{ \ }

\begin{document}

\title{Combining Spatial and Temporal Logics:\\ Expressiveness vs.
Complexity}

\author{\name David Gabelaia \email gabelaia@dcs.kcl.ac.uk \\
        \name Roman Kontchakov \email romanvk@dcs.kcl.ac.uk \\
        \name Agi Kurucz \email kuag@dcs.kcl.ac.uk \\
        \addr Department of Computer Science, King's College London \\
              Strand, London WC2R 2LS, U.K.
        \AND
        \name Frank Wolter \email frank@csc.liv.ac.uk \\
        \addr Department of Computer Science, University of Liverpool \\
              Liverpool L69 7ZF, U.K.
        \AND
        \name Michael Zakharyaschev \email mz@dcs.kcl.ac.uk \\
        \addr Department of Computer Science, King's College London \\
              Strand, London WC2R 2LS, U.K.
}


\maketitle

\begin{abstract}
In this paper, we construct and investigate a hierarchy of
spatio-temporal formalisms that result from various combinations of
propositional spatial and temporal logics such as the propositional
temporal logic $\PTL$, the spatial logics $\RCCE$, $\BRCCE$, $\SFU$
and their fragments. The obtained results give a clear picture of
the trade-off between expressiveness and `computational
realisability' within the hierarchy. We demonstrate how different
combining principles as well as spatial and temporal primitives can
produce NP-, PSPACE-, EXPSPACE-, 2EXPSPACE-complete, and even
undecidable spatio-temporal logics out of components that are at
most NP- or PSPACE-complete.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Introduction}

Qualitative representation and reasoning has been quite successful
in dealing with both time and space.  There exists a wide spectrum
of temporal logics (see, e.g.,
\citeR{Allen83,Clarke&Emerson81,Manna&Pnueli92,Gabbayetal94,Benthem95}).
There is a variety of spatial formalisms (e.g.,
\citeR{Clarke81,Egenhofer&Franzosa91,Randelletal92,Asher&Vieu95,Lemon&Pratt98}).
In both cases determining the computational complexity of the
respective reasoning problems has been one of the most important
research issues. For example, Renz and Nebel
\citeyear{Renz&Nebel99} analysed the complexity of $\RCCE$, a
fragment of the \emph{region connection calculus} $\mathcal{RCC}$
with eight jointly exhaustive and pairwise disjoint base relations
between spatial regions introduced by Egenhofer and Franzosa
\citeyear{Egenhofer&Franzosa91} and Randell and his colleagues
\citeyear{Randelletal92}; Nebel and B\"urckert
\citeyear{Nebel&Buerckert95} investigated the complexity of
Allen's interval algebra; numerous results on the computational
complexity of the point-based propositional linear temporal logic
$\PTL$ over various flows of time were obtained by Sistla and
Clarke \citeyear{Sistla&Clarke85} and Reynolds
\citeyear{Reynolds03,Reynolds99}. In many cases these
investigations resulted in the development and implementation of
effective reasoning algorithms (see, e.g.,
\citeR{Wolper85,Smith&Park92,Egenhofer&Sharma93,Schwendimann98b,Fisheretal01,Renz&Nebel01,Hustadt&Konev03}).

The next apparent and natural step is to \emph{combine} these two
kinds of reasoning. Of course, there have been attempts to
construct spatio-temporal hybrids. For example, the intended
interpretation of Clarke's \citeyear{Clarke81,Clarke85}
region-based calculus was spatio-temporal.  Region connection
calculus $\mathcal{RCC}$ \cite{Randelletal92} contained a function
$\mathsf{space}(X,t)$ for representing the space occupied by
object $X$ at moment of time $t$. Muller \citeyear{Muller98}
developed a first-order theory for reasoning about motion of
spatial entities. However, all of these formalisms turn out to be
`too expressive' from the computational point of view: they are
\emph{undecidable}. Moreover, as far as we know, no serious
attempts to investigate and implement partial (say, incomplete)
algorithms capable of spatio-temporal reasoning with these logics
have been made.

The problem of constructing spatio-temporal logics with better
algorithmic properties and analysing their computational
complexity was first attacked by Wolter and Zakharyaschev
\citeyear{Wolter&Z00kr}; see also the `popular' and extended
version \cite{Wolter&Z02millenium} of that conference paper, as
well as \cite{Bennett&Cohn99,Bennettetal02,Gerevini&Nebel02ecai}.

\begin{figure}[t]
\begin{center}
\unitlength=0.18mm%
\begin{picture}(480, 420)%
\put(0,60){\includegraphics{pic1}}

\put(33,670){%
%\textcolor{Red}{
\put(0,0){%
\put(30,-640){\vector(1,0){420}}%
\put(440,-630){$\mathfrak{F}$}%
\put(425,-663){time}%
\put(5,-380){\begin{rotate}{90}space\end{rotate}}%
%
\multiput(30,-643)(120,0){4}{\line(0,1){6}}%
\put(26,-665){$0$}%
\put(146,-665){$1$}%
\put(266,-665){$2$}%
\put(386,-665){$3$}%
}
%
\put(20,-485){$X$}%
\put(145,-410){$X$}%
\put(260,-390){$X$}%
\put(385,-380){$X$}%
\put(385,-500){$X$}%
%}
}%
\end{picture}
\end{center}
\caption{Topological temporal model.}\label{fig:ttmodel}
\end{figure}


The main idea underlying all these papers is to consider various
combinations of `well-behaved' spatial and temporal logics. The
intended spatio-temporal structures can be regarded then as the
Cartesian products of the intended time-line and topological (or
some other) spaces that are used to model the spatial dimension.
Figure~\ref{fig:ttmodel} shows such a product (of the flow of time
$\mathfrak{F} = \auf \mathbb N,<\zu$ and the two-dimensional
Euclidean space $\mathfrak{T}$) with a moving spatial object $X$.
The moving object can be viewed either as a 3D spatio-temporal
entity (in this particular case) or as the collection of the
`snapshots' or slices of this entity at each moment of time; for a
discussion see, e.g., \cite{Muller98a} and references therein. In
this paper, we use the snapshot terminology and understand by a
\emph{moving spatial object} (or, more precisely, interpret such
an object as) any set of pairs $\auf X, t\zu$ where, for each
point $t$ of the flow of time, $X$ is a subset of the topological
space---the \emph{state of the spatial object at moment} $t$.

The expressive power (and consequently the computational
complexity) of the combined spatio-temporal formalisms obviously
depends on three parameters:
%
\begin{enumerate}
\item the expressivity of the spatial component,
%
\item the expressivity of the temporal component, and
%
\item the \emph{interaction} between the two components allowed in
the combined logic.
\end{enumerate}
%
Regardless of the chosen component languages, the minimal
requirement for a spatio-temporal combination to be useful is its
ability to
%
\medskip

\emph{express changes in time of the truth-values of purely
spatial propositions}.\hfill ({\bf PC})

\medskip\noindent
%
Typical examples of logics meeting this \emph{spatial
propositions' truth change principle} are the combinations of
$\RCCE$ and Allen's interval calculus
\cite{Bennettetal02,Gerevini&Nebel02ecai} and those combinations
of $\RCCE$ and $\PTL$ introduced by Wolter and Zakharyaschev
\citeyear{Wolter&Z00kr} that allow applications of temporal
operators to Boolean combinations of $\RCCE$ relations. Languages
satisfying (PC) can capture, for instance, some aspects of the
\emph{continuity of change principle} \cite<see, e.g.,>{Cohn97}
such as
%
\begin{itemize}
\item[(A)] if two images on the computer screen are disconnected
now, then they either remain disconnected or become externally
connected in one quantum of the computer's time.
\end{itemize}
%
Another example is the following statement about the geography of
Europe:
%
\begin{itemize}
\item[(B)] Kaliningrad is disconnected from the EU until the
moment when Poland becomes a tangential proper part of the EU,
after which Kaliningrad and the EU will be externally connected
forever.
\end{itemize}

However, languages meeting (PC) do not necessarily satisfy our
second fundamental \emph{spatial object change principle}
according to which we should be able to

\medskip

\emph{express changes or evolutions of spatial objects in
time}.\hfill ({\bf OC})

\medskip\noindent
%
In logical terms, (PC) refers to the change of truth-values of
propositions, while (OC) to the change of extensions of
predicates; see Fig.~\ref{fig:nexttime} where $\nxt X$ at moment
$t$ denotes the state of $X$ at moment $t+1$. Here are some
examples motivating (OC):
%
\begin{figure}[t]
\begin{center}
\unitlength=0.15mm%
\begin{picture}(640, 570)
\put(0,40){\includegraphics{pic2}}

%\textcolor{Red}{%
\put(-177,570){%
\put(273,-400){$X$}%
\put(482,-295){$X$}%
\put(695,-250){$X$}%
\put(262,-278){$\nxt X$}%
\put(255,-190){$\nxt\nxt X$}%
%\put(215,-340){$\Diamond X$}
%
\put(180,-530){\vector(1,0){630}}%
\put(800,-520){$\mathfrak{F}$}
\put(790,-558){time}%
\put(200,-200){\begin{rotate}{90}space\end{rotate}}%
%
\multiput(280,-534)(210,0){3}{\line(0,1){8}}%
\put(275,-560){$t$}%
\put(470,-560){$t+1$}%
\put(680,-560){$t+2$}%
%
\multiput(220,-460)(210,0){3}{$\mathfrak{T}$}
}%
%}%
\end{picture}
\end{center}
\caption{Temporal operators on regions.}\label{fig:nexttime}
\end{figure}
%
\begin{itemize}
\item[(C)] Continuity of change: `the cyclone's current position
overlaps its position in an hour.'
%
\item[(D)] Two physical objects cannot occupy the same space:
`if tomorrow object $X$ is at the place where object $Y$ is today,
then $Y$ will have to move by tomorrow.'
%
\item[(E)] Geographic regions change: `the space occupied by Europe
never changes.'
%
\item[(F)] Geographic regions change: `in two years the EU will be
extended with Romania and Bulgaria.'
%
\item[(G)] Fairness conditions on regions: `it will be raining over
every part of England ever and ever again.'
%
\item[(H)] Mutual exclusion: `if Earth consists
of water and land, and the space occupied by water expands, then
the space occupied by land shrinks.'
\end{itemize}
%
It should be clear that to represent these statements we have to
refer to the evolution of spatial objects in time (say, to compare
objects $X$ and $\nxt\nxt X$)---it is not enough to only take into
account the change of the truth-values of propositions speaking
about spatial objects.

\medskip

\emph{The main aim of this paper is to investigate the trade-off
between the expressive power and the computational behaviour of
spatio-temporal hybrids satisfying the {\rm(PC)} and {\rm(OC)}
principles and interpreted in various spatio-temporal structures.
Our purpose is to show what computational obstacles one can expect
if the application domain requires this or that kind of
interactions between temporal and spatial operators.}


\medskip

The spatio-temporal logics we consider below are combinations of
fragments of $\PTL$ interpreted over different flows of time with
fragments of the propositional spatial logic $\SFU$ (equipped with
the interior and closure operators, the universal and existential
quantifiers over points in space as well as the Booleans)
interpreted in topological spaces. This choice is motivated by the
following reasons:
%
\begin{itemize}
\item The component logics are well understood and established in
temporal and spatial knowledge representation; all of them are
supported by reasonably effective reasoning procedures.
%
\item By definition, implicit or explicit temporal quantification is
necessary to capture (OC), and fragments of $\PTL$ are the weakest
languages with such quantification we know of. Allen's interval
calculus, for example, does not provide means for any
quantification over intervals. It is certainly suitable for
spatio-temporal hybrids satisfying (PC)
\cite<see>{Bennettetal02,Gerevini&Nebel02ecai} but there is no
natural conservative way of combining it with spatial formalisms
to meet (OC). On the other hand, it is embedded in $\PTL$
\cite{Blackburn92}. A natural alternative to $\PTL$ would be the
extension of Allen's calculus by means of quantification over
intervals introduced by Halpern and Shoham
\citeyear{Halpern&Shoham86}, but unfortunately this temporal logic
turns out to be highly undecidable.
%
\item Although the logic $\SFU$ was originally introduced in the
realm of modal logic (see below for details), the work of Bennett
\citeyear{Bennett94}, Nutt \citeyear{Nutt99}, Renz
\citeyear{Renz02} and Wolter and Zakharyaschev
\citeyear{Wolter&Z00ecai} showed that it can be regarded as a
unifying language that contains many spatial formalisms like
$\RCCE$, $\BRCCE$ or the 9-intersections of Egenhofer and Herring
\citeyear{Egenhofer&Herring91} as fragments.
\end{itemize}

Apart from the choice of component languages and the level of
their interaction, the expressive power and the computational
complexity of spatio-temporal logics strongly depend on the
restrictions we may want to impose on the intended spatio-temporal
structures and the interpretations of spatial objects.
%
\begin{itemize}
\item We can choose among different flows of time (say,
discrete or dense, infinite or finite)
%
\item and among different topological spaces (say, arbitrary,
Euclidean or Aleksandrov).
%
\item At each time point we can interpret spatial objects as
arbitrary subsets of the topological space, as regular closed (or
open) ones, as polygons, etc.
%
\item To represent the assumption that everything eventually
comes to an end, we only do not know when, one can restrict the
class of intended models by imposing the \emph{finite change
assumption} which states that no spatial object can change its
spatial configuration infinitely often, or the more `liberal'
\emph{finite state assumption} according to which every spatial
object can have only finitely many possible states (although it
may change its states infinitely often).
\end{itemize}

The paper is organised as follows. In
Section~\ref{section:components} we introduce in full detail the
component spatial and temporal logics to be combined later on. In
particular, besides the standard spatial logics like $\RCCE$ or
the 9-intersections of Egenhofer and Herring
\citeyear{Egenhofer&Herring91}, we consider their generalisations
in the framework of $\SFU$ and investigate the computational
complexity. For example, we show that the maximal fragment of
$\SFU$ dealing with regular closed spatial objects turns out to be
PSPACE-complete, while a natural generalisation of the
9-intersections is still in NP. In
Section~\ref{section:combinations} we introduce a hierarchy of
spatio-temporal logics outlined above, provide them with a
topological-temporal semantics, and analyse their computational
properties. First we show that spatio-temporal logics satisfying
only the (PC) principle are not more complex than their
components. Then we consider `maximal' combinations of $\SFU$ with
(fragments of) $\PTL$ meeting both (PC) and (OC) and see that this
straightforward approach does not work: the resulting logics turn
out to be undecidable. Finally, we systematically investigate the
trade-off between expressivity and complexity of spatio-temporal
formalisms and construct a hierarchy of decidable logics
satisfying (PC) and (OC) whose complexity ranges from PSPACE to
2EXPSPACE. These and other results, possible implementations as
well as open problems are discussed in Section~\ref{conclusion}.
For the reader's convenience most important (un)decidability and
complexity results obtained in this paper are summarised in
Table~\ref{table:complexity} on page~\pageref{table:complexity}.
All technical definitions and detailed proofs can be found in the
appendices.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\section{Propositional Logics of Space and
Time}\label{section:components}



We begin by introducing and discussing the spatial and the
temporal formalisms we are going to combine later on in this
paper.


\subsection{Logics of Space}\label{space}


We will be dealing with a number of logics suitable for
qualitative spatial representation and reasoning: the well-known
$\RCCE$, $\BRCCE$ and $\SFU$, as well as certain fragments of the
last one. The intended interpretations for all of these logics are
topological spaces.

A \emph{topological space} is a pair $\mathfrak{T}=\auf
U,\mathbb{I}\zu$ in which $U$ is a nonempty set, the
\emph{universe} of the space, and $\mathbb I$ is the
\emph{interior operator} on $U$ satisfying the standard
\emph{Kuratowski axioms}: for all $X,Y\subseteq U$,
%
\begin{equation*}
\mathbb{I} (X\cap Y) = \mathbb{I}X\cap \mathbb{I}Y,\quad
\mathbb{I}X \subseteq \mathbb{I}\mathbb{I} X,\quad \mathbb{I}X
\subseteq X\quad\text{and}\quad \mathbb{I} U = U.
\end{equation*}
%
The operator dual to $\mathbb I$ is called the \emph{closure
operator} and denoted by $\mathbb C$: for every $X\subseteq U$, we
have $\mathbb{C}X=U-\mathbb{I}(U-X)$. Thus, $\mathbb I X$ is the
\emph{interior} of a set $X$, while $\mathbb C X$ is its
\emph{closure}. $X$ is called \emph{open} if $X=\mathbb{I}X$ and
\emph{closed} if $X=\mathbb{C}X$. The complement of an open set is
closed and vice versa. The \emph{boundary} of a set $X\subseteq U$
is defined as $\mathbb{C}X - \mathbb{I}X$. Note that $X$ and $U -
X$ have the same boundary.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection[S4u]{$\SFU$}

Our most expressive spatial formalism is $\SFU$---i.e., the
propositional modal logic $\SF$ extended with the universal
modalities. The `pedigree' of this logic is quite unusual. $\SF$
was introduced independently by Orlov \citeyear{Orlov28}, Lewis
\cite<in>{Lewis&Langford32}, and G\"odel \citeyear{Godel33a}
without any intention to reason about space. Orlov and G\"odel
understood it as a logic of `provability' (in order to provide a
classical interpretation for the intuitionistic logic of Brouwer
and Heyting) and Lewis as a logic of necessity and possibility,
that is, as a \emph{modal logic}. Besides the Boolean connectives
and propositional variables, the language of $\SF$ contains two
modal operators: $\IB$ (it is necessary or provable) and $\CB$,
the dual of $\IB$ (it is possible or consistent). In other words,
the formulas of $\SF$ can be defined as follows:
%
\begin{equation}\label{S4:defin}
\tau \quad::= \quad p \quad|\quad \overline{\tau} \quad|\quad
\tau_1\sqcap\tau_2 \quad|\quad \IB\tau,
\end{equation}
%
where the $p$ are variables. Set
$\CB\tau=\overline{\IB\overline{\tau}}$. We denote the modal
operators by $\IB$ and $\CB$ (rather than the conventional $\Box$
and $\Diamond$) because we understand, following an observation
made by several logicians in the late thirties and early forties
\cite{Stone37,Tarski38,Tsao-Chen38,McKinsey41}, $\SF$ as a
\emph{logic of topological spaces}: if we interpret the
propositional variables as subsets of a topological space, the
Booleans as the standard set-theoretic operations, and $\IB$ and
$\CB$ as, respectively, the interior and the closure operators on
the space, then an $\SF$-formula is modally consistent if and only
if it is satisfiable in a topological space---i.e., its value is
not empty under some interpretation.\!\footnote{Moreover,
according to McKinsey \citeyear{McKinsey41} and McKinsey and
Tarski \citeyear{McKinsey&Tarski44}, any $n$-dimensional
\emph{Euclidean} space, for $n\ge 1$, is enough to satisfy all
consistent $\SF$-formulas.}

More precisely, a \emph{topological model} is a pair of the form
$\mathfrak{M} = \auf \mathfrak{T}, \mathfrak{U}\zu$, where
$\mathfrak{T} = \auf U, \mathbb{I}\zu$ is a topological space and
$\mathfrak{U}$, a \emph{valuation}, is a map associating with
every variable $p$ a set $\mathfrak{U}(p)\subseteq U$. Then the
valuation $\mathfrak{U}$ is inductively extended to arbitrary
$\SF$-formulas by taking:
%
\begin{equation*}
\mathfrak{U}(\overline{\tau}) = U - \mathfrak{U}(\tau),\qquad
\mathfrak{U}(\tau_1\sqcap\tau_2) = \mathfrak{U}(\tau_1)
  \cap \mathfrak{U}(\tau_2),\qquad
\mathfrak{U}(\IB\tau) = \mathbb{I}\mathfrak{U}(\tau).
\end{equation*}
%
Expressions $\tau$ of the form \eqref{S4:defin} are interpreted as
subsets of topological spaces; that is why we will call them
\emph{spatial terms}. In particular, propositional variables of
$\SF$ will be understood as \emph{spatial variables}.

The language of $\SFU$ extends $\SF$ with the \emph{universal} and
the \emph{existential quantifiers} $\Ubox$ and $\Udiamond$,
respectively (known in modal logic as the \emph{universal
modalities}). Given a spatial term $\tau$, we write
$\Udiamond\tau$ to say that the part of space (represented by)
$\tau$ is not empty (there is at least one point in $\tau$);
$\Ubox\tau$ means that $\tau$ occupies the whole space (all points
belong to $\tau$). By taking Boolean combinations of such
expressions we arrive at what will be called \emph{spatial
formulas}. A BNF definition looks as follows:%
%
\footnote{Formally, the language of $\SFU$ as defined above is
weaker than the standard one, say, that of Goranko and Passy
\citeyear{Goranko&Passy92}. However, one can easily show that they
have precisely the same expressive power: see, e.g.,
\cite{Hughes&Cresswell96} or \cite{Aiello&Benthem02}.}
%
\begin{align*}
\varphi \quad ::= \quad & \Ubox\tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2,
\end{align*}
%
where the $\tau$ are spatial terms. Set $\Udiamond\tau
=\neg\Ubox\overline{\tau}$. Spatial formulas can be either true or
false in topological models. The \emph{truth-relation} $\mathfrak
M\models\varphi$---a spatial formula $\varphi$ is true in a
topological model $\mathfrak{M}$---is defined in the standard way:
%
\begin{itemize}
\item ${\mathfrak M}\models \Ubox\tau$\quad iff\quad
$\mathfrak{U}(\tau) = U$,
%
\item ${\mathfrak M}\models \neg\varphi$\quad iff\quad
${\mathfrak M}\not\models \varphi$,
%
\item ${\mathfrak M}\models \varphi_1\land\varphi_2$\quad iff\quad
${\mathfrak M}\models \varphi_1$ and ${\mathfrak M}\models
\varphi_{2}$.
\end{itemize}
%
Say that a spatial formula $\varphi$ is \emph{satisfiable} if
there is a topological model $\mathfrak{M}$ such that ${\mathfrak
M}\models \varphi$.

The seemingly simple `query language' $\SFU$ can express rather
complex relations between sets in topological spaces. For example,
the formula
%
\begin{equation*}
\Ubox(q \imp p)\; \land\; \Ubox(p\imp \CB q )\; \land\;\Udiamond p\; \land\;\neg\Udiamond\IB q %
\end{equation*}
%
says that a set $q$ is dense in a nonempty set $p$, but has no
interior (here $\tau_1 \imp \tau_2$ is an abbreviation for
$\overline{\tau_1 \sqcap \overline{\tau_2}}$).

\smallskip

The following `folklore' complexity result has been proved in
different settings \cite<see, e.g.,>{Nutt99,Arecesetal00}:

\begin{theorem}\tlabel{s4u:pspace}
\textup{(i)} $\SFU$ enjoys the exponential finite model property; i.e.,
every satisfiable spatial formula $\varphi$ is satisfiable in a topological space whose size
is at most exponential in the size of $\varphi$.

\textup{(ii)} Satisfiability of spatial formulas in topological
models is {\cc PSPACE}-complete.
\end{theorem}

One way of proving this theorem is first to observe that every
satisfiable spatial formula is satisfied in an \emph{Aleksandrov
model}, i.e., a model based on an Aleksandrov topological
space---alias a standard Kripke frame for $\SF$ \cite<see,
e.g.,>{McKinsey&Tarski44,Goranko&Passy92}.

We remind the reader that a topological space is called an
\emph{Aleksandrov
space}\label{def:Aleksandrov}~\cite{Alexandroff37} if arbitrary
(not only finite) intersections of open sets are open. A
\emph{Kripke frame} (or simply a \emph{frame}) for $\SF$ is a pair
the form $\mathfrak{G}=\auf V, R\zu$, where $V$ is a nonempty set
and $R$ a transitive and reflexive relation (i.e., a
\emph{quasi-order}) on $V$. Every such frame $\mathfrak{G}$
induces the interior operator $\mathbb{I}_\mathfrak{G}$ on $V$:
for every $X \subseteq V$,
%
\begin{equation*}
\mathbb{I}_\mathfrak{G} X = \{ x\in X \mid \forall y\in V\ (xRy
\to y\in X) \}.
\end{equation*}
%
In other words, the open sets of the topological space
$\mathfrak{T}_\mathfrak{G} = \auf V, \mathbb{I}_\mathfrak{G}\zu$
are the \emph{upward closed} (or $R$-\emph{closed}) subsets of
$V$. The \emph{minimal neighbourhood} of a point $x$ in
$\mathfrak{T}_\mathfrak{G}$ (that is the minimal open set to
contain $x$) consists of all those points that are $R$-accessible
from $x$. It is well-known \cite<see, e.g.,>{Bourbaki66} that
$\mathfrak{T}_\mathfrak{G}$ is an Aleksandrov space and,
conversely, every Aleksandrov space is induced by a quasi-order.

Now, to complete the proof, it suffices to recall that $\SF$ is
PSPACE-hard \cite{Ladner77} and use, say, the standard tableau
technique to establish the exponential finite model property and
construct a PSPACE satisfiability checking algorithm for spatial
formulas.

\smallskip

Although being of the same computational complexity as $\SF$, the
logic $\SFU$ is more expressive. A standard example is that
spatial formulas can distinguish between arbitrary and
connected\footnote{We remind the reader that a topological space
is \emph{connected} if its universe cannot be represented as the
union of two disjoint nonempty open sets.} topological spaces.
Consider, for instance, the formula
%
\begin{equation}\label{eq:disconnected}
\Ubox(\CB p \imp p)\; \land\; \Ubox(p \imp \IB p)\; \land\;
\Udiamond p\; \land\; \neg\Ubox p %
\end{equation}
%
saying that $p$ is both closed and open, nonempty and does not
coincide with the whole space. It can be satisfied only in a model
whose underlying topological space is not connected, while all
satisfiable $\SF$-formulas are satisfied in connected (e.g.,
Euclidean) spaces.

Another example illustrating the expressive power of $\SFU$ is the
formula
\begin{equation}\label{eq:pathological}
\Udiamond p\land\Ubox(p \imp \CB \overline{p})
\land\Ubox(\overline{p}
\imp \CB p) %
\end{equation}
%
defining a nonempty set $p$ such that both $p$ and $\overline{p}$
have empty interiors. In fact, the second and the third conjuncts
say that $p$ and $\overline{p}$ consist of boundary points only.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection[RCC-8 as a Fragment of  S4u]{$\RCCE$ as a Fragment of $\SFU$}\label{rcc8}

In qualitative spatial representation and reasoning, it is quite
often assumed that spatial terms can only be interpreted by regular
closed (or open) sets of topological spaces \cite<see,
e.g.,>{Davis90,Asher&Vieu95,Gotts96a}. One of the reasons for
imposing this restriction is to exclude from consideration such
`pathological' sets as $p$ in \eqref{eq:pathological}. Recall that a
set $X$ is \emph{regular closed} if $X = \mathbb{C}\mathbb{I} X$,
which clearly does not hold for any set $p$
satisfying~\eqref{eq:pathological}. Another reason is to ensure that
the space occupied by a physical body is homogeneous in the sense
that it does not contain parts of `different dimensionality.' For
example, the subset $X$ of the Euclidean plane in
Fig.~\ref{fig:regclosed} consists of three parts: a 2D ellipse with
a hole, a 2D circle, and a 1D curve connecting them. This curve
disappears if we form the set $\mathbb{C}\mathbb{I} X$, which is
regular closed because $\mathbb{C}\mathbb{I}\mathbb{C}\mathbb{I} X =
\mathbb{C}\mathbb{I} X$, for every $X$ and every topological space.

\begin{figure}%[ht]
\begin{center}
\unitlength=0.055cm
\begin{picture}(235,27)
\put(0,0){\includegraphics{sptmpic2bis}}
%
\put(-10,3){%
\put(0,12){$X$}%
\put(90,12){$\mathbb{I}X$}%
\put(176,12){$\mathbb{CI}X$}%
}%
\end{picture}
\end{center}
\caption{Regular closure.}\label{fig:regclosed}
\end{figure}
%

In this paper, we will consider several fragments of $\SFU$
dealing with \emph{regular closed sets}. From now on we will call
such sets \emph{regions}. Perhaps, the best known language devised
for speaking about regions is $\RCCE$ which was introduced in the
area of Geographical Information Systems
\cite<see>{Egenhofer&Franzosa91,Smith&Park92} and as a decidable
subset of Region Connection Calculus $\mathcal{RCC}$
\cite{Randelletal92}. The syntax of $\RCCE$ contains eight binary
predicates,
%
\begin{itemize}
\item ${\sf DC}(X, Y)$ --- regions $X$ and $Y$ are disconnected,
\item ${\sf EC}(X,Y)$ --- $X$ and $Y$ are externally connected,
\item ${\sf EQ}(X,Y)$ --- $X$ and $Y$ are equal,
\item ${\sf PO}(X,Y)$ --- $X$ and $Y$ partially overlap,
\item ${\sf TPP}(X,Y)$ --- $X$ is a tangential proper part of $Y$,
\item ${\sf NTPP}(X,Y)$ --- $X$ is a nontangential proper part of
$Y$,
\item the inverses of the last two---${\sf TPPi}(X,Y)$ and ${\sf
NTPPi}(X,Y)$,
\end{itemize}
%
which can be combined using the Boolean connectives. For example,
given a spatial database
describing the geography of Europe, we can query whether the
United Kingdom and the Republic of Ireland share a common border.
The answer can be found by checking whether the $\RCCE$ formula
${\sf EC}(\textit{UK},\textit{RoI})$ follows from the database.

The arguments of the $\RCCE$ predicates are called \emph{region
variables}; they are interpreted by regular closed sets---i.e.,
regions---of topological spaces. The satisfiability problem for
$\RCCE$ formulas under such interpretations is
NP-complete~\cite{Renz&Nebel99}.

The expressive power of $\RCCE$ is rather limited. It only
operates with `simple' regions and does not distinguish between
connected and disconnected ones, regions with and without holes,
etc.\ \cite{Egenhofer&Herring91}. Nor can $\RCCE$ represent
complex relations between more than two regions. Consider, for
example, three countries (say, Russia, Lithuania and Poland) such
that not only each one of them is adjacent to the others, but
there is a point where all the three meet. To express this fact we
may need a ternary predicate like
%
\begin{equation}\label{R2}
{\sf EC3}(\textit{Russia},\textit{Lithuania},\textit{Poland}).
\end{equation}
%

To analyse possible ways of extending the expressive power of
$\RCCE$, it will be convenient to view it as a fragment of $\SFU$
\cite<that $\RCCE$ can be embedded into $\SFU$ was first shown
by>{Bennett94}. Observe first that, for every spatial variable
$p$, the spatial term
%
\begin{equation}\label{eq:cip}
\CB\IB p
\end{equation}
%
is interpreted as a regular closed set in every topological model.
So, with every region variable $X$ of $\RCCE$ we can associate the
spatial term $\varrho_X = \CB\IB p_X$, where $p_X$ is a spatial
variable, and then translate the $\RCCE$ predicates into spatial
formulas by taking:
%
\begin{align*}
{\sf EC}(X, Y) = &\ \Udiamond(\varrho_X \sqcap \varrho_Y) \land
\neg\Udiamond(\IB\varrho_X\sqcap\IB\varrho_Y),\\
{\sf DC}(X, Y) = &\ \neg\Udiamond(\varrho_X\sqcap\varrho_Y),\\
{\sf EQ}(X, Y) = &\ \Ubox(\varrho_X\imp\varrho_Y)\land
\Ubox(\varrho_Y\imp\varrho_X), \\
{\sf PO}(X, Y) = &\ \Udiamond(\IB\varrho_X\sqcap\IB\varrho_Y)\land
\neg\Ubox(\varrho_X\imp\varrho_Y)\land\neg\Ubox(\varrho_Y\imp\varrho_X), \\
{\sf TPP}(X, Y) = &\ \Ubox(\varrho_X\imp\varrho_Y)\land\neg
\Ubox(\varrho_Y\imp\varrho_X)\land
\neg\Ubox(\varrho_X\imp\IB\varrho_Y), \\
{\sf NTPP}(X, Y) = &\ \Ubox(\varrho_X\imp\IB\varrho_Y)\land
\neg\Ubox(\varrho_Y\imp\varrho_X) %
\end{align*}
%
(${\sf TPPi}$ and ${\sf NTPPi}$ are the mirror images of ${\sf
TPP}$ and ${\sf NTPP}$, respectively). The first of these
formulas, for instance, says that two regions are externally
connected iff the intersection of the regions is not empty,
whereas the intersection of their interiors is. It should be clear
that an $\RCCE$ formula is satisfiable in a topological space if
and only if its translation into $\SFU$ defined above is
satisfiable in a topological model.

This translation also shows that in $\RCCE$ any two regions can be
related in terms of truth/falsity of atomic spatial formulas of
the form
%
\begin{equation*}
\Ubox(\overline{\varrho_1\sqcap\varrho_2}),\qquad
\Ubox(\overline{\IB\varrho_1\sqcap\IB\varrho_2}),\qquad  %
\Ubox(\varrho_1\imp\varrho_2) \quad\text{and}\quad
\Ubox(\varrho_1\imp\IB\varrho_2),
\end{equation*}
%
where $\varrho_1$ and $\varrho_2$ are spatial terms of the form
\eqref{eq:cip}. For example, the first of these formulas says that
the intersection of two regions is empty, whereas the last one
states that one region is contained in the interior of another
one. In other words, $\RCCE$ can be regarded as part of the
following fragment of $\SFU$:
%
\begin{align*}
\varrho\quad ::= \quad  & \CB\IB p,\\
\tau \quad ::= \quad & \overline{\varrho_1\sqcap\varrho_2}
\quad|\quad \overline{\IB\varrho_1\sqcap\IB\varrho_2} \quad|\quad
\varrho_1\imp\varrho_2 \quad|\quad
\varrho_1\imp\IB\varrho_2 ,\\
%
\varphi \quad ::= \quad & \Ubox \tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2.
\end{align*}
%
Here we distinguish between two types of spatial terms. Those of
the form $\varrho$ will be called \emph{atomic region
terms}---they represent the (regular closed) regions we want to
compare. Spatial terms of the form $\tau$ are used to relate
regions to each other (note that their extensions are not
necessarily regular closed).

Actually, the fragment introduced above is a bit more expressive
than $\RCCE$: for example, it contains (appropriately modified)
formula~\eqref{eq:disconnected} which can be satisfied only in
disconnected topological spaces, while all satisfiable $\RCCE$
formulas are satisfiable in any Euclidean space \cite{Renz98}.
However, it will be convenient for us {\em not to distinguish}
between these two spatial logics. First, it will turn out that the
same technical results regarding their computational complexity
hold for them even when combined with temporal logics. And second,
the more intuitive and concise language of $\RCCE$ is more
suitable for illustrations. For instance, we do not distinguish
between the region variable $X$ and the region term $\varrho_X$
and use ${\sf DC}(\varrho_1, \varrho_2)$ as an abbreviation for
$\neg\Udiamond(\varrho_1\sqcap\varrho_2)$.

The definition above suggests two ways of increasing the
expressive power of $\RCCE$ (while keeping all regions regular
closed):
%
\begin{itemize}
\item[(i)] by allowing more complex region terms $\varrho$,
and
%
\item[(ii)] by allowing more ways of relating them (i.e.,
more complex terms $\tau$).
\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\subsubsection[BRCC-8 as a Fragment of S4u]{$\BRCCE$ as a Fragment of $\SFU$}

The language $\BRCCE$ of Wolter and Zakharyaschev
\citeyear{Wolter&Z00ecai} \cite<see also>{Balbianietal04} extends
$\RCCE$ in direction (i). It uses the same eight binary predicates
as $\RCCE$ and allows not only atomic regions but also their
intersections, unions and complements. For instance, in $\BRCCE$
we can express the fact that a region (say, the Swiss Alps) is the
intersection of two other regions (Switzerland and the Alps in
this case):
%
\begin{equation}\label{swissalps}
{\sf EQ}(\textit{SwissAlps},
\textit{Switzerland}\sqcap\textit{Alps}).
\end{equation}
%
We can embed $\BRCCE$ to $\SFU$ by using almost the same
translation as in the case of $\RCCE$. The only difference is that
now, since Boolean combinations of regular closed sets are not
necessarily regular closed, we should prefix compound spatial
terms with $\CB\IB$. This way we can obtain, for example, the
spatial term
%
\begin{equation*}
\CB\IB\, (\textit{Switzerland} \sqcap \textit{Alps})
\end{equation*}
%
representing the Swiss Alps. In the same manner we can treat other
set-theoretic operations, which leads us to the following
definition of \emph{Boolean region terms}:
%
\begin{equation*}
\varrho \quad ::= \quad  \CB\IB p \quad|\quad
\CB\IB\overline{\varrho} \quad|\quad
\CB\IB(\varrho_1\sqcap\varrho_2).
\end{equation*}
%
In other words, Boolean region terms denote precisely the members
of the well-known Boolean algebra of regular closed sets. (The
union $\sqcup$ is expressible via intersection and complement in
the usual way.) To simplify notation, given a spatial term $\tau$,
we write $\reg{\tau}$ to denote the result of prefixing $\CB\IB$
to every subterm of $\tau$; in particular,
%
\begin{equation*}
\reg{p} = \CB\IB p,\qquad %
\reg{\overline{\tau}} = \CB\IB \overline{\reg{\tau}}\quad
\text{and}\quad %
\reg{\tau_1\sqcap\tau_2} = \CB\IB (\reg{\tau_1}\sqcap\reg{\tau_2}). %
\end{equation*}
%
Note that $\reg{\tau}$ is (equivalent to) a Boolean region term,
for every spatial term $\tau$. Now the Swiss Alps from the example
above can be represented as
$\reg{\textit{Switzerland}\sqcap\textit{Alps}}$.

It is of interest to note that Boolean region terms do not
increase the complexity of reasoning in arbitrary topological
models: the satisfiability problem for $\BRCCE$ formulas is still
NP-complete (however, it becomes PSPACE-complete if all intended
models are based on connected spaces). On the other hand, $\BRCCE$
allows some restricted comparisons of more than two regions as,
e.g., in \eqref{swissalps}.
%
Nevertheless, as we shall see below, ternary relations like
\eqref{R2} are still unavailable in $\BRCCE$: they require
different ways of comparing regions; cf.\ (ii).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection[RC]{$\RC$}

Egenhofer and Herring \citeyear{Egenhofer&Herring91} proposed to
relate any \emph{two} regions in terms of the
9-inter\-sections---$3\times 3$-matrix specifying
emptiness/nonemptiness of all (nine) possible intersections of the
interiors, boundaries and exteriors of the regions. Recall that,
for a region $X$, these three disjoint parts of the space $\auf
U,\mathbb I\zu$ can be represented as
%
\begin{equation*}
\mathbb I X,\qquad X\cap (U-\mathbb I X) \qquad
\text{and}\qquad U-X,%
\end{equation*}
%
respectively. By generalising this approach to any finite number
of regions, we obtain the following fragment $\RC$ of $\SFU$:
%
\begin{align*}
\varrho \quad ::= \quad & \mbox{Boolean region terms},\\
\tau \quad ::= \quad & \varrho \quad|\quad \IB\varrho \quad|\quad
\overline{\tau} \quad|\quad \tau_1\sqcap\tau_2,\\
%
\varphi \quad ::= \quad & \Ubox \tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2.
\end{align*}
%
In other words, in $\RC$ we can define relations between regions
in terms of emptiness/non\-emptiness of sets formed by using
arbitrary set-theoretic operations on regions and their interiors.
However, nested applications of the topological operators are not
allowed (an example where such applications are required can be
found in the next section).

Clearly, both $\RCCE$ and $\BRCCE$ are fragments of $\RC$.
Moreover, unlike $\BRCCE$, the language of $\RC$ allows us to
consider more complex relations between regions. For instance, the
ternary relation required in~\eqref{R2} can now be defined as
follows:
%
\begin{equation*}
\mathsf{EC3}(X, Y, Z) =
\Udiamond(\varrho_X\sqcap\varrho_Y\sqcap\varrho_Z)\land
\neg\Udiamond(\IB\varrho_X\sqcap\IB\varrho_Y) \land
\neg\Udiamond(\IB\varrho_Y\sqcap\IB\varrho_Z) \land
\neg\Udiamond(\IB\varrho_Z\sqcap\IB\varrho_X).
\end{equation*}
%
Another, more abstract, example is the formula
%
\begin{equation*}
\Udiamond\bigl(
\varrho_1\sqcap\dots\sqcap\varrho_i\land%
\IB\varrho_{i+1}\sqcap\dots\sqcap\IB\varrho_j\sqcap%
\overline{\varrho_{j+1}}\sqcap\dots\sqcap\overline{\varrho_k}\sqcap%
\overline{\IB\varrho_{k+1}}\sqcap\dots\sqcap\overline{\IB\varrho_n}\bigr)
\end{equation*}
%
which says that

\bigskip
\noindent\mbox{}\hfill\parbox{0.85\textwidth}{%
%
\emph{regions $\varrho_1,\dots,\varrho_i$ \underline{meet}
somewhere \underline{inside} the region occupied jointly by all
$\varrho_{i+1},\dots,\varrho_j$, but \underline{outside} the
regions $\varrho_{j+1},\dots,\varrho_k$ and \underline{not inside}
$\varrho_{k+1},\ldots,\varrho_n$.}
%
}\hfill\mbox{}%

\bigskip

\noindent%

Although $\RC$ is more expressive than both $\RCCE$ and $\BRCCE$,
reasoning in this language is still of the same computational
complexity:
%
\begin{theorem}\tlabel{theorem:rc:np}
The satisfiability problem for $\RC$-formulas in arbitrary
topological models is {\cc NP}-complete.
\end{theorem}
%

\begin{figure}
\begin{center}
\setlength{\unitlength}{.08cm}
\begin{picture}(170,30)
\put(0,-7){%
\multiput(40,25)(20,0){4}{\circle{1.5}}%
\put(70,10){\circle{1.5}}%
\put(71,11.5){\vector(2,3){8}}%
\put(72,11){\vector(2,1){26}}%
\put(69,11.5){\vector(-2,3){8}}%
\put(68,11){\vector(-2,1){26}}%
%
\put(120,8){depth 1}%
\put(120,23){depth 0}%
}%
\end{picture}
\end{center}
\caption{$n$-broom (for $n=4$).}\label{fig:nbroom}
\end{figure}

This result will be proved in Appendix \ref{appendix:spatial}.
Lemma~\ref{s4u:simula2} shows that every satisfiable $\RC$-formula
can be satisfied in a model based on the Aleksandrov space that is
induced by a disjoint union of \emph{$n$-brooms}---i.e.,
quasi-orders of the form depicted in Fig.~\ref{fig:nbroom}.
Topological spaces of this kind have a rather primitive structure
satisfying the following property:
%
\begin{itemize}
\item[(rc)] only the roots of $n$-brooms can be boundary points, and
the minimal neighbourhood of every boundary point---i.e., the
$n$-broom containing this point---must contain \emph{at least one}
internal point and \emph{at least one} external point.
\end{itemize}
%
For example, spatial formula \eqref{eq:pathological} cannot be
satisfied in a model with this property, and so it is not in
$\RC$.

By Lemma~\ref{s4u:simula3}, the size of such a satisfying model is
polynomial (in fact, quadratical) in the length of the input
$\RC$-formula, and so we have a nondeterministic polynomial time
algorithm. Actually, the proof is a straightforward generalisation
of the complexity proof for $\BRCCE$ given by Wolter and
Zakharyaschev \citeyear{Wolter&Z00ecai}: the only difference is
that in the case of $\BRCCE$ it is sufficient to consider only
2-brooms (which were called \emph{forks}). This means, in
particular, that ternary relation \eqref{R2}---which is
satisfiable only in a model with an $n$-broom, for $n\ge 3$---is
indeed not expressible in $\BRCCE$.

\begin{remark}
In topological terms, $n$-brooms are examples of so-called
\emph{door spaces} where every subset is either open or closed.
However, the modal theory of $n$-brooms defines a wider and more
interesting topological class known as \emph{submaximal spaces} in
which every dense subset is open. Submaximal spaces have been
around since the early 1960s and have generated interesting and
challenging problems in topology. For a survey and a systematic
study of these spaces see \cite{AC95} and references
therein.
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection[RCmax]{$\RCp$}

One could go even further in direction (ii) and impose no
restrictions whatsoever on the ways of relating Boolean region
terms. This leads us to the \emph{maximal} fragment $\RCp$ of
$\SFU$ in which spatial terms are interpreted by regular closed
sets. Its syntax is defined as follows:
%
\begin{align*}
\varrho \quad ::= \quad & \mbox{Boolean region terms},\\
\tau \quad ::= \quad & \varrho \quad|\quad
\overline{\tau} \quad|\quad \tau_1\sqcap\tau_2\quad|\quad \IB\tau,\\
%
\varphi \quad ::= \quad & \Ubox \tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2,
\end{align*}
%
To understand the difference between $\RC$ and $\RCp$, consider
the $\RCp$-formula
%
\begin{equation}\label{eq:nestedI}
\Udiamond \big(\reg{q_1} \sqcap \overline{\IB\reg{q_1}}\big)\;
\land\; \Ubox\Bigl(\big(\reg{q_1} \sqcap
\overline{\IB\reg{q_1}}\big) \imp \CB\bigl(\IB\reg{q_1} \sqcap
\reg{q_2} \sqcap \overline{\IB\reg{q_2}}\bigr)\Bigr).
\end{equation}
%
It says that the boundary of $\reg{q_1}$ is not empty and that
every neighbourhood of every point in this boundary contains an
internal point of $\reg{q_1}$ that belongs to the boundary of
$\reg{q_2}$ (compare with property (rc) above). The simplest
Aleksandrov model satisfying this formula is of depth 2; it is
shown in Fig.~\ref{fig:tree}.

\begin{figure}
\begin{center}
\setlength{\unitlength}{.08cm}
\begin{picture}(170,40)
\put(0,-10){%
\put(60,10){\circle{1.5}}%
\put(61,11.5){\vector(2,3){18}}%
\put(59,11.5){\vector(-2,3){8}}%
%
\put(50,25){\circle{1.5}}%
\put(51,26.5){\vector(2,3){8}}%
\put(49,26.5){\vector(-2,3){8}}%
%
\put(40,40){\circle{1.5}}%
\put(60,40){\circle{1.5}}%
\put(80,40){\circle{1.5}}%
%
\put(47,9){$q_1\,q_2$}%
\put(37,24){$q_1\,q_2$}%
\put(35,44){$q_1\,q_2$}%
\put(55.5,44){$q_1\;\overline{q_2}$}%
\put(75,44){$\overline{q_1}\;\;\overline{q_2}$}%
%
\put(120,8){depth 2}%
\put(120,23){depth 1}%
\put(120,38){depth 0}%
}%
\end{picture}
\end{center}
\caption{Model satisfying formula
\eqref{eq:nestedI}.}\label{fig:tree}
\end{figure}


The price we have to pay for this expressivity is that the
complexity of $\RCp$ is the same as that of full $\SFU$:

\begin{theorem}\tlabel{fullreg}
The satisfiability problem for $\RCp$-formulas is {\rm
PSPACE}-complete.
\end{theorem}

The upper bound follows from Theorem~\ref{s4u:pspace} and the
lower bound is proved in Appendix~\ref{appendix:spatial}, where we
construct a sequence of $\RCp$-formulas such that each of them is
satisfiable in an Aleksandrov space of cardinality at least
exponential in the length of the formula. The first formula of the
sequence is similar to \eqref{eq:nestedI} above.

It is of interest to note, however, that $\RCp$ is still not
expressive enough to define such `pathological' sets as $p$ in
\eqref{eq:pathological} which is clearly not regular closed.

To conclude this section, we summarise the inclusions between the
spatial languages introduced above:
%
\begin{equation*}
\RCCE \quad\subsetneqq\quad \BRCCE \quad\subsetneqq\quad \RC
\quad\subsetneqq\quad \RCp \quad\subsetneqq\quad \SFU. %
\end{equation*}
%
For more discussions of spatial logics of this kind we refer the
reader to the paper \cite{PrattHartmann02}.








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Temporal Logics}


As was said in the introduction, the temporal components of our
spatio-temporal hybrids are (fragments of) the \emph{propositional
temporal logic} $\PTL$ interpreted in various \emph{flows of time}
which are modelled by strict linear orders ${\mathfrak F}=\auf
W,<\zu$, where $W$ is a nonempty set of \emph{time points} and $<$
is a (connected, transitive and irreflexive) \emph{precedence}
relation on $W$.

The language $\PTL$ is based on the following alphabet:
%
\begin{itemize}
\item propositional variables $p_0,p_1,\dots$,
\item the Booleans $\neg$ and $\land$, and
\item the binary temporal operators $\until$ (`until') and
$\since$ (`since').
\end{itemize}
%
The set of $\PTL$-\emph{formulas} is defined in the standard way:
%
\begin{equation*}
\varphi \quad ::= \quad p \quad|\quad \neg\varphi \quad|\quad
\varphi_1\land\varphi_2 \quad|\quad \varphi_1\until\varphi_2
\quad|\quad \varphi_1\since\varphi_2.
\end{equation*}
%
$\PTL$-\emph{models} are pairs of the form $\mathfrak{M} = \auf
\mathfrak{F}, \mathfrak{V}\zu$ such that $\mathfrak{F} = \auf W,
<\zu$ is a flow of time and $\mathfrak{V}$, a \emph{valuation}, is
a map associating with each variable $p$ a set
$\mathfrak{V}(p)\subseteq W$ of time points (where $p$ is supposed
to be true). The \emph{truth-relation} $(\mathfrak M, w) \models
\varphi$, for an arbitrary $\PTL$-formula $\varphi$ and $w\in W$,
is defined inductively as follows, where $(u, v)$ denotes the open
interval $\{ w\in W \mid u < w < v\}$:
%
\begin{itemize}
\item $(\mathfrak{M}, w) \models p\quad \text{iff} \quad w \in
\mathfrak{V}(p)$,
%
\item $(\mathfrak{M}, w) \models \neg\varphi \quad \text{iff} \quad
(\mathfrak{M}, w) \not\models\varphi$,
%
\item $(\mathfrak{M}, w) \models \varphi_1\land\varphi_2 \quad
\text{iff}\quad (\mathfrak{M}, w) \models \varphi_1 \ \
\text{and}\ \ (\mathfrak M, w) \models \varphi_2$,
%
\item $(\mathfrak{M}, w)\models\varphi_1\until\varphi_2$ \quad iff \quad
there is $v > w$ such that $(\mathfrak{M}, v)\models\varphi_2$ and
$(\mathfrak M, u) \models\varphi_1$ for all $u\in(w, v)$,
%
\item $(\mathfrak{M}, w)\models\varphi_1\since\varphi_2$ \quad iff \quad
there is $v < w$ such that $(\mathfrak{M}, v)\models\varphi_2$ and
$(\mathfrak{M}, u)\models\varphi_1$ for all $u\in(v, w)$.
\end{itemize}
%
A $\PTL$-formula $\varphi$ is \emph{satisfied} in $\mathfrak{M}$
if $(\mathfrak{M}, w)\models \varphi$ for some $w\in W$.

We took the operators $\until$ and $\since$ as primitive simply
because all other important temporal operators can be defined via
them. For example, $\Diamond_F$ (`sometime in the future') and
$\Box_F$ (`always in the future') are expressible via $\until$ as
%
\begin{equation*}
\Diamond_F\varphi = \top\until\varphi, \qquad \Box_F\varphi =
\neg\Diamond_F\neg\varphi,
\end{equation*}
%
($\top$ is the logical constant `true') which means that
%
\begin{itemize}
\item $(\mathfrak{M}, w)\models \Diamond_F\varphi$ \quad iff \quad there
is $v > w$ such that $(\mathfrak{M}, v)\models\varphi$,
%
\item $(\mathfrak{M}, w)\models\Box_F\varphi$ \quad iff \quad
$(\mathfrak{M}, v)\models\varphi$ for all $v>w$.
\end{itemize}
%
As our intended flows of time are \emph{strict} linear orders, the
`next-time' operator $\nxt$ is also definable via $\until$ by
taking
%
\begin{equation*}
\nxt\varphi = \bot\until\varphi
\end{equation*}
%
($\bot$ is the logical constant `false') which perfectly reflects
our intuition: if $\mathfrak{F}$ is discrete then
%
\begin{itemize}
\item $(\mathfrak M, w)\models \nxt\psi$ \quad iff \quad $(\mathfrak M,
w+1)\models\psi$,
\end{itemize}
%
where $w+1$ is the immediate successor of $w$ in $\mathfrak{F}$.
The reader should not have problems in defining the `past'
versions of $\Diamond_F$, $\Box_F$ and $\nxt$.

The following results are due to Sistla and Clarke
\citeyear{Sistla&Clarke85} and Reynolds
\citeyear{Reynolds03,Reynolds99}:

\begin{theorem}\tlabel{complextempPSP}
The satisfiability problem for $\PTL$-formulas is {\cc
PSPACE}-complete for each of the following classes of flows of
time: all strict linear orders, all finite strict linear orders,
$\auf\mathbb{N},<\zu$, $\auf\mathbb{Z},<\zu$,
$\auf\mathbb{Q},<\zu$, $\auf\mathbb{R},<\zu$.
\end{theorem}

Note, however, that reasoning becomes somewhat simpler if we take
$\Diamond_F$, $\Box_F$ and their past counterparts (but no $\nxt$,
$\until$ and $\since$) as the only temporal primitives. Denote by
$\PTL_{\Box}$ the corresponding fragment of $\PTL$. Then,
according to the results of Ono and Nakamura
\citeyear{Ono&Nakamura80}, Sistla and Clarke
\citeyear{Sistla&Clarke85}, and Wolter \citeyear{Wolter96e}, we
have:

\begin{theorem}\tlabel{complextempNP}
The satisfiability problem for $\PTL_{\Box}$-formulas is {\cc
NP}-complete for each of the classes of flows of time mentioned in
Theorem~\ref{complextempPSP}.
\end{theorem}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Combinations of Spatial and Temporal
Logics}\label{section:combinations}

In this section we introduce and discuss various ways of combining
logics of space and time. First we construct spatio-temporal
logics satisfying only the (PC) principle (see the introduction)
and show that they inherit good computational properties of their
components. Being encouraged by these results, we then consider
`maximal' combinations of $\SFU$ with (fragments of) $\PTL$
meeting both (PC) and (OC) and see that such a straightforward
approach does not work: we end up with undecidable logics. This
leads us to a systematic investigation of the trade-off between
expressivity and computational complexity of spatio-temporal
formalisms. The result is a hierarchy of decidable logics
satisfying (PC) and (OC) whose complexity ranges from PSPACE to
2EXPSPACE.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Spatio-Temporal Logics with (PC)}\label{section:temporalisation}

We begin our investigation of combinations of the spatial and
temporal logics introduced above by considering the language
$\PTL[\SFU]$ in which the temporal operators can be applied to
spatial formulas but not to spatial terms (this way of
`temporalising' a logic was first introduced by Finger and Gabbay,
\citeyearR{Finger&Gabbay92}). A precise syntactic definition of
$\PTL[\SFU]$-\emph{terms} $\tau$ and $\PTL[\SFU]$-\emph{formulas}
$\varphi$ is as follows:
%
\begin{align*}
\tau \quad ::= \quad & p \quad|\quad \overline{\tau}
\quad|\quad \tau_1\sqcap\tau_2 \quad|\quad \IB\tau,\\
%
\varphi \quad ::= \quad & \Ubox\tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2 \quad|\quad
\varphi_1\until\varphi_2 \quad|\quad \varphi_1\since\varphi_2.%
\end{align*}
%
Note that the definition of $\PTL[\SFU]$-terms coincides with the
definition of spatial terms in $\SFU$ which reflects the fact that
$\PTL[\SFU]$ cannot capture the change of spatial objects in time.
We have imposed no restrictions upon the temporal operators in
formulas---so the combined language still has the full expressive
power of $\PTL$. (Clearly, $\SFU$ is a fragment of $\PTL[\SFU]$.)

In a similar way we can introduce spatio-temporal logics based on
all other spatial languages we are dealing with:  $\RCCE$,
$\BRCCE$, $\RC$ and $\RCp$. For example, the temporalisation
$\PTL[\BRCCE]$ of $\BRCCE$ (denoted by $\mathcal{ST}_0$ in the
hierarchy of Wolter and Zakharyaschev
\citeyearR{Wolter&Z02millenium}) allows applications of the
temporal operators to $\RCCE$ predicates but not to Boolean region
terms. These languages can be regarded as fragments of
$\PTL[\SFU]$ in precisely the same way as their spatial components
were treated as fragments of $\SFU$.

We illustrate the expressive power of $\PTL[\RCCE]$ by formalising
sentences (A) and (B) from the introduction:
%
\begin{align}
\tag{A}\label{f:A} %
& {\sf DC}(\textit{Image}_1, \textit{Image}_2) \to \nxt{\sf
DC}(\textit{Image}_1, \textit{Image}_2) \lor \nxt{\sf
EC}(\textit{Image}_1, \textit{Image}_2),\\[4pt]
%
\tag{B}\label{f:B}%
& \bigl({\sf DC}(\textit{Kaliningrad}, \textit{EU}) \until
{\sf TPP}(\textit{Poland}, \textit{EU})\bigr) \land\mbox{}\\ \nonumber %
& \hspace*{5cm}\Box_F\bigl({\sf TPP}(\textit{Poland}, \textit{EU})
\to {\sf EC}(\textit{Kaliningrad}, \textit{EU})\bigr).
\end{align}
%
Sentences \eqref{f:C}--\eqref{f:H} cannot be expressed in this
language (or even in $\PTL[\SFU]$): they require comparisons of
states of spatial objects at different time instants.

The intended semantics of $\PTL[\SFU]$ (and all other
spatio-temporal logics considered in this paper) is rather
straightforward. A \emph{topological temporal model} (a
\emph{tt-model}, for short) is a triple of the form $\mathfrak{M}
= \auf \mathfrak{F}, \mathfrak{T}, \mathfrak{U}\zu$, where
$\mathfrak{F} = \auf W, <\zu$ is a flow of time, $\mathfrak{T} =
\auf U, \mathbb{I}\zu$ a topological space, and $\mathfrak{U}$, a
\emph{valuation}, is a map associating with every spatial variable
$p$ and every time point $w\in W$ a set $\mathfrak{U}(p,
w)\subseteq U$---the `space' occupied by $p$ at moment $w$; see
Fig.~\ref{fig:ttmodel}. The valuation $\mathfrak{U}$ is
inductively extended to arbitrary $\PTL[\SFU]$-terms (i.e.,
spatial terms) in precisely the same way as for $\SFU$, we only
have to add a time point as a parameter:
%
\begin{equation*}
\mathfrak{U}(\overline{\tau}, w) = U - \mathfrak{U}(\tau,
w),\qquad
%
\mathfrak{U}(\tau_1\sqcap\tau_2, w) = \mathfrak{U}(\tau_1, w) \cap
\mathfrak{U}(\tau_2, w),\qquad
%
\mathfrak{U}(\IB\tau, w) = \mathbb{I}\mathfrak{U}(\tau, w).
\end{equation*}
%
The truth-values of $\PTL[\SFU]$-formulas are defined in the same
way as for $\PTL$:
%
\begin{itemize}
\item $(\mathfrak{M},w)\models\Ubox\tau$\quad iff \quad
$\mathfrak{U}(\tau, w) = U$,
%
\item $(\mathfrak{M},w)\models\neg\varphi$ \quad iff \quad
$(\mathfrak{M}, w) \not\models\varphi$,
%
\item $(\mathfrak{M},w)\models\varphi_1\land\varphi_2$ \quad
iff \quad $(\mathfrak{M},w)\models\varphi_1$ \quad and \quad
$(\mathfrak{M},w)\models\varphi_2$,
%
\item $(\mathfrak{M},w)\models\varphi_1\until\varphi_2$\quad iff \quad
there is $v > w$ such that $(\mathfrak{M},v)\models\varphi_2$ and
$(\mathfrak{M},u) \models\varphi_1$ for all $u\in(w, v)$,
%
\item $(\mathfrak{M},w)\models\varphi_1\since\varphi_2$\quad iff \quad
there is $v < w$ such that $(\mathfrak{M},v)\models\varphi_2$ and
$(\mathfrak{M},u)\models\varphi_1$ for all $u\in(v, w)$.
\end{itemize}
%
And as in the pure temporal case, the operators $\Box_F$,
$\Diamond_F$, $\nxt$ as well as their past counterparts can be
defined in terms of $\until$ and $\since$.

A $\PTL[\SFU]$-formula $\varphi$ is said to be \emph{satisfiable}
if there exists a tt-model $\mathfrak M$ such that
$(\mathfrak{M},w)\models\varphi$ for some time point $w$.


\medskip

The following optimal complexity result will be obtained in
Appendix~\ref{appendix:temporalisation}:

\begin{theorem}\tlabel{PST0:pspace:text}
The satisfiability problem for $\PTL[\SFU]$-formulas in tt-models
based on arbitrary flows of time, {\rm(}arbitrary{\rm)} finite
flows of time, $\auf\mathbb{N},<\zu$, $\auf\mathbb{Z},<\zu$,
$\auf\mathbb{Q},<\zu$, or $\auf\mathbb{R},<\zu$, is {\cc
PSPACE}-complete.
\end{theorem}

The proof of this theorem is based on the fact that the
interaction between spatial and temporal components of
$\PTL[\SFU]$ is very restricted. In fact, for every
$\PTL[\SFU]$-formula $\varphi$ one can construct a $\PTL$-formula
$\varphi^\ast$ by replacing every occurrence of a (spatial)
subformula $\Ubox\tau$ in $\varphi$ with a fresh propositional
variable $p_\tau$. Then, given a $\PTL$-model $\mathfrak{N}=\auf
\mathfrak F, \mathfrak V\zu$ for $\varphi^\ast$ and a moment of
time $w$, we take the set
%
\begin{equation*}
\Phi_w = \{ \Ubox\tau \mid (\mathfrak{N},w)\models p_{\tau} \}\;
\cup\; \{ \neg\Ubox\tau \mid (\mathfrak{N},w)\models \neg p_{\tau}
\}
\end{equation*}
%
of spatial formulas. It is not hard to see that if $\Phi_w$ is
satisfiable for every $w$ in $\mathfrak F$, then there is a
tt-model satisfying $\varphi$ and based on the flow $\mathfrak F$.
Now, to check whether $\varphi$ is satisfiable, it suffices to use
a suitable nondeterministic algorithm \cite<see,
e.g.,>{Sistla&Clarke85,Reynolds03,Reynolds99} which guesses a
$\PTL$-model for $\varphi^\ast$ and then, for each time point $w$,
to check satisfiability of $\Phi_w$. This can be done using
polynomial space in the length of $\varphi$.

\smallskip

Theorem~\ref{PST0:pspace:text} (together with
Theorem~\ref{complextempPSP}) shows that all spatio-temporal
logics of the form $\PTL[\mathcal{L}]$, for $\mathcal{L} \in
\{\RCCE, \BRCCE, \RC, \RCp\}$, are also PSPACE-complete over the
standard flows of time.



Now let us consider temporalisations of spatial logics with the
({\cc NP}-complete) fragment $\PTL_{\Box}$ of $\PTL$. By
Theorems~\ref{fullreg} and \ref{PST0:pspace:text}, both
$\PTL_{\Box}[\SFU]$ and $\PTL_{\Box}[\RCp]$ are PSPACE-complete.
However, for simpler ({\cc NP}-complete) spatial components we
obtain a better result:

\begin{theorem}\tlabel{getlost}
The satisfiability problem for $\PTL_{\Box}[\RC]$-formulas in
tt-models based on each of the classes of flows of time mentioned
in Theorem~\ref{PST0:pspace:text} is {\cc NP}-complete.
\end{theorem}

The proof is essentially the same as that of
Theorem~\ref{PST0:pspace:text}, but now nondeterministic
polynomial-time algorithms for the component logics are available.
It follows from Theorem~\ref{getlost} that $\PTL_{\Box}[\RCCE]$
and $\PTL_{\Box}[\BRCCE]$ are {\cc NP}-complete as well.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{`Maximal' Combinations with (PC) and (OC)}\label{PST}

As we saw in the previous section, the computational complexity of
spatio-temporal logics without (OC) is the maximum of the
complexity of their components, which reflects the very limited
interaction between spatial and temporal operators in languages
without any means of expressing (OC).

A `maximalist' approach to constructing spatio-temporal logics
capable of capturing both (PC) and (OC) is to allow unrestricted
applications of the Booleans, the topological and the temporal
operators to form spatio-temporal terms.

Denote by $\PTL\times\SFU$ the spatio-temporal language given by
the following definition:
%
\begin{align*}
\tau \quad ::= \quad & p \quad|\quad \overline{\tau} \quad|\quad
\tau_1\sqcap\tau_2 \quad|\quad \IB\tau \quad| \quad
\tau_1\until\tau_2 \quad|\quad \tau_1\since\tau_2,
\\
%
\varphi \quad ::= \quad & \Ubox\tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2 \quad|\quad
\varphi_1\until\varphi_2 \quad|\quad \varphi_1\since\varphi_2.
\end{align*}
%
Expressions of the form $\tau$ will be called
\emph{spatio-temporal terms}. Unlike the previous section, these
terms can be time-dependent. The definition of expressions of the
form $\varphi$ is the same as for $\PTL[\SFU]$; they will be
called $\PTL\times\SFU$-\emph{formulas}. All of the languages from
Section~\ref{section:temporalisation}, including $\PTL[\SFU]$, are
clearly fragments of $\PTL\times\SFU$.

As before, we can introduce the temporal operators $\Box_F$,
$\Diamond_F$, $\nxt$ as well as their past counterparts applicable
to formulas. Moreover, these operators can now be used to form
spatio-temporal terms: for example,
%
\begin{equation*}
\Diamond_F\tau = \top\until\tau, \qquad \Box_F\tau =
\neg\Diamond_F\overline{\tau} \quad\text{and}\quad\nxt \tau =
\bot\until\tau,
\end{equation*}
%
where $\bot$ denotes the empty set and $\top$ the whole space.

Spatio-temporal formulas are supposed to represent propositions
speaking about moving spatial objects represented by
spatio-temporal terms. The truth-values of propositions in
spatio-temporal structures can vary in time, but do not depend on
points of spaces---they are defined in precisely the same way as
in the case of $\PTL[\SFU]$. But how to understand temporalised
terms?

The meaning of $\nxt\tau$ should be clear: at moment $w$, it
denotes the space occupied by $\tau$ at the next moment $w+1$ (see
Fig.~\ref{fig:nexttime}). For example, we can write
%
\begin{align}\tag{C}\label{f:C}%
& \Udiamond\big(\IB\reg{\textit{Cyclone}}\sqcap\IB
\reg{\nxt\textit{Cyclone}}\big)
\end{align}
%
to say that regions $\textit{Cyclone}$ and $\nxt\textit{Cyclone}$
overlap (thereby formalising sentence \eqref{f:C} from the
introduction). The formula
%
\begin{align}\tag{F}\label{f:F}%
& \mathsf{EQ}(\nxt\nxt\textit{EU}, \textit{EU}\sqcup
\textit{Romania}\sqcup\textit{Bulgaria})
\end{align}
%
says that in two years the EU (as it is today) will be extended
with Romania and Bulgaria. Note that
$\nxt\nxt\mathsf{EQ}(\textit{EU}, \textit{EU}\sqcup
\textit{Romania}\sqcup\textit{Bulgaria})$ has a different meaning
because the EU may expand or shrink in a year. It is also not hard
to formalise sentences \eqref{f:D}, \eqref{f:E} and \eqref{f:H}
from the introduction:
%
\begin{align}
& \tag{D}\label{f:D}%
{\sf EQ}(\nxt X, Y) \rightarrow \neg{\sf EQ}(Y,\nxt Y),\\[4pt] %
%
& \tag{E}\label{f:E}%
\Box_F{\sf EQ}(\nxt \textit{Europe},\textit{Europe}),\\[4pt]  %
%
& \tag{H}\label{f:H}%
{\sf EQ}(\textit{Earth},W \sqcup L)\land%
{\sf EC}(W, L) \land {\sf P}(W,\nxt W) \to {\sf P}(\nxt L, L), %
\end{align}
%
where ${\sf P}(X,Y)$---`$X$ is a part of $Y$'---denotes the
disjunction of ${\sf EQ}(X,Y)$, ${\sf TPP}(X,Y)$ and ${\sf
NTPP}(X,Y)$.

The intended interpretation of terms of the form $\Diamond_F\tau$,
$\Box_F\tau$ (and their past counterparts) is a bit more
sophisticated. It reflects the standard temporal meanings of
propositions `$\Diamond_F\, x\in\tau$' and `$\Box_F\, x\in\tau$,'
for all points $x$ in the topological space:
%
\begin{itemize}
\item at moment $w$, term $\Diamond_F\tau$ is interpreted as the
union of all spatial extensions of $\tau$ at moments $v>w$;
%
\item at moment $w$, term $\Box_F\tau$ is interpreted as the
intersection of all spatial extensions of $\tau$ at moments $v>w$.
\end{itemize}
%
For example, consider Fig.~\ref{fig:nexttime} with moving spatial
object $X$ depicted on it at three consecutive moments of time (it
does not change after $t+2$). Then $\Diamond_F X$ at $t$ is the
union of $\nxt X$ and $\nxt\nxt X$ at $t$ and $\Box_F X$ at $t$ is
the intersection of $\nxt X$ and $\nxt\nxt X$ at $t$ (i.e., $\nxt
X$).

As another example, take the spatial object $\textit{Rain}$. Then
%
\begin{itemize}
\item $\Diamond_F\textit{Rain}$ at moment $w$ occupies the space
where it will be raining at \emph{some} time points $v>w$ (which
may be different for different places). $\Box_{F}\textit{Rain}$ at
$w$ occupies the space where it will \emph{always} be raining
after $w$.
%
\item $\Box_F\Diamond_F\textit{Rain}$ at $w$ is the space
where it will be raining ever and ever again after $w$, while
$\Diamond_F\Box_F\textit{Rain}$ comprises all places where it will
always be raining starting from some future moments of time.
\end{itemize}
%
This interpretation shows how to formalise sentence \eqref{f:G} from
the introduction:
%
\begin{align}\tag{G}\label{f:G}
& {\sf P}(\textit{England},\Box_{F}\Diamond_{F} \textit{Rain}). %
\end{align}

Now, what can be the meaning of
$\textit{Rain}\until\textit{Snow}$? Similarly to the readings of
$\Box_F\tau$ and $\Diamond_F\tau$ above, we adopt the following
definition:
%
\begin{itemize}
%
\item at moment $w$, the spatial extension of $\tau_1\until\tau_2$
consists of those points $x$ of the topological space for which
there is $v>w$ such that $x$ belongs to $\tau_2$ at moment $v$ and
$x$ is in $\tau_1$ at all $u$ whenever $w<u<v$.
\end{itemize}
%
The past counterpart of $\until$---i.e., the operator `since'
$\since$---can be used to say that the part of Russia that has
been remaining Russian since 1917 is not connected to the part of
Germany (K\"onigsberg) that became Russian after the Second World
War (Kaliningrad):
%
\begin{equation*}
{\sf DC}(\textit{Russia}\since
\textit{Russian\_Empire},\textit{Russia}\since \textit{Germany}).
\end{equation*}

The models $\mathfrak{M} = \auf \mathfrak{F}, \mathfrak{T},
\mathfrak{U}\zu$ for $\PTL\times\SFU$ are precisely the same
topological temporal models we introduced for $\PTL[\SFU]$.
However, now we need additional clauses defining extensions of
spatio-temporal terms:
%
\begin{itemize}
\item $\displaystyle\mathfrak{U}(\tau_1\until\tau_2, w) = \bigcup_{v > w}
\Big(\mathfrak{U}(\tau_2,v)\cap\bigcap_{u\in
(w,v)}\mathfrak{U}(\tau_1,u)\Big)$,
%
\item $\displaystyle\mathfrak{U}(\tau_1\since\tau_2, w) = \bigcup_{v < w}
\Big(\mathfrak{U}(\tau_2,v)\cap\bigcap_{u\in
(v,w)}\mathfrak{U}(\tau_1,u)\Big)$.
\end{itemize}
%
Then we also have:
%
\begin{equation*}
\mathfrak{U}(\Diamond_F\tau, w) = \bigcup\limits_{v >
w}\mathfrak{U}(\tau, v)%
\qquad\text{and}\qquad%
\mathfrak{U}(\Box_F\tau, w) = \bigcap\limits_{v
> w}\mathfrak{U}(\tau, v),
\end{equation*}
%
and, for discrete $\mathfrak{F}$,
%
\begin{equation*}
\mathfrak{U}(\nxt\tau,w) = \mathfrak{U}(\tau,w+1).
\end{equation*}
%
The truth-values of $\PTL\times\SFU$-formulas are computed in
precisely the same way as in the case of $\PTL[\SFU]$. A
$\PTL\times\SFU$-formula $\varphi$ is called \emph{satisfiable} if
there exists a tt-model $\mathfrak M$ such that $(\mathfrak M,
w)\models\varphi$ for some time point $w$.

At first sight it may appear that the computational properties of
the constructed logic should not be too bad---after all, its
spatial and temporal components are PSPACE-complete. It turns out,
however, that this is not the case:

\begin{theorem}\tlabel{PSTOundecidable}
The satisfiability problem for $\PTL\times\SFU$-formulas in
tt-models based on the flows of time $\auf\mathbb{N},<\zu$ or
$\auf\mathbb{Z},<\zu$ is undecidable.
\end{theorem}

Without going into details of the proof of this theorem, one might
immediately conjecture that it is the use of the \emph{infinitary}
operators $\until$, $\Box_F$ and $\Diamond_F$ in the construction
of spatio-temporal terms that makes the logic `over-expressive.'
Moreover, the whole idea of topological temporal models based on
\emph{infinite} flows of time may look counterintuitive in the
context of spatio-temporal representation and reasoning (unlike,
say, models used to represent the behaviour of reactive computer
systems).

There are different approaches to avoid infinity in tt-models. The
most radical one is to allow only finite flows of time. A more
cautious approach is to impose the following \emph{finite change
assumption} on models (based on infinite flows of time):

\smallskip

{\bf FCA\ } {\em No term can change its spatial extension
infinitely often.}

\smallskip

\noindent This means that under \FCA{} we consider only those
valuations $\mathfrak U$ in tt-models $\auf\mathfrak F, \mathfrak
T, \mathfrak U\zu$ that satisfy the following condition: for every
spatio-temporal term $\tau$, there are pairwise disjoint intervals
$I_{1},\dots,I_{n}$ of ${\mathfrak F}=\auf W,<\zu$ such that
$W=I_{1}\cup \dots \cup I_{n}$ and the state of $\tau$ remains
constant on each $I_{j}$, i.e., $\mathfrak{U}(\tau,u) =
\mathfrak{U}(\tau,v)$ for any $u,v\in I_j$. It turns out, however,
that in the case of \emph{discrete} flows of time \FCA{} does not
give us anything new as compared to arbitrary finite flows of
time. More precisely, one can easily show that the satisfiability
problem for $\PTL\times\SFU$-formulas in tt-models satisfying
\FCA{} and based on $\auf\mathbb{N},<\zu$ or $\auf\mathbb{Z},<\zu$
is polynomially reducible to satisfiability in tt-models based on
finite flows of time, and the other way round.
%
Note also that for the flows of time mentioned above, \FCA{} can
be captured by the formulas $\Diamond_F\Box_F{\sf EQ}(\tau,\nxt_F
\tau)$ (and its past counterpart for $\auf\mathbb{Z},<\zu$), for
every spatio-temporal term $\tau$.

A more `liberal' way of reducing infinite unions and intersections
to finite ones is to adopt the \emph{finite state assumption}:

\bigskip

{\bf FSA\ } \parbox[t]{12cm}{\em Every term may have only finitely
many possible states $($although it may change its states
infinitely often$)$.}

\bigskip

\noindent Say that a tt-model $\auf\mathfrak F,\mathfrak T,
\mathfrak U\zu$ satisfies \FSA{} if, for every spatio-temporal
term $\tau$, there are finitely many sets $A_1,\dots,A_m$ in the
space $\mathfrak T$ such that $\{\mathfrak U(\tau,w) \mid w\in
W\}= \{A_1,\dots,A_m\}$. Such models can be used, for instance, to
capture periodic fluctuations due to season or climate changes,
say, a daily tide. Similarly to \FCA{} finitising the flow of
time, \FSA{} virtually makes the underlying topological space
finite. The following proposition will be proved in
Appendix~\ref{section:sptm:on:s4u}:
%
\begin{proposition}\label{proposition:fsa}
A $\PTL\times\SFU$-formula is satisfiable in a tt-model with
\FSA{} and based a flow of time $\mathfrak{F}$ iff it is
satisfiable in a tt-model based on $\mathfrak{F}$ and a finite
\textup{(}Aleksandrov\textup{)} topological space.
\end{proposition}

Unfortunately, none of these approaches works for
$\PTL\times\SFU$---we still have:

\begin{theorem}\tlabel{PST01undecidable}
{\rm (i)} The satisfiability problem for $\PTL\times\SFU$-formulas
in tt-models based on {\rm(}arbitrary{\rm)} finite flows of time
is undecidable.

{\rm (ii)} The satisfiability problem for
$\PTL\times\SFU$-formulas in tt-models based on the flows of time
$\auf\mathbb{N},<\zu$ or $\auf\mathbb{Z},<\zu$ and satisfying
\FSA{} is undecidable.
\end{theorem}


The next-time operator $\nxt$ does not look so `harmful' as the
infinitary $\until$, $\Box_F$, $\Diamond_F$, and still can capture
some aspects of (OC) (see formulas \eqref{f:C}, \eqref{f:D},
\eqref{f:F} and \eqref{f:G} above). So let us consider the fragment
$\PTL\circ\SFU$ of $\PTL\times\SFU$ with spatio-temporal terms of
the form:
%
\begin{equation*}
\tau \quad ::= \quad p \quad|\quad \overline{\tau} \quad|\quad
\tau_1\sqcap\tau_2 \quad|\quad \IB\tau \quad| \quad \nxt\tau.
\end{equation*}
%
In other words, $\PTL\circ\SFU$ does not allow applications of
temporal operators different from $\nxt$ to form spatio-temporal
terms (but they are still available as formula constructors). This
means that we can compare the states of a spatial object $X$ over
a bounded set of time points only: for any time point $t$ and any
natural numbers $n,m\ge 0$, we can compare at $t$ the state of $X$
at $t+n$ with its state at $t+m$.

This fragment is definitely less expressive than full
$\PTL\times\SFU$. For instance, according to Lemma~\ref{simula1},
$\PTL\circ\SFU$-formulas do not distinguish between arbitrary
tt-models and those based on Aleksandrov topological spaces---we
will call them \emph{Aleksandrov} tt-models. On the other hand,
the set of $\PTL\times\SFU$-formulas satisfiable in Aleksandrov
models is a proper subset of those satisfiable in arbitrary
tt-models. Consider, for example, the $\PTL\times\SFU$-formula
%
\begin{equation*}%\label{eq:PST:distinguish}
\Ubox(\Box_F\IB  p\; \imp\; \IB\Box_F p).
\end{equation*}
%
One can readily see that it is true in every Aleksandrov tt-model,
but its negation can be satisfied in a topological model. For it
suffices to take the flow $\mathfrak{F} = \auf \mathbb{N}, <\zu$
and the topology $\mathfrak{T} = \auf \mathbb{R}, \mathbb{I}\zu$
with the standard interior operator $\mathbb{I}$ on the real line,
select a sequence $X_n$ of open sets such that
$\bigcap_{n\in\mathbb{N}} X_n$ is not open, e.g., $X_n = (-1/n,
1/n)$, and put $\mathfrak{U}(p, n) = X_n$.

However, even this seemingly weak interaction between topological
and temporal operators turns out to be dangerous:
%
\begin{theorem}\tlabel{PST1undecidable}
The satisfiability problem for $\PTL\circ\SFU$-formulas in
tt-models based on the flows of time $\auf\mathbb{N},<\zu$ or
$\auf\mathbb{Z},<\zu$ is undecidable. It is undecidable as well
for tt-models satisfying \FSA{} or based on {\rm(}arbitrary{\rm)}
finite flows of time.
\end{theorem}


Theorem~\ref{complextempNP} might suggest considering the fragment
$\PTL_{\Box}\times\SFU$ with $\Box_F$ and its past counterpart
$\Box_P$ as the only temporal primitives applicable both to
formulas and terms:
%
\begin{align*}
\tau \quad ::= \quad & p \quad|\quad \overline{\tau} \quad|\quad
\tau_1\sqcap\tau_2 \quad|\quad \IB\tau \quad|\quad \Box_F\tau
\quad|\quad \Box_P\tau,
\\
%
\varphi \quad ::= \quad & \Ubox\tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2 \quad|\quad \Box_F\varphi
\quad|\quad \Box_P\varphi.
\end{align*}
%
Yet again the result is `negative:'

\begin{theorem}\tlabel{PSTBundecidable}
The satisfiability problem for $\PTL_{\Box}\times\SFU$-formulas in
tt-models {\rm(}with or without \FSA{}{\rm)} based on the flows of
time $\auf\mathbb{N},<\zu$ or $\auf\mathbb{Z},<\zu$ is
undecidable. It is undecidable as well for tt-models based on
{\rm(}arbitrary{\rm)} finite flows of time.
\end{theorem}

These undecidability results (the strongest ones,
Theorems~\ref{PST1undecidable} and \ref{PSTBundecidable}, to be
more precise) will be proved in Appendix~\ref{appendix:product} by
a reduction of Post's correspondence problem which is known to be
undecidable \cite{Post46}. As we will see from the proofs, these
theorems actually hold for the `future fragments' of the
corresponding languages.

%The reduction essentially uses the interaction between spatial and
%temporal operators as well as the fact that, at each moment of
%time, the spatio-temporal variables can be interpreted by
%\emph{arbitrary} subsets of the topological space, e.g., similar
%to the set defined by formula~\eqref{eq:pathological}.

%Rewrite:

%1. Arbitrary valuations: sets vs regions.

%2. Arbitrary comparisons of terms vs restricted.

%3. Careful with interaction? Elaborate!!!!!!!!!!!!!!



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{Decidable Spatio-Temporal Logics with (PC) and (OC)}

An important lesson we learn from (the proofs of) the `negative'
results of Section~\ref{PST} is that full $\SFU$ is too expressive
for computationally well-behaved combinations with fragments of
$\PTL$. On the other hand, as was said in Section~\ref{rcc8},
qualitative spatial representation and reasoning often requires
extensions of spatial variables to be regular closed (i.e.,
regions). This restriction is very important for constructing
decidable spatio-temporal logics with (PC) and (OC). First, the
undecidability proofs from Appendix~\ref{appendix:product} do not
go through in this case. And second, as will be shown below,
decidable combinations of $\PTL$ and some of the fragments of
$\SFU$ introduced in Section~\ref{space} do exist. In fact, we
will construct a hierarchy of decidable spatio-temporal logics of
different computational complexity by imposing various
restrictions on regions themselves, the ways they can be compared,
and the interactions between spatial and temporal constructors.

We begin by considering the simplest combination of $\PTL$ and
$\RCCE$ capturing (PC) and (OC). This logic called
$\PTL\circ\RCCE$ (it was introduced under the name
$\mathcal{ST}_1^-$ by Wolter and Zakharyaschev,
\citeyearR{Wolter&Z02millenium}) operates with
\emph{spatio-temporal region terms} of the form
%
\begin{equation*}
\varrho \quad ::= \quad  \CB\IB p \quad|\quad \CB\IB \nxt\varrho.
\end{equation*}
%
To relate these terms, we are allowed to use the eight binary
predicates of $\RCCE$; then arbitrary temporal operators and Boolean
connectives can be applied to produce \mbox{$\PTL\circ\RCCE$}
\emph{formulas}. Typical examples of such formulas are \eqref{f:A},
\eqref{f:B}, \eqref{f:D} and \eqref{f:E} above. Note that
\eqref{f:C} can be regarded as a $\PTL\circ\RCCE$ formula as well
(two regions overlap iff they are neither disconnected nor
externally connected). On the other hand, \eqref{f:F}, \eqref{f:H}
and \eqref{f:G} are not $\PTL\circ\RCCE$ formulas because the first
two use the $\sqcup$ operation on region terms and~\eqref{f:G} uses
temporal operators $\Box_F$ and $\Diamond_F$ on region terms.

As before, $\PTL\circ\RCCE$ formulas are interpreted in
topological temporal models (or tt-models). However, only discrete
flows of time do make sense for this language. Although the
interaction between topological and temporal operators is similar
to that in $\PTL\circ\SFU$ (clearly, $\PTL\circ\RCCE$ is a
fragment of $\PTL\circ\SFU$), we have the following rather
unexpected and encouraging result:
%
\begin{theorem}\tlabel{st1m:pspace}
The satisfiability problem for $\PTL\circ\RCCE$ formulas in
tt-models based on $\auf\mathbb{N},<\zu$, $\auf\mathbb{Z},<\zu$ or
{\rm(}arbitrary{\rm)} finite flows of time is {\cc
PSPACE}-complete.
\end{theorem}
%
This theorem will be proved in Appendix~\ref{appendix:pspace}. The
idea of the proof is similar to that of
Theorem~\ref{PST0:pspace:text}: we consider the spatial and the
temporal parts of a given formula separately. However, to take
into account the interaction between these parts, we use the
so-called `completion property' of $\RCCE$
\cite<cf.>{Balbiani&Condotta02} with respect to a certain class
$\mathfrak{C}$ of models: given a satisfiable set $\Phi$ of
$\RCCE$ formulas and a model in $\mathfrak{C}$ satisfying a subset
of $\Phi$, one can extend this `partial' model to a model in
$\mathfrak{C}$ satisfying the whole $\Phi$.


\smallskip


What happens if we extend the expressive power of the spatial
component by allowing Boolean operators on spatio-temporal region
terms, i.e., jump from $\RCCE$ to $\BRCCE$? Define
\emph{spatio-temporal Boolean region terms} by taking
%
\begin{equation*}
\varrho \quad ::= \quad  \CB\IB p \quad|\quad
\CB\IB\overline{\varrho} \quad|\quad
\CB\IB(\varrho_1\sqcap\varrho_2) \quad|\quad \CB\IB \nxt\varrho.
\end{equation*}
%
Denote by $\PTL\circ\BRCCE$ the language obtained from
$\PTL\circ\RCCE$ by allowing spatio-temporal Boolean region terms as
arguments of the $\RCCE$ predicates (this language was called
$\mathcal{ST}_1$ by Wolter and Zakharyaschev,
\citeyearR{Wolter&Z02millenium}). Formulas \eqref{f:A}--\eqref{f:F}
and \eqref{f:H} belong to $\PTL\circ\BRCCE$, but \eqref{f:G} uses
the $\Box_F$ and $\Diamond_F$ operators on regions and so is not in
$\PTL\circ\BRCCE$.

Now, another surprise is that the replacement of $\RCCE$ with
$\BRCCE$ in our temporal context results in an exponential jump of
the computational complexity (remember that both $\RCCE$ and
$\BRCCE$ are NP-complete):
%
\begin{theorem}\tlabel{st1:expspace}
The satisfiability problem for $\PTL\circ\BRCCE$ formulas in
tt-models based on the flows of time $\auf\mathbb{N},<\zu$ or
$\auf\mathbb{Z},<\zu$ is {\cc EXPSPACE}-complete. It is {\cc
EXPSPACE}-complete as well for models satisfying \FSA{} or based
on {\rm(}arbitrary{\rm)} finite flows of time.
\end{theorem}
%
The EXPSPACE upper bound (see Appendix~\ref{appendix:embedding})
is proved by a polynomial embedding of $\PTL\circ\BRCCE$ into the
one-variable fragment $\QTL^1$ of first-order temporal logic,
which is known to be {\cc EXPSPACE}-complete
\cite{Hodkinsonetal03}. To construct this embedding, we first show
that $\PTL\circ\BRCCE$ is complete with respect to Aleksandrov
tt-models. In fact, we prove that every satisfiable formula of the
more expressive logic $\PTL\circ\SFU$ introduced in
Section~\ref{PST} can be satisfied in an Aleksandrov tt-model (see
Lemma~\ref{simula1} and the discussion above).
Lemma~\ref{rc:simula2} then shows that to satisfy a
$\PTL\circ\BRCCE$ formula, it suffices to take an Aleksandrov
tt-model based on a partial order of depth 1. By
Lemma~\ref{rc:simula3}, the width of the partial order can be
bounded by $2$ (just as in the case of $\BRCCE$), and therefore
unions of forks (or 2-brooms) are enough to satisfy
$\PTL\circ\BRCCE$ formulas. These Aleksandrov tt-models based on
unions of forks can be encoded by means of unary predicates of
$\QTL^1$.

The EXPSPACE lower bound is proved in
Appendix~\ref{appendix:RC:lowerI} by encoding the corridor tiling
problem. It can also be established by a direct polynomial
embedding of $\QTL^1$ into $\PTL\circ\BRCCE$. To illustrate the
idea, consider the $\QTL^1$-formula $\forall x\,(P(x) \lor \nxt
P(x))$ saying that, for every point of the space, either it is in
$P$ now or will be there tomorrow. The same statement can be
expressed in $\PTL\circ\BRCCE$ by the formula ${\sf EQ}(P \sqcup
\nxt P, \overline{E})\; \land\; {\sf DC}(E, E)$, where the last
conjunct makes $E$ empty.


Now let us make one more step in space and extend $\BRCCE$ to
$\RC$, thus obtaining the spatio-temporal language $\PTL\circ\RC$
with the following syntax:
%
\begin{align*}
\varrho \quad ::= \quad & \CB\IB p \quad|\quad
\CB\IB\overline{\varrho}
\quad|\quad \CB\IB(\varrho_1\sqcap\varrho_2)\quad|\quad \CB\IB\nxt\varrho,\\
%
\tau \quad ::= \quad & \varrho \quad|\quad \IB\varrho \quad|\quad
\overline{\tau}\quad|\quad \tau_1\sqcap\tau_2,\\
%
\varphi \quad ::= \quad & \Ubox \tau \quad|\quad \neg\varphi
\quad|\quad \varphi_1\land\varphi_2\quad|\quad
\varphi_1\until\varphi_2 \quad|\quad \varphi_2\since\varphi_2.%
\end{align*}
%
The reader should not be surprised now (although the authors were)
that the extra expressivity results in one more exponential gap:
%
\begin{theorem}\tlabel{theorem:rc:2expspace}
The satisfiability problem for $\PTL\circ\RC$-formulas in
tt-models based on the flows of time $\auf\mathbb{N},<\zu$ or
$\auf\mathbb{Z},<\zu$ is {\cc 2EXPSPACE}-complete. It is {\cc
2EXPSPACE}-complete as well for models satisfying \FSA{} or based
on {\rm(}arbitrary{\rm)} finite flows of time.
\end{theorem}
%
The lower bound is established in
Appendix~\ref{appendix:RC:lowerI} and the upper bound in
Appendix~\ref{appendix:quasimodels}.



Perhaps, it is proper time now to have a closer look at the
emerging landscape. What exactly causes these exponential `jumps'?
Can we locate more precise borders in the ladder
PSPACE--EXPSPACE--2EXPSPACE?

By analysing the proof of Theorem~\ref{st1m:pspace} (see
Appendix~\ref{appendix:pspace}), we note that not so much can be
added to $\RCCE$. In fact, the maximal spatio-temporal logic
(denoted by $\PTL\circ\RC_2$) for which this proof goes through is
based on spatio-temporal terms of the form\label{def:rc2}
%
\begin{align*}
\varrho \quad ::= \quad & \CB\IB p \quad|\quad\CB\IB\nxt\varrho,\\
%
\theta \quad ::= \quad & \varrho \quad|\quad \IB\varrho
\quad|\quad
\overline{\varrho} \quad|\quad \overline{\IB\varrho},\\
%
\tau \quad ::= \quad & \overline{\theta_1 \sqcap\theta_2}.
\end{align*}
%
On the other hand, even the addition of predicates of the form
$\textsf{EQ}(X,Y\sqcup Z)$ is enough to make the logic
EXPSPACE-hard (see Remark~\ref{remark:rcc8:expspace}). Thus,
$\PTL\circ\RCCE$ (or rather its extension $\PTL\circ\RC_2$) is
located pretty close to the border between PSPACE and EXPSPACE
spatio-temporal logics.

The following fragment $\RCm$ of $\RC$ indicates where the border
between EXPSPACE and 2EXPSPACE may lie:
%
\begin{align*}
\varrho \quad::=\quad & \text{Boolean region terms},\\
\delta \quad::=\quad &\varrho \quad|\quad \overline{\sigma}, \\
\sigma \quad::=\quad &\IB\varrho \quad|\quad \overline{\delta} \quad|\quad \sigma_1\sqcap\sigma_2, \\
\tau \quad::=\quad &\overline{\delta_1\sqcap\dots\sqcap\delta_m}
\quad|\quad \overline{\delta\sqcap\sigma} \quad|\quad \overline{\sigma}. %
\end{align*}
%
Intuitively, the $\delta$ and the $\sigma$ are spatial terms
interpreted by regular closed and regular open\footnote{Remember
that a set $X$ is \emph{regular open} if $\mathbb{I}\mathbb{C}X =
X$.} sets, respectively (the interior of a region is regular open,
the complement of a regular closed set is regular open (and vice
versa), regular closed sets are closed under unions and regular
open ones are closed under intersections). Thus, $\delta$ can be
regarded as a generalisation of region terms and $\sigma$ as a
generalisation of the interiors of regions. In other words, $\RCm$
is the fragment of $\RC$ in which only the following ways of
relating regions are available:
%
\begin{itemize}
\item there is a point where some regions meet;
\item a region intersects the interior of another one;
\item the interior of a region is not empty.
\end{itemize}
%
It is readily checked that $\BRCCE$ is a fragment of $\RCm$.
Moreover, it is a proper fragment because \eqref{R2} belongs to
the latter but not to the former. The formula
%
\begin{equation}\label{eq:expgap}
\Ubox\big((\reg{NorthKorea}\sqcap\reg{SouthKorea})\imp\reg{DmZone}\big)
\end{equation}
%
(saying that the demilitarised zone between the North Korea and
the South Korea consists of the border between them along with
some adjacent territories) shows that $\RCm$ is a proper subset of
$\RC$:
%
\begin{equation*}
\BRCCE\quad \subsetneqq\quad \RCm\quad \subsetneqq\quad \RC.
\end{equation*}

Although $\RCm$ extends $\BRCCE$, it gives rise to the
spatio-temporal logic of the same computational complexity:
%
\begin{theorem}\tlabel{theorem:rcm:expspace}
The satisfiability problem for $\PTL\circ\RCm$-formulas in
tt-models based on the flows of time $\auf\mathbb{N},<\zu$ or
$\auf\mathbb{Z},<\zu$ is {\cc EXPSPACE}-complete. It is {\cc
EXPSPACE}-complete as well for models satisfying \FSA{} or based
on {\rm(}arbitrary{\rm)} finite flows of time.
\end{theorem}
%
The lower bound follows immediately from
Theorem~\ref{st1:expspace} and the proof of the upper bound is
similar to that of Theorem~\ref{st1:expspace} (see
Appendix~\ref{appendix:embedding}). Again, due to the restriction
on possible ways of relating regions, we can polynomially bound
the width $n$ of $n$-brooms that are required to satisfy
$\PTL\circ\RCm$-formulas (cf.\ Lemma~\ref{rc:simula3}). In fact,
we need formulas similar to \eqref{eq:expgap} in order to increase
complexity to 2EXPSPACE.

\smallskip

The constructed hierarchy of decidable spatio-temporal logics
still leaves at least one important question: do there exist
decidable spatio-temporal logics that allow applications of the
temporal operators $\until$, $\Box_F$, $\Diamond_F$ to region
terms and what is their complexity? Consider the languages
$\PTL\times\mathcal{L}$, for $\mathcal{L}\in \{\BRCCE,
\RCm,\RC\}$, which differ from $\PTL\circ\mathcal{L}$ only in the
definition of spatio-temporal \emph{region} terms:
%
\begin{equation*}
\varrho \quad::=\quad \CB\IB p \quad|\quad \CB\IB\overline{\varrho} \quad|\quad \CB\IB(\varrho_1\sqcap\varrho_2) \quad|\quad \CB\IB(\varrho_1\until\varrho_2) \quad|\quad \CB\IB(\varrho_1\since\varrho_2). %
\end{equation*}
%
The following two theorems provide a positive (though partial)
answer to this question:

\begin{theorem}\tlabel{theorem:brcc8:expspace:full}
The satisfiability problem for $\PTL\times\BRCCE$ and
$\PTL\times\RCm$-formulas in tt-models based on
$\auf\mathbb{N},<\zu$ or $\auf\mathbb{Z},<\zu$ and satisfying
\FSA{}, or based on {\rm(}arbitrary{\rm)} finite flows of time is
{\cc EXPSPACE}-complete.
\end{theorem}

\begin{theorem}\tlabel{theorem:rc:2expspace:full}
The satisfiability problem for $\PTL\times\RC$-formulas in
tt-models  based on $\auf\mathbb{N},<\zu$ or $\auf\mathbb{Z},<\zu$
and satisfying \FSA{}, or based on {\rm(}arbitrary{\rm)} finite
flows of time is {\cc 2EXPSPACE}-complete.
\end{theorem}

The upper bounds mentioned in these two theorems are proved in
Appendices~\ref{appendix:embedding} and
\ref{appendix:quasimodels}, respectively. The lower bounds follow
from the results for $\PTL\circ\BRCCE$
(Theorem~\ref{st1:expspace}) and $\PTL\circ\RCm$
(Theorem~\ref{theorem:rc:2expspace}).


To appreciate the following theorem, the reader should recall that
both $\PTL_\Box$ and $\RCm$ are NP-complete:

\begin{theorem}\tlabel{theorem:brcc8:expspace:box}
The satisfiability problem for $\PTL_\Box\times\BRCCE$ and
$\PTL_\Box\times\RCm$-formulas in tt-models based on
$\auf\mathbb{N},<\zu$ or $\auf\mathbb{Z},<\zu$ and satisfying
\FSA{}, or based on {\rm(}arbitrary{\rm)} finite flows of time is
{\cc EXPSPACE}-complete.
\end{theorem}
%
Actually it is a consequence of the EXPSPACE-hardness of $\QTL^1$
with the sole temporal operator $\Box_F$
\cite<see>{Hodkinsonetal03}.

Unfortunately, very little is known about the complexity of our
spatio-temporal languages interpreted in tt-models based on dense
or arbitrary flows of time. In fact, the only result we know of
can be proved using the recent work
\cite{Hodkinson03guarded,Hodkinsonetal03}:

\begin{theorem}\tlabel{theorem:brcc8:expspace:box:dense}
The satisfiability problem for $\PTL\times\BRCCE$ and
$\PTL\times\RCm$-formulas in tt-models satisfying \FSA{} and based
on $\auf\mathbb{Q},<\zu$, $\auf\mathbb{R},<\zu$ or arbitrary flows
of time belongs to {\cc 2EXPTIME} and is {\cc EXPSPACE}-hard.
\end{theorem}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conclusion}\label{conclusion}



%\begin{sidewaystable}
%\small\mbox{}\hfill\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
%\hline%
%spatial & spatial & \multicolumn{7}{c}{spatio-temporal logics} \vline \\ \cline{3-9} %
%lang. & reasoning & %
%$\PTL_\Box[S]$ & $\PTL[S]$  & \multicolumn{2}{c}{$\PTL\circ S$} \vline & $\PTL_\Box\times S$ & \multicolumn{2}{c}{$\PTL\times S$} \vline \\  \cline{3-9}%
%& & %
%\multicolumn{2}{c}{$\mathcal{A},\mathcal{F},\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$}
%\vline & $\mathbb{N},\mathbb{Z}$ & $\mathcal{F}$ or $\mathbb{N},\mathbb{Z}_{\FSA}$ %
%& $\mathbb{N},\mathbb{Z}_{\FSA}$ & $\mathcal{F}$ or $\mathbb{N},\mathbb{Z}_{\FSA}$ & $\mathcal{A},\mathbb{Q},\mathbb{R}_{\FSA}$\\  \hline %
%%
%%
%$\SFU$ & %
%\cmplx{PSPACE}{(Thm.~\ref{s4u:pspace})}{} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%\cmplx{undecidable}{(Thm.~\ref{PST1undecidable})}{(Thm.~\ref{theorem:fullspacetime})} & %
%\cmplx{undecidable}{(Thm.~\ref{PST1undecidable})}{(Thm.~\ref{theorem:fullspacetime})} & %
%\cmplx{undecidable}{(Thm.~\ref{PSTBundecidable})}{(Thm.~\ref{theorem:fullspacetime:box})} & %
%\cmplx{undecidable}{(Thm.~\ref{PST01undecidable})}{(Thm.~\ref{theorem:fullspacetime})} & %
%? \\\hline  %
%%
%$\RCp$ & %
%\cmplx{PSPACE}{(Thm.~\ref{fullreg})}{(Lemma~\ref{s4u:reg:pspace})} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%? &  ? & ? & ? &  ? \\\hline %
%
%$\RC$ & %
%\cmplx{NP}{(Thm.~\ref{theorem:rc:np})}{(Lemma~\ref{s4u:simula3}?)} & %
%\cmplx{NP}{(Thm.~\ref{getlost})}{(Thm.~\ref{PST0:pspace}*)} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%\cmplx{2EXPSPACE}{(Thm.~\ref{theorem:rc:2expspace})}{(Thm.~\ref{rc:2expspace},\\Thm.~\ref{theorem:rc:upper:algorithm}~(i))} & %
%\cmplx{2EXPSPACE}{(Thm.~\ref{theorem:rc:2expspace})}{(Thm.~\ref{rc:2expspace},\\Thm.~\ref{theorem:rc:upper:algorithm}~(ii))} & %
%\cmplx{$\left\{\begin{array}{c}\leq {\cc 2EXPSPACE}\\\geq {\cc EXPSPACE}\end{array}\right.$}{}{(Thm.~\ref{theorem:ptlbox:brcce:expspace},\\Thm.~\ref{theorem:rc:upper:algorithm}~(ii))} &  %
%\cmplx{2EXPSPACE}{(Thm.~\ref{theorem:rc:2expspace:full})}{(Thm.~\ref{rc:2expspace},\\Thm.~\ref{theorem:rc:upper:algorithm}~(ii))} & %
%? %$\leq {\cc 3EXPTIME}$ %
%\\\hline %
%%
%$\BRCCE$ & %
%NP & %
%\cmplx{NP}{(Thm.~\ref{getlost})}{(Thm.~\ref{PST0:pspace}*)} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%\cmplx{EXPSPACE}{(Thm.~\ref{st1:expspace})}{(Thm.~\ref{brcc8:lowbound},\\ Thm.~\ref{theorem:rc:upper} (i))} & %
%\cmplx{EXPSPACE}{(Thm.~\ref{st1:expspace})}{(Thm.~\ref{brcc8:lowbound},\\ Thm.~\ref{theorem:rc:upper} (ii))} & %
%\cmplx{EXPSPACE}{(Thm.~\ref{theorem:brcc8:expspace:box})}{(Thm.~\ref{theorem:ptlbox:brcce:expspace},\\ Thm.~\ref{theorem:rc:upper} (ii))} & %
%\cmplx{EXPSPACE}{(Thm.~\ref{theorem:brcc8:expspace:full})}{(Thm.~\ref{brcc8:lowbound},\\ Thm.~\ref{theorem:rc:upper} (ii))} & %
%\cmplx{$\left\{\begin{array}{c}\!\leq {\cc 2EXPTIME}\\\!\geq {\cc EXPSPACE}\end{array}\right.$\hspace*{-0.5em}}{(Thm.~\ref{theorem:brcc8:expspace:box:dense})}{(Thm.~\ref{theorem:ptlbox:brcce:expspace},\\ Thm.~\ref{theorem:rc:upper} (iii))}%
%\\\hline  %
%
%$\RCCE$ & %$
%NP  & %
%\cmplx{NP}{(Thm.~\ref{getlost})}{(Thm.~\ref{PST0:pspace}*)} & %
%\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{(Thm.~\ref{PST0:pspace})} & %
%\cmplx{PSPACE}{(Thm.~\ref{st1m:pspace})}{(Thm.~\ref{theorem:rc2:pspace})} & %
%\cmplx{$\left\{\begin{array}{c}\!\leq {\cc EXPSPACE}\\\geq {\cc PSPACE}\end{array}\right.$\hspace*{-0.5em}}{}{(Thm.~\ref{theorem:rc:upper} (ii))} & %
%\cmplx{\hspace*{-0.5em}$\left\{\begin{array}{c}\!\leq {\cc EXPSPACE}\\\geq {\cc NP}\end{array}\right.$\hspace*{-0.5em}}{}{(Thm.~\ref{theorem:rc:upper} (ii))} & %
%\cmplx{$\left\{\begin{array}{c}\!\leq {\cc EXPSPACE}\\\geq {\cc PSPACE}\end{array}\right.$\hspace*{-0.5em}}{}{(Thm.~\ref{theorem:rc:upper} (ii))} & %
%\cmplx{$\left\{\begin{array}{c}\!\leq {\cc 2EXPTIME}\\\geq {\cc PSPACE}\end{array}\right.$\hspace*{-0.5em}}{}{(Thm.~\ref{theorem:rc:upper} (iii))} %
%\\\hline%
%\end{tabular}\hfill\mbox{}
%\caption{Complexity of pure spatial and spatio-temporal logics.}
%\end{sidewaystable}



\begin{table}
\small\mbox{}\hfill\begin{tabular}{|c|c||c|c|c|c|c|} \hline%
\rule{4pt}{0pt}\begin{rotate}{90}\hspace*{-6ex}language\end{rotate}& \raisebox{-3pt}[0pt][0pt]{flow} & \multicolumn{5}{c}{\rule[-8pt]{0pt}{22pt}spatial component $\mathcal{L}$}\vline\\\cline{3-7} %
& \raisebox{2pt}[0pt][0pt]{of time} & \rule[-8pt]{0pt}{22pt} $\RCCE$ & $\BRCCE$ & $\RC$ & $\RCp$ & $\SFU$ \\\hhline{|=|=#=|=|=|=|=|}  %
%
\rule{4pt}{0pt}\begin{rotate}{90}$\mathcal{L}$\end{rotate} & n/a & %
\multicolumn{2}{c}{NP} \vline & %
%NP & %
\cmplx{NP}{(Thm.~\ref{theorem:rc:np})}{} & %
\cmplx{PSPACE}{(Thm.~\ref{fullreg})}{} & %
\cmplx{PSPACE}{(Thm.~\ref{s4u:pspace})}{} \\ \hline %
%
\rule{4pt}{0pt}\rule[-5ex]{0pt}{12ex}\begin{rotate}{90}\hspace*{-3ex}$\PTL_\Box[\mathcal{L}]$\end{rotate} & \hspace*{-1em}\begin{tabular}{c}$\mathbb{N},\mathbb{Z}$,$\mathbb{Q},\mathbb{R}$,\\finite\\or\\arbitrary\end{tabular}\hspace*{-1em} & %
\multicolumn{3}{c}{\cmplx{NP}{(Thm.~\ref{getlost})}{}} \vline & %
\multicolumn{2}{c}{\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{}} \vline \\\hline %
%
\rule{4pt}{0pt}\rule[-5ex]{0pt}{12ex}\begin{rotate}{90}\hspace*{-3ex}$\PTL[\mathcal{L}]$\end{rotate} & \hspace*{-1em}\begin{tabular}{c}$\mathbb{N},\mathbb{Z}$,$\mathbb{Q},\mathbb{R}$,\\finite\\or\\arbitrary\end{tabular}\hspace*{-1em} & %
\multicolumn{5}{c}{\cmplx{PSPACE}{(Thm.~\ref{PST0:pspace:text})}{}} \vline \\\hline %
%
\rule{4pt}{0pt}\begin{rotate}{90}\hspace*{-8ex}$\PTL\circ\mathcal{L}$\end{rotate} & $\mathbb{N},\mathbb{Z}$ & %
\cmplx{PSPACE}{(Thm.~\ref{st1m:pspace})}{} & %
%\cmplx{EXPSPACE}{(Thm.~\ref{st1:expspace})}{}
& %
%\cmplx{2EXPSPACE}{(Thm.~\ref{theorem:rc:2expspace})}{}
& %
%?
& %
%\cmplx{undecidable}{(Thm.~\ref{PST1undecidable})}{}
\\ \cline{2-3}%
%
& \hspace*{-1em}\begin{tabular}{c}finite\\or\\ $\mathbb{N},\mathbb{Z}$+\scriptsize\FSA{}\end{tabular}\hspace*{-1em} & %
\cmplx{\hspace*{-0.2em}$\left\{\begin{array}{c}\!\leq {\cc EXPSPACE}\\\geq {\cc PSPACE}\end{array}\right.$\hspace*{-0.8em}}{}{} & %
\raisebox{4ex}[0mm][0mm]{\cmplx{EXPSPACE}{(Thm.~\ref{st1:expspace})}{}} & %
\raisebox{4ex}[0mm][0mm]{\cmplx{2EXPSPACE}{(Thm.~\ref{theorem:rc:2expspace})}{}} & %
\raisebox{3ex}[0mm][0mm]{?} & %
\raisebox{4ex}[0mm][0mm]{\cmplx{undecidable}{(Thm.~\ref{PST1undecidable})}{}} \\\hline %
%
\rule{4pt}{0pt}\begin{rotate}{90}\hspace*{-15ex}$\PTL_\Box\times\mathcal{L}$\end{rotate} & \hspace*{-1em}\begin{tabular}{c}$\mathbb{N},\mathbb{Z}$\end{tabular}\hspace*{-1em} & %
\multicolumn{4}{c}{?} \vline & %
%? & %
%? &  %
%? & %
\rule[-8pt]{0pt}{24pt}%\cmplx{undecidable}{(Thm.~\ref{PSTBundecidable})}{}
\\\cline{2-6} %
%
& \hspace*{-1em}\begin{tabular}{c}finite\\or\\ $\mathbb{N},\mathbb{Z}$+\scriptsize\FSA{}\end{tabular}\hspace*{-1em} & %
\cmplx{\hspace*{-0.2em}$\left\{\begin{array}{c}\!\leq {\cc EXPSPACE}\\\geq {\cc NP}\end{array}\right.$\hspace*{-0.8em}}{}{} & %
\cmplx{EXPSPACE}{(Thm.~\ref{theorem:brcc8:expspace:box})}{} & %
\cmplx{\hspace*{-0.2em}$\left\{\begin{array}{c}\leq {\cc 2EXPSPACE}\\\geq {\cc EXPSPACE}\end{array}\right.$\hspace*{-0.8em}}{}{} &  %
? & %
\raisebox{4ex}[0mm][0mm]{\cmplx{undecidable}{(Thm.~\ref{PSTBundecidable})}{}} \\\cline{2-7} %
%
& \hspace*{-1em}\begin{tabular}{c}arbitrary\\or\\ $\mathbb{Q},\mathbb{R}$\\with \scriptsize\FSA{}\end{tabular}\hspace*{-1em} & %
\cmplx{$\left\{\begin{array}{c}\!\leq {\cc 2EXPTIME}\\\geq {\cc NP}\end{array}\right.$\hspace*{-0.8em}}{}{} & %
\cmplx{$\left\{\begin{array}{c}\!\leq {\cc 2EXPTIME}\\\!\geq {\cc EXPSPACE}\end{array}\right.$\hspace*{-0.8em}}{}{} & %
? &  %
? & %
? \\\hline %
%
\rule{4pt}{0pt}\begin{rotate}{90}\hspace*{-14ex}$\PTL\times\mathcal{L}$\end{rotate} & \hspace*{-1em}\begin{tabular}{c}$\mathbb{N},\mathbb{Z}$\end{tabular}\hspace*{-1em} & %
\multicolumn{4}{c}{?} \vline & %
%? & %
%? & %
%? & %
\cmplx{undecidable}{(Thm.~\ref{PSTOundecidable})}{}\\\cline{2-7} %
%
& \hspace*{-1em}\begin{tabular}{c}finite\\ or\\ $\mathbb{N},\mathbb{Z}$+\scriptsize\FSA{}\end{tabular}\hspace*{-1em} & %
\cmplx{$\left\{\begin{array}{c}\!\leq {\cc EXPSPACE}\\\geq {\cc PSPACE}\end{array}\right.$\hspace*{-0.8em}}{}{} & %
\cmplx{EXPSPACE}{(Thm.~\ref{theorem:brcc8:expspace:full})}{} & %
\cmplx{2EXPSPACE}{(Thm.~\ref{theorem:rc:2expspace:full})}{} & %
? & %
\cmplx{undecidable}{(Thm.~\ref{PST01undecidable})}{}\\\cline{2-7} %
%
& \hspace*{-1em}\begin{tabular}{c}arbitrary\\ or\\ $\mathbb{Q},\mathbb{R}$\\ with \scriptsize\FSA{}\end{tabular}\hspace*{-1em} & %
\cmplx{$\left\{\begin{array}{c}\!\leq {\cc 2EXPTIME}\\\geq {\cc PSPACE}\end{array}\right.$\hspace*{-0.8em}}{}{} & %
\cmplx{$\left\{\begin{array}{c}\!\leq {\cc 2EXPTIME}\\\!\geq {\cc EXPSPACE}\end{array}\right.$\hspace*{-0.8em}}{(Thm.~\ref{theorem:brcc8:expspace:box:dense})}{} & %
? & %
? & %
? \\\hline %
\end{tabular}\hfill\mbox{}
\caption{Complexity of the satisfiability problem for spatial and
spatio-temporal logics.}\label{table:complexity}
\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



We have provided an in-depth analysis of the computational
complexity of various spatio-temporal logics interpreted in
Cartesian products of flows of time and topological spaces. Some
of these results are collected in Table~\ref{table:complexity}.
The design of the languages was driven by the idea to cover the
most basic features of spatio-temporal hybrids combining standard
logics of time and mereotopology, with the aim being to see how
complex reasoning with these hybrids could be.  We did not try to
fine-tune the languages for real-world applications.  On the
contrary, we tried to keep them as `pure' and representative as
possible and determine computational challenges which \emph{any}
multi-dimensional approach to reasoning about space and time would
face.  With this research objective in mind, we discuss now some
conclusions that can be drawn from Table~\ref{table:complexity}.

The conclusion to be drawn from the undecidability results is
easy: do not try to implement a sound, complete and terminating
algorithm which is supposed to decide the satisfiability problem
for $\PTL\times\SFU$, $\PTL\circ\SFU$ or
$\PTL_\Box\times\SFU$---you will never succeed. If decision
procedures are required, then alternative languages have to be
devised.

The interpretation of the complexity results for decidable logics
is not so transparent: it is well-known that such results do not
provide us with immediate conclusions regarding the behaviour of
implemented systems. For example, sometimes algorithms running in
exponential time in the worst-case perform better on
\emph{practical problems} than worst-case optimal algorithms that
run in polynomial time. Indeed, the complexity results should be
analysed \emph{together with their proofs}---if significant
conclusions are required (cf.\ \citeR{Nebel96}).  Only the proofs
show where the sources of the complexity are and whether they
could be relevant for practical problems and the implementation of
algorithms.

In this respect our proofs are actually rather informative. The
decidability proof for $\PTL[\SFU]$ immediately provides us with a
modular algorithm combining any known procedures for the
components. The EXPSPACE-completeness results for
$\PTL\times\BRCCE$ (with \FSA) and $\PTL\circ\BRCCE$ show an
extremely close link between the spatio-temporal languages and the
\emph{one-variable fragment of first-order temporal logic}. The
algorithmic problems investigated in the context of first-order
temporal logic are, therefore, of the same character as those we
deal with in the spatio-temporal context. Thus, the experience of
working with algorithms for (fragments of) first-order temporal
logics \cite{Hodkinsonetal00,Degtyarevetal03,Kontchakovetal02}
about which we have a pretty good knowledge by now almost directly
translates to insights into possible algorithms for
spatio-temporal logics.  The PSPACE-completeness result for
$\PTL\circ\RCCE$ is obtained by means of a reduction (modulo
$\RCCE$ reasoning) to $\PTL$. So we can conclude from the proof
that it will be sufficient to have good solvers for $\RCCE$
\emph{and} $\PTL$ to obtain a reasonable prover for
$\PTL\circ\RCCE$. The interaction between the two components
turned out to be rather weak.

In conclusion, the complexity proofs clearly show the algorithmic
problems to be solved when dealing with the spatio-temporal logics
presented in this paper. In particular, devising algorithms for
these logics should be conceived as part of the more general
enterprise of developing algorithms for propositional and the
one-variable fragment of first-order temporal logic.

Here are some comments on and explanations of the most important
results in Table~\ref{table:complexity}:
%
\begin{enumerate}
\item The undecidability result for $\PTL\times\SFU$, $\PTL\circ\SFU$
and $\PTL_\Box\times\SFU$ solves a major open problem of Wolter
and Zakharyaschev \citeyear{Wolter&Z02millenium}. It shows that,
while $\SFU$ is a suitable candidate for efficient pure spatial
reasoning \cite{Bennett96,Renz&Nebel98,Aiello&Benthem00}, its
temporal extensions satisfying both (PC) and (OC) are not suitable
for practical \emph{spatio-temporal} representation and reasoning.
%
\item Logics like $\PTL\times\BRCCE$ may turn out to be
undecidable when interpreted in \emph{arbitrary topological
temporal models}. One of the main origins of their expressive
power is a possibility to form infinite intersections and unions
of regions. However, we can `tame' the computational behaviour of
these logics by imposing natural restrictions on the classes of
admissible models such as \FSA.
%
\item The PSPACE upper bound for $\PTL\circ\RCCE$ and the EXPSPACE lower
bound for $\PTL\circ\BRCCE$ solve two other major open problems of
Wolter and Zakharyaschev \citeyear{Wolter&Z02millenium}. It is of
interest to note that the spatial fragments of $\PTL\circ\RCCE$
and $\PTL\circ\BRCCE$ have the same computational complexity: both
are NP-complete over arbitrary topological spaces. Thus the
additional Boolean connectives on spatial regions interacting with
the next-time operator $\nxt$ can make the logic substantially
more complex.
%
\item The 2EXPSPACE-completeness result for
$\PTL\times\RC$ with \FSA{} and $\PTL\circ\RC$ is another example
when a seemingly tiny increase of expressiveness results in a
significant jump of complexity.
%
\item PSPACE-completeness of $\PTL\circ\RCCE$ is a particularly `good
news,' since it shows that this combination of $\PTL$ and $\RCCE$
has the same computational complexity as $\PTL$ itself, for which
surprisingly fast systems have been implemented
\cite{Schwendimann98b,Hustadt&Konev03}. This gives us hope that
`practical' algorithms for $\PTL\circ\RCCE$ can be implemented.
Indeed, the proof shows that it may be possible to encode the
satisfiability problem for $\PTL\circ\RCCE$ into the
satisfiability problem for $\PTL$ and then use $\PTL$ provers. We
note that this complexity result has been conjectured by Demri and
D'Souza \citeyear{Demri&Souza02} and that our proof uses some
ideas of Balbiani and Condotta \citeyear{Balbiani&Condotta02}.
%
\item On the other hand, the EXPSPACE lower bounds for
$\PTL\times\BRCCE$ with \FSA{} and $\PTL\circ\BRCCE$ do not
necessarily mean that reasoning with these logics is hopeless. In
fact, we show that both of them can be regarded as fragments of
the one-variable first-order temporal logic, for which tableau-
and resolution-based decision procedures have been developed and
implemented \cite{Degtyarevetal03,Kontchakovetal02}.
\end{enumerate}
%
Of course, there are many directions of further research in
spatio-temporal knowledge representation and reasoning. Here we
mention only some of them that are closely related to the logics
we have considered above.
%
\begin{itemize}
\item In this paper, we confined ourselves to considering
linear flows of time. It may be of interest, however, to
investigate the computational properties of spatio-temporal logics
based on the branching time paradigm (see, e.g.,
\citeR{Clarke&Emerson81,Emerson&Halpern85}) in order to model
uncertainty about the future. Recent results by Hodkinson, Wolter
and Zakharyaschev
\citeyear{Hodkinsonetal01lpar,Hodkinsonetal02lics} give hope that
such logics can be decidable.
%
\item We confined ourselves to considering only mereotopological
formalisms for the spatial dimension. It would be also of interest
to consider spatial logics of directions \cite{Ligozat98}, shape
\cite{Galton&Meathrel99}, size \cite{Zimmermann95}, position
\cite{Clementinietal97}, or even their hybrids
\cite{Gerevini&Renz02}.  We note that some results in this
direction have been recently obtained by Balbiani and Condotta
\citeyear{Balbiani&Condotta02} and Demri and D'Souza
\citeyear{Demri&Souza02}.
%
\item Another interesting and important perspective in both
spatial and spatio-temporal representation and reasoning is to
move from arbitrary topological spaces to those induced by
\emph{metric spaces} and introduce explicit and/or implicit
numerical parameters. First encouraging steps in this direction
have been made in the work \cite{Kutzetal03}.
\end{itemize}

\noindent
%
We conclude the paper with a number of open problems:
%
\begin{enumerate}
\item What is the precise computational complexity of $\PTL\times\BRCCE$
with \FSA{} over dense flows of time and arbitrary strict
linear orders?
%
\item Are logics of the form $\PTL\times\mathcal{L}$ and
$\PTL_\Box\times\mathcal{L}$, for $\mathcal{L} \in \{ \RC, \BRCCE,
\RCCE\}$, decidable without \FSA{}?
%
\item Are combinations of $\PTL$ and $\PTL_\Box$ with $\RCp$
(satisfying both (PC) and (OC)) decidable?
%
\item Is $\PTL\times\SFU$ undecidable over dense flows of time and
arbitrary strict linear orders?
%
\item Is $\PTL\times\RCCE$ with \FSA{} decidable in PSPACE?
\end{enumerate}

\section*{Acknowledgments}

The work on this paper was partially supported by  U.K.\ EPSRC
grants no.\ GR/R45369/01, GR/R42474/01, GR/S61966/01 and
GR/S63182/01. The work of the third author was also partially
supported by Hungarian Foundation for Scientific Research grants
T30314 and 035192.

Special thanks are due to the referees of the first version of
this paper whose remarks, criticism and constructive suggestions
have led to many days of intensive and exciting research, new
results and, hopefully, a better paper.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix

\section{Complexity of Spatial Logics}\label{appendix:spatial}

In this appendix we prove Theorems \ref{theorem:rc:np} and
\ref{fullreg}. In these proofs we use the fact that $\SFU$ (as
well as its fragments) is complete with respect to (finite)
Aleksandrov topological spaces
\cite{McKinsey&Tarski44,Goranko&Passy92}. Recall from
p.~\pageref{def:Aleksandrov} that an \emph{Aleksandrov
\textup{(}topological\textup{)} model} is a pair of the form
$\mathfrak M = \auf \mathfrak G, \mathfrak V \zu$, where
$\mathfrak G = \auf V,R \zu$ is a quasi-order and $\mathfrak V$ is
a map from the set of spatial variables into $2^V$. It will be
more convenient for us to unify notation for spatial formulas and
spatial terms and write $(\mathfrak M,x)\models\tau$ instead of
$x\in\mathfrak V(\tau)$, for $\tau$ a spatial term and $x$ a point
in $V$. In particular, by the definition of the interior and
closure operators in Aleksandrov spaces,
%
\begin{align*}
(\mathfrak M,x)\models\IB\tau\quad &\text{iff}\quad \forall y\in V
\big(xRy \to (\mathfrak M,y)\models\tau\big), \\
(\mathfrak M,x)\models\CB\tau\quad &\text{iff}\quad \exists y\in V
\big(xRy \land (\mathfrak M,y)\models\tau\big).
\end{align*}

By the \emph{length} $\ell(\varphi)$ of a formula $\varphi$ we
understand the number of subformulas and subterms occurring in
$\varphi$.


\begin{proof}[\proofof{Theorem \ref{theorem:rc:np}}]
The proof follows from Lemmas~\ref{s4u:simula2} and
\ref{s4u:simula3} below which show together that every satisfiable
$\RC$-formula can be satisfied in an Aleksandrov model of size
polynomial (in fact, quadratical) in the length of the input
formula (in other words, $\RC$ has the polynomial finite model
property). Thus, we have a nondeterministic polynomial time
algorithm for the satisfiability problem.
\end{proof}


In fact, Lemma~\ref{s4u:simula2} shows that $\RC$ is complete with
respect to a subclass of Aleksandrov spaces, namely, finite
disjoint unions of finite brooms. Recall from
p.~\pageref{fig:nbroom} that a \emph{broom} is a partial order
$\mathfrak{b}$ of the form $\auf \{r\}\cup V_0, R\zu$, where $R$
is the reflexive closure of $\{r\}\times V_0$
(see~Fig.~\ref{fig:nbroom}). We call $r$ the \emph{root} of
$\mathfrak{b}$ and points in $V_0$ the \emph{leaves} of
$\mathfrak{b}$; they are also referred to as \emph{points of
depth} $1$ and $0$, respectively. A broom $\mathfrak{b}$ is said
to be a \emph{$\kappa$-broom}, $\kappa\le \omega$, if $|V_0| \le
\kappa$. In particular, we call a broom \emph{finite} if it is an
$n$-broom, for some $n < \omega$.

\begin{lemma}\label{s4u:simula2}
Every satisfiable $\RC$-formula is satisfied in an Aleksandrov
model based on a finite disjoint union of finite brooms.
\end{lemma}
%
\begin{proof}
As is well-known, if an $\RC$-formula $\varphi$ is satisfiable
then it can be satisfied in a finite Aleksandrov model $\mathfrak
M = \auf \mathfrak G, \mathfrak V \zu$, $\mathfrak G = \auf V,R
\zu$. Define a new relation $R'$ on $V$ by taking $R'$ to be the
reflexive closure of $R \cap (V_1 \times V_0)$, where
%
\begin{equation*}
V_0 = \{ x \in V \mid \neg \exists y\: (xRy \wedge \neg\; yRx)\}
\quad\text{and}\quad V_1=V-V_0.
\end{equation*}
%
(Without loss of generality we may assume that $V_1\neq\emptyset$
and no $y\in V_0$ has more than one proper $R$-predecessor.) Let
$\mathfrak G'=\auf V,R'\zu$ and $\mathfrak{M}'= \auf \mathfrak G',
\mathfrak V\zu$. Clearly, $\mathfrak G'$ is a partial order as
required. We prove that, for every $\RC$-formula $\psi$,
%
\begin{equation}\label{eq:rc:subf}
\mathfrak{M}\models\psi\qquad\text{iff}\qquad\mathfrak{M}'\models\psi.
\end{equation}
%
First we show that, for every Boolean region term $\varrho$ and
every $x\in V$,
%
\begin{equation}\label{s4u:ci}
(\mathfrak{M}',x)\models\varrho\qquad\text{iff}\qquad(\mathfrak{M},x)\models\varrho.
\end{equation}
%
By definition, $(\mathfrak{M}',x) \models p$ iff
$(\mathfrak{M},x)\models p$, for every spatial variable $p$. It is
readily seen that for every $y\in V_0$ and every spatial term
$\tau$, we have $(\mathfrak{M}',y)\models \tau$ iff
$(\mathfrak{M},y)\models \tau$. Now, if $\varrho$ is a Boolean
region term then $\varrho = \CB\IB\tau$ for some spatial term
$\tau$, and we clearly have:
%
\begin{align*}
(\mathfrak{M},x)\models \CB\IB\tau\quad%
&\text{iff}\quad \exists y\in V\ \bigl(xRy \text{ and } \forall z\in V\ (yRz\to (\mathfrak{M},z)\models \tau)\bigr)\\
&\text{iff}\quad \exists y\in V_0 \ \bigl(xR'y \text{ and } (\mathfrak{M},y)\models \tau\bigr)\\
&\text{iff}\quad \exists y\in V_0 \ \bigl(xR'y \text{ and } (\mathfrak{M}',y)\models \tau\bigr)\\
&\text{iff}\quad \exists y\in V_0 \ \bigl(xR'y \text{ and } (\mathfrak{M}',y)\models \IB\tau\bigr)\\
&\text{iff}\quad (\mathfrak{M}', x)\models \CB\IB\tau.
\end{align*}
%
Next, we extend \eqref{s4u:ci} to spatial terms of the form
$\IB\varrho$ where $\varrho$ is a Boolean region term. If
$(\mathfrak{M},x)\models\IB\varrho$ then
$(\mathfrak{M},y)\models\varrho$ whenever $xRy$, and so, by
$R'\subseteq R$, we have $(\mathfrak{M}',x)\models\IB\varrho$.
Conversely, suppose $(\mathfrak{M}',x)\models\IB\varrho$. Take any
$y$ with $xRy$ and any $z\in V_0$ with $yRz$. We claim that
$(\mathfrak{M},z)\models\varrho$. Indeed, if $x\in V_1$ then this
follows by IH from $xR'z$. If $x\in V_0$ then $zRx$. Since
$(\mathfrak{M}',x)\models\varrho$, by IH and $\varrho=\CB\IB\tau$,
we obtain $(\mathfrak{M},z)\models\varrho$. Now
$(\mathfrak{M},y)\models\varrho$ follows by $yRz$ and
$\varrho=\CB\IB\tau$. Thus, $(\mathfrak{M},x)\models\IB\varrho$.

Finally, we can easily extend \eqref{s4u:ci} to arbitrary spatial
terms and formulas of $\RC$ because both are constructed from
spatial terms of the form $\varrho$ and $\IB\varrho$, with
$\varrho$ a Boolean region term, using operators that do not
depend on the structure of the underlying partial order. Thus we
have \eqref{eq:rc:subf}.
\end{proof}




\begin{lemma}\label{s4u:simula3}
Every satisfiable $\RC$-formula $\varphi$ is satisfied in an
Aleksandrov model based on a disjoint union of at most
$\ell(\varphi)$ many $2\ell(\varphi)$-brooms.
\end{lemma}
%
\begin{proof}
Remember that every $\RC$-formula $\varphi$ is (equivalent to) a
Boolean combination of spatial formulas from some set
$\Sigma_\varphi = \{
\Udiamond\tau_1,\dots, \Udiamond\tau_m\}$.\!\footnote{%
%
In the following proof we consider {\normalsize$\Udiamond$} as
primary and {\normalsize$\Ubox\tau$} as an abbreviation for
{\normalsize$\neg\Udiamond\overline{\tau}$}.
}\ %
%
For each $\Udiamond\tau\in\Sigma_\varphi$, the spatial term $\tau$
is also a Boolean (or rather set-theoretic) combination of some
$\varrho_1, \dots, \varrho_k,\; \IB\varrho'_1, \dots,
\IB\varrho'_m$, where the $\varrho_i$ and the $\varrho_i'$ are
Boolean region terms.

It follows from Lemma~\ref{s4u:simula2} that $\varphi$ is
satisfied in an Aleksandrov model $\mathfrak{M}=\auf \mathfrak{G},
\mathfrak{V}\zu$, where $\mathfrak{G}= \auf V,R\zu$ is a finite
disjoint union of finite brooms.  For every $\Udiamond\tau\in
\Sigma_\varphi$ with $\mathfrak{M}\models \Udiamond\tau$, fix a
point $x_\tau\in V$ such that $(\mathfrak{M}, x_\tau)\models
\tau$. We may assume that the $x_\tau$ are pairwise distinct and
that the roots of all brooms are the points of the form $x_\tau$
for $\Udiamond\tau\in\Sigma_\varphi$. Therefore, $\mathfrak{G}$ is
a disjoint union of $\leq \ell(\varphi)$ many finite brooms
$\mathfrak{b}_\tau$, for $\Udiamond\tau\in\Sigma_\varphi$.

Let us construct a new model $\mathfrak{M}'$ as follows. For each
broom $\mathfrak{b}_\tau$, $\Udiamond\tau\in\Sigma_\varphi$, and
each $\varrho\in\Xi_{\tau}$, we pick
%
\begin{itemize}
\item a leaf $y_{\tau,\varrho}$ of $\mathfrak{b}_\tau$ (if any) such that
$(\mathfrak{M},y_{\tau,\varrho})\models\varrho$,
%
\item a leaf $y_{\tau,\overline{\varrho}}$ of $\mathfrak{b}_\tau$ (if any) such that
$(\mathfrak{M},y_{\tau,\overline{\varrho}})\models\overline{\varrho}$
\end{itemize}
%
and remove the other leaves of $\mathfrak{b}_\tau$. Denote by
$\mathfrak{b}'_\tau$ the resulting broom. Clearly, it is a
$2\ell(\varphi)$-broom. Let $\mathfrak{G}' = \auf V',R'\zu$ be the
disjoint union of all $\mathfrak{b}'_\tau$, for
$\Udiamond\tau\in\Sigma_\varphi$, and $\mathfrak M' = \auf
\mathfrak G', \mathfrak V\zu$. It is easy to see that
$\mathfrak{G}'$ is as required.

Now, to show that $\varphi$ is satisfied in $\mathfrak{M}'$, it
suffices to prove that, for all $\Udiamond \tau\in\Sigma_\varphi$,
%
\begin{equation}\label{eq:rc:terms}
\mathfrak{M}'\models\Udiamond\tau\qquad\text{iff}\qquad
\mathfrak{M}\models \Udiamond\tau.
\end{equation}
%
By definition of $\mathfrak{M}'$, for all leaves $y$ of
$\mathfrak{G}'$ and all spatial terms $\tau$,
%
\begin{equation*}%\label{justincase}
(\mathfrak{M}',y)\models\tau\qquad\text{iff}\qquad
(\mathfrak{M},y)\models\tau .
\end{equation*}
%
Next, for every root $x_{\tau}$ of $\mathfrak{b}_\tau$, every
$\Udiamond\tau\in\Sigma_\varphi$ and every $\varrho\in\Xi_{\tau}$,
we have $(\mathfrak{M},x_\tau)\models\varrho$ iff there is a leaf
$y$ such that $x_\tau Ry$ and $(\mathfrak{M},y)\models\varrho$
(simply because $\varrho=\CB\IB\delta$, for some $\delta$). It
follows from the construction of $\mathfrak M'$ that
$(\mathfrak{M},x_\tau)\models\varrho$ iff
$(\mathfrak{M}',x_\tau)\models\varrho$, for every
$\varrho\in\Xi_\tau$. It also follows that
$(\mathfrak{M},x_\tau)\models\IB\varrho$ implies
$(\mathfrak{M}',x_\tau)\models\IB\varrho$. Conversely, if
$(\mathfrak{M}',x_\tau)\models\IB\varrho$, but
$(\mathfrak{M},x_\tau)\not\models\IB\varrho$ then there is a leaf
$y$ such that $x_\tau Ry$ and $(\mathfrak{M},y)\not\models\varrho$
which is a contradiction. Since intersection and complement do not
depend on the structure of the underlying frame, we have
$(\mathfrak{M}',x_\tau)\models\tau$ iff
$(\mathfrak{M},x_\tau)\models\tau$, for every root $x_\tau$ of
$\mathfrak{b}_\tau$, which proves~\eqref{eq:rc:terms}.
\end{proof}



\begin{proof}[\proofof{Theorem \ref{fullreg}}]
The PSPACE upper bound follows from Theorem~\ref{s4u:pspace}. The
proof of PSPACE-hardness is by reduction of the validity problem
for quantified Boolean formulas which is known to be
PSPACE-complete \cite{Stockmeyer87}. We will slightly modify the
proof of Ladner \citeyear{Ladner77} (that shows the
PSPACE-hardness of $\SF$), in order to take into account that the
variables in $\RCp$-formulas are always prefixed by $\CB\IB$.

We may assume that quantified Boolean formulas are of the form
%
\begin{equation*}
\varphi = Q_1 p_1 \dots Q_n p_n\; \varphi',
\end{equation*}
%
where $Q_i\in\{\forall, \exists\}$ and $\varphi'$ is a Boolean
formula with variables $p_1,\dots,p_n$. As is well known, all
possible truth assignments to $p_1,\dots,p_n$ can be arranged as
the leaves of a full binary tree of depth $n$. The left subtree of
the root contains all truth assignments with $p_1$ true and the
right subtree those with $p_1$ false; then we branch on $p_2$,
then $p_3$, and so on. We can determine whether $\varphi$ is valid
by pruning this full binary tree: whenever $Q_i$ is $\forall$,
then we keep both subtrees at the $i$th level, and whenever $Q_i$
is $\exists$ then only one of them. If this way we can end up with
a tree such that all its leaves evaluate $\varphi'$ to true, then
$\varphi$ is valid, otherwise not.

We will `generate' the leaves of this binary tree in Aleksandrov
models with the help of an $\RCp$-formula. More precisely, we will
construct an $\RCp$-formula $\varphi^*$ such that
\begin{itemize}
\item
its length is polynomial in the length of $\varphi$, and
\item
$\varphi^*$ is satisfied in an Aleksandrov model iff $\varphi$ is
valid.
\end{itemize}
%
Take fresh spatial variables $q_0, \dots, q_{n}$, and put, for
$i=0,\dots,n$,
%
\begin{equation*}
\lambda_i=\begin{cases}
 \ \reg{q_0}\sqcap\reg{q_1} & \text{if}\quad i=0;\\[4pt]
 \ \overline{\reg{q_{i-1}}}\sqcap\reg{q_i}\sqcap\reg{q_{i+1}},\quad & \text{if}\quad 0<i<n;\\[4pt]
 \ \overline{\reg{q_{n-1}}}\sqcap\reg{q_n}, & \text{if}\quad i=n.
 \end{cases}
\end{equation*}
%
Now consider the variables $p_1\dots,p_n$ of $\varphi$ as spatial
variables, and let $\varphi''$ be the result of replacing every
occurrence of $p_i$ with $\reg{p_i}$ in $\varphi'$.  Put
%
\begin{equation*}
\varphi^* = \Udiamond\lambda_0 \ \land\ %
\bigwedge_{Q_i=\exists}\!\Ubox\bigl(\lambda_{i-1} \imp (\tau^-_i\sqcup\tau^+_i)\bigr) \ \land\ %
\bigwedge_{Q_i=\forall}\!\Ubox\bigl(\lambda_{i-1} \imp (\tau^-_i\sqcap\tau^+_i)\bigr) \ \land\ %
\Ubox\bigl(\lambda_n\imp \varphi''\bigr), %
\end{equation*}
%
where, for $i=1,\dots,n$,
%
\begin{equation*}
\tau^-_i = \CB\bigl(\lambda_i\sqcap \overline{\reg{p_i}}\bigr)\qquad\text{and}\qquad %
\tau^+_i = \CB\bigl(\lambda_i\sqcap \IB \reg{p_i}\bigr). %
\end{equation*}
%
Clearly, $\varphi^\ast$ is an $\RCp$-formula and its length is
polynomial in the length of $\varphi$.

Suppose first that $\varphi$ is valid. Then
Fig.~\ref{fig:rcplus:model} shows the structure of a possible
Aleksandrov model satisfying $\varphi^\ast$.

\begin{figure}
\begin{center}
\begin{picture}(260,140)(20,-5)
%\put(0,0){\line(1,0){260}}\put(0,0){\line(0,1){140}}%
%
\thicklines%
\put(150,0){\put(0,0){\circle*{3}}\put(20,0){\circle*{3}}\put(0,0){\vector(1,0){18}}}%
\put(150,0){\vector(2,1){78}}\put(150,0){\vector(-2,1){78}}%
\put(70,40){\put(0,0){\circle*{3}}\put(20,0){\circle*{3}}\put(0,0){\vector(1,0){18}}}%
\put(230,40){\put(0,0){\circle*{3}}\put(20,0){\circle*{3}}\put(0,0){\vector(1,0){18}}}%
\put(70,40){\vector(0,1){38}}\put(230,40){\vector(0,1){38}}%
\put(70,80){\put(0,0){\circle*{3}}\put(20,0){\circle*{3}}\put(0,0){\vector(1,0){18}}}%
\put(230,80){\put(0,0){\circle*{3}}\put(20,0){\circle*{3}}\put(0,0){\vector(1,0){18}}}%
\put(70,80){\vector(1,1){38}}\put(70,80){\vector(-1,1){38}}%
\put(230,80){\vector(1,1){38}}\put(230,80){\vector(-1,1){38}}%
\put(30,120){\put(0,0){\circle*{3}}}%
\put(110,120){\put(0,0){\circle*{3}}}%
\put(190,120){\put(0,0){\circle*{3}}}%
\put(270,120){\put(0,0){\circle*{3}}}%
%\put(30,120){\vector(0,1){38}}\put(110,120){\vector(0,1){38}}\put(190,120){\vector(0,1){38}}\put(270,120){\vector(0,1){38}} %
%\put(30,160){\circle*{3}}\put(110,160){\circle*{3}}\put(190,160){\circle*{3}}\put(270,160){\circle*{3}} %
\put(-1,10){%
\put(32,123){${}^{\varphi''}$}%
\put(112,123){${}^{\varphi''}$}%
\put(192,123){${}^{\varphi''}$}%
\put(272,123){${}^{\varphi''}$}
\put(28,93){${}^{\lambda_3}$}%
\put(108,93){${}^{\lambda_3}$}%
\put(188,93){${}^{\lambda_3}$}%
\put(268,93){${}^{\lambda_3}$}
\put(20,113){${}^{q_3,\,p_1,\,p_2,\,p_3}$}%
\put(100,113){${}^{q_3,\,p_1,\,p_2}$}%
\put(180,113){${}^{q_3,\,p_2,\,p_3}$}%
\put(260,113){${}^{q_3,\,p_2}$}%
\put(96,70){${}^{q_2,\,p_1,\,p_2}$}%
\put(258,70){${}^{q_2,\,p_2}$}%
\put(56,65){${}^{\lambda_2}$}%
\put(218,65){${}^{\lambda_2}$}%
\put(96,26){${}^{q_1,\,p_1}$}%
\put(258,26){${}^{q_1}$}%
\put(56,26){${}^{\lambda_1}$}%
\put(218,28){${}^{\lambda_1}$}%
\put(175,-18){${}^{q_0}$}%
\put(136,-21){${}^{\lambda_0}$}
}%
%
\end{picture}
\end{center}
\caption{An Aleksandrov model that may satisfy $\varphi^\ast$,
for $\varphi = \forall p_1 \exists p_2 \forall p_3\;
\varphi'$.}\label{fig:rcplus:model}%
\end{figure}

The converse direction is similar to that of Ladner's
proof~\citeyear{Ladner77}. Suppose that $\varphi^\ast$ is
satisfied in an Aleksandrov model $\mathfrak M$. Then, for each
`necessary' sequence of truth values for
$\reg{p_1},\dots,\reg{p_n}$, there is a point in $\mathfrak M$
`reflecting' this sequence (we do not use the `structure' of the
spatial terms $\lambda_i$ here). Since, by the last conjunct of
$\varphi^\ast$, $\varphi''$ holds in $\mathfrak M$ at all these
points, we obtain that the quantified Boolean formula $\varphi$
must be valid.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section[Spatio-Temporal Logics Based on S4u]{Spatio-Temporal Logics Based on $\SFU$}\label{section:sptm:on:s4u}

In this appendix we prove Theorems \ref{PST0:pspace:text},
\ref{getlost}, \ref{PST1undecidable} and \ref{PSTBundecidable} as
well as Proposition~\ref{proposition:fsa}. Then
Theorems~\ref{PSTOundecidable} and \ref{PST01undecidable} are
immediate corollaries of Theorem~\ref{PST1undecidable}. But first,
some general results are established to be used later on.

We remind the reader that by an \emph{Aleksandrov tt-model} we mean
a tt-model based on an Aleksandrov (topological) space. Every such
model can be regarded as a triple of the form $\mathfrak{K} =\auf
\mathfrak{F}, \mathfrak{G}, \mathfrak{V}\zu$, where $\mathfrak F =
\auf W,<\zu$ is a flow of time, $\mathfrak G = \auf V,R\zu$ a
quasi-order, and $\mathfrak V$ is a map associating with every
spatial variable $p$ and every time point $w\in W$ a set $\mathfrak
V (p,w)\subseteq V$. As in Appendix~\ref{appendix:spatial}, instead
of $x\in \mathfrak V(\tau,w)$ we write $(\mathfrak{K},\auf w,x\zu)
\models \tau$ to unify notation for spatio-temporal formulas and
terms.

Given\label{def:subformulas:subterms} a spatio-temporal formula
$\varphi$, we denote by $\textit{sub}\,\varphi$ the set of all its
subformulas and by $\textit{term}\,\varphi$ the set of all
spatio-temporal terms occurring in $\varphi$.

\begin{lemma}\label{simula1}
{\rm (i)} If a $\PTL\times\SFU$-formula $\varphi$ is satisfied in
a tt-model with \FSA{} and based on a flow of time $\mathfrak{F}$,
then $\varphi$ is satisfied in an Aleksandrov tt-model with \FSA{}
and based on~$\mathfrak{F}$.

{\rm (ii)} If a $\PTL\circ\SFU$-formula $\varphi$ is satisfied in
a tt-model based on a flow of time $\mathfrak{F}$, then $\varphi$
is satisfied in an Aleksandrov tt-model based on $\mathfrak{F}$ as
well.

Moreover, in both cases we can choose an Aleksandrov tt-model
$\mathfrak{K} =\auf \mathfrak{F}, \mathfrak{G}, \mathfrak{V}\zu$
satisfying $\varphi$ {\rm(}with $\mathfrak{F}=\auf W,<\zu$ and
$\mathfrak{G}= \auf V,R\zu${\rm)} in such a way that for all $w\in
W$, $x\in V$ and spatio-temporal terms $\tau$, the set
%
\begin{equation*}
A_{w,x,\tau}=\{ y\in V \mid x R y \text{ and } (\mathfrak{K},\auf
w,y\zu) \models \tau\}
\end{equation*}
%
contains an $R$-maximal point\,\footnote{A point $z$ is said to be
$R$-{\em maximal} in $A\subseteq V$ if, for every $z'\in A$, we
have $z'Rz$ whenever $zRz'$.} $($provided of course that
$A_{w,x,\tau} \neq \emptyset$$)$.
\end{lemma}
%
\begin{proof}
The proof uses the Stone--J\'onsson--Tarski representation of
topological Boolean algebras (in particular, topological spaces)
in the form of general frames (see, e.g., \citeR{Goldblatt76a} or
\citeR{Chagrov&Z97}).

(i) Suppose that $\varphi$ is satisfied in a tt-model
$\mathfrak{M}=\auf \mathfrak{F}, \mathfrak{T}, \mathfrak{U}\zu$
with \FSA{} and based on a topological space $\mathfrak{T}=\auf
U,\mathbb{I}\zu$.
%An \emph{ultrafilter $\boldsymbol{x}$ over} $U$
%is a subset of the powerset $2^{U}$ of $U$ such that
%
%\begin{itemize}
%\item $U\in \boldsymbol{x}$;
%\item if $A\in \boldsymbol{x}$ and $A\subseteq B$, then $B\in \boldsymbol{x}$;
%\item $A\cap B\in \boldsymbol{x}$ iff $A,B\in \boldsymbol{x}$, for all
%  $A,B\subseteq U$;
%\item $A\cup B\in \boldsymbol{x}$ iff either $A\in \boldsymbol{x}$ or
%  $B\in \boldsymbol{x}$, for all $A,B\subseteq U$;
%\item $A\in \boldsymbol{x}$ iff $U-A\notin \boldsymbol{x}$, for every
%  $A\subseteq U$.
%\end{itemize}
%
Denote by $V$ the set of all ultrafilters over $U$.
%This set is
%not empty: as is well known, for every set
%$\boldsymbol{y}\subseteq 2^{U}$ with the \emph{finite intersection
%property} (i.e., such that $A_1\cap \dots \cap A_k \neq\emptyset$
%for any $A_1,\dots,A_k\in \boldsymbol{y}$ and $k<\omega$) there
%exists an ultrafilter $\boldsymbol{x}\supseteq \boldsymbol{y}$.
%(For instance, the set $\{ A \subseteq U \mid u\in A\}$ is an
%ultrafilter, for every $u\in U$.) So if a set $A \subseteq U$ is
%in \emph{every} ultrafilter over $U$, then $A$ must be $U$ itself.
%
For any two ultrafilters $\boldsymbol{x}_1,\boldsymbol{x}_2\in V$,
put $\boldsymbol{x}_1 R \boldsymbol{x}_2$ iff $\forall A \subseteq
U\ (\mathbb{I}A \in \boldsymbol{x}_1 \to A \in \boldsymbol{x}_2)$.
It is easy to see that $R$ is a quasi-order on $V$. Define an
Aleksandrov tt-model $\mathfrak{K}=\auf\mathfrak{F}, \mathfrak{G},
\mathfrak{V}\zu$ by taking $\mathfrak{G}=\auf V,R\zu$ and
$\mathfrak{V}(p,w) = \{ \boldsymbol{x}\in V \mid \mathfrak{U}(p,w)
\in \boldsymbol{x}\}$. We show by induction on the construction of a
spatio-temporal term $\tau$ that, for all $w \in W$ and
$\boldsymbol{x}\in V$,
%
\begin{equation}\label{eq:PST:model}
(\mathfrak{K},\auf w,\boldsymbol{x}\zu)\models\tau
\qquad\text{iff}\qquad%
\mathfrak{U}(\tau,w) \in \boldsymbol{x}.
\end{equation}
%
The basis of induction and the case of the Booleans are trivial.
The case of $\tau = \IB\tau'$ is standard (consult
\citeR{Goldblatt76a} or \citeR{Chagrov&Z97}).

%For
%propositional variables \eqref{eq:PST:model} follows from the
%definition of $\mathfrak{K}$.

%\smallskip

%{\em Case} $\tau = \tau_1\sqcap \tau_2$. We have:
%$(\mathfrak{K},\auf w,\boldsymbol{x}\zu)\models\tau$ iff
%$(\mathfrak{K},\auf w,\boldsymbol{x}\zu)\models\tau_1$ and
%$(\mathfrak{K},\auf w,\boldsymbol{x}\zu)\models\tau_2$ iff (by IH)
%$\mathfrak{U}(\tau_1,w) \in \boldsymbol{x}$ and
%$\mathfrak{U}(\tau_2,w) \in \boldsymbol{x}$ iff (by the definition
%of ultrafilter) $\mathfrak{U}(\tau_1,w)\cap \mathfrak{U}(\tau_2,w)
%\in \boldsymbol{x}$ iff $\mathfrak{U}(\tau_1\sqcap \tau_2,w) \in
%\boldsymbol{x}$.

%\smallskip

%{\em Case} $\tau = \overline{\tau'}$. In this case,
%$(\mathfrak{K},\auf w,\boldsymbol{x}\zu)\models\overline{\tau'}$
%iff $(\mathfrak{K},\auf w,\boldsymbol{x}\zu)\not\models\tau'$ iff
%(by IH) $\mathfrak{U}(\tau',w) \notin \boldsymbol{x}$ iff
%$U-\mathfrak{U}(\tau',w)\in \boldsymbol{x}$ iff
%$\mathfrak{U}(\overline{\tau'},w)\in \boldsymbol{x}$.

%\smallskip

%{\em Case} $\tau = \IB\tau'$. Suppose that $(\mathfrak{K},\auf
%w,\boldsymbol{x}\zu)\models \IB \tau'$, but $\mathfrak{U}(\IB
%\tau',w)\notin \boldsymbol{x}$. Then
%$\mathbb{I}\mathfrak{U}(\tau',w) \notin \boldsymbol{x}$ which
%means that $\mathbb{C}(U - \mathfrak{U}(\tau',w))\in
%\boldsymbol{x}$. Observe that the set
%
%\begin{equation*}
%\boldsymbol{y}_0= \{U - \mathfrak{U}(\tau',w)\} \cup \{ A
%\subseteq U \mid \mathbb{I}A\in \boldsymbol{x}\}
%\end{equation*}
%
%has the finite intersection property. Indeed, otherwise we would
%have sets $A_1,\dots,A_k\subseteq U$ such that $\mathbb{I}A_i\in
%\boldsymbol{x}$, for $1 \leq i \leq k$, and $(U -
%\mathfrak{U}(\tau',w))\cap A_1\cap \dots \cap A_k=\emptyset$. But
%then
%
%\begin{equation*}
%\emptyset \;=\;  \mathbb{C}(U - \mathfrak{U}(\tau',w))\cap
%\mathbb{I}(A_1\cap \dots \cap A_k) \;=\; \mathbb{C}(U -
%\mathfrak{U}(\tau',w))\cap \mathbb{I}A_1\cap \dots \cap
%\mathbb{I}A_k\in\boldsymbol{x},
%\end{equation*}
%
%which is impossible, since in this case $U\notin \boldsymbol{x}$.
%Take an ultrafilter $\boldsymbol{y} \supseteq \boldsymbol{y}_0$.
%Then we have $\boldsymbol{x} R \boldsymbol{y}$ and so
%$(\mathfrak{K},\auf w,\boldsymbol{y}\zu)\models\tau'$, i.e., by
%IH, $\mathfrak{U}(\tau',w) \in \boldsymbol{y}$, contrary to
%$(U-\mathfrak{U}(\tau',w))\in\boldsymbol{y}_0
%\subseteq\boldsymbol{y}$.

%Conversely, let $(\mathfrak{K},\auf
%w,\boldsymbol{x}\zu)\not\models \IB\tau'$. Then we can find
%$\boldsymbol{y}$ such that $\boldsymbol{x} R \boldsymbol{y}$ and
%$(\mathfrak{K},\auf w,\boldsymbol{y}\zu)\not\models \tau'$. By
%IH, $\mathfrak{U}(\tau',w)\notin \boldsymbol{y}$, and so, by the
%definition of $R$, $\mathbb{I}\mathfrak{U}(\tau',w)\notin
%\boldsymbol{x}$, which means that $\mathfrak{U}(\tau,w)\notin
%\boldsymbol{x}$.

%\smallskip

{\em Case} $\tau=\tau_1\until\tau_2$. Assume that
$(\mathfrak{K},\auf
w,\boldsymbol{x}\zu)\models\tau_1\until\tau_2$. Then there is
$v>w$ such that $(\mathfrak{K},\auf
v,\boldsymbol{x}\zu)\models\tau_2$ and $(\mathfrak{K},\auf
u,\boldsymbol{x}\zu)\models\tau_1$ for all $u$ in the interval
$(w,v)$. By IH, $\mathfrak{U}(\tau_2,v)\in \boldsymbol{x}$ and
$\mathfrak{U}(\tau_1,u)\in \boldsymbol{x}$ for all $u\in (w,v)$.
Since
%
\begin{equation*}
\mathfrak{U}(\tau_1\until\tau_2,w) \supseteq \mathfrak{U}(\tau_2,v) \cap \bigcap_{u\in (w,v)}\mathfrak{U}(\tau_1,u), %
\end{equation*}
%
we shall have $\mathfrak{U}(\tau_1\until\tau_2,w)\in
\boldsymbol{x}$ if we show that
%
\begin{equation}\label{eq:PST:s13.1inx}
\mathfrak{U}(\tau_2,v) \cap \bigcap_{u\in (w,v)}\mathfrak{U}(\tau_1,u) \in \boldsymbol{x}. %
\end{equation}
%
In view of \FSA{}, we can find time points $u_1,\dots, u_l\in
(w,v)$ such that
%
\begin{equation*}
\mathfrak{U}(\tau_1,u_1)\cap \dots \cap \mathfrak{U}(\tau_1,u_l)= \bigcap_{u\in (w,v)} \mathfrak{U}(\tau_1,u), %
\end{equation*}
%
which yields (\ref{eq:PST:s13.1inx}) because ultrafilters are
closed under \emph{finite} intersections.

Conversely, let $\mathfrak{U}(\tau_1\until\tau_2,w) \in
\boldsymbol{x}$. By \FSA{}, there are time points $v_1,\dots v_l$
such that
%
\begin{equation*}
\mathfrak{U}(\tau_1\until\tau_2,w) = \bigcup_{1\leq i \leq l}
\Big(\mathfrak{U}(\tau_2,v_i) \cap \bigcap_{u\in (w,v_i)}
\mathfrak{U}(\tau_1,u)\Big).
\end{equation*}
%
And since $\boldsymbol{x}$ is an ultrafilter,
%
\begin{equation*}
\mathfrak{U}(\tau_2,v_i) \cap \bigcap_{u\in (w,v_i)}
\mathfrak{U}(\tau_1,u)\in \boldsymbol{x},
\end{equation*}
%
for some $i$, $1\leq i\leq l$. Therefore, by IH,
$(\mathfrak{K},\auf v_i,\boldsymbol{x}\zu)\models \tau_2$ and
$(\mathfrak{K},\auf u,\boldsymbol{x}\zu)\models\tau_1$ for all
$u\in (w,v_i)$. Hence $(\mathfrak{K},\auf
w,\boldsymbol{x}\zu)\models \tau_1\until\tau_2$.

%\smallskip

{\em Case} $\tau=\tau_1\since\tau_2$ is considered analogously.

Now, we show that, for all $w \in W$ and spatio-temporal terms
$\tau$,
%
\begin{equation*}
(\mathfrak{K}, w)\models\Ubox\tau\qquad\text{iff}\qquad
\mathfrak{U}(\tau,w) = U.
\end{equation*}
%
Suppose that $(\mathfrak{K}, w)\models \Ubox\tau$. Then
$(\mathfrak{K},\auf w,\boldsymbol{y}\zu)\models\tau$ for all
$\boldsymbol{y}\in V$, and so, by IH, $\mathfrak{U}(\tau,w)\in
\boldsymbol{y}$ for all $\boldsymbol{y}\in V$. But then
$\mathfrak{U}(\tau,w)=U$. Conversely, if $\mathfrak{U}(\tau,w)=U$
then $\mathfrak{U}(\tau,w)\in \boldsymbol{y}$ for all
$\boldsymbol{y}\in V$, from which, by IH, $(\mathfrak{K},
w)\models\Ubox\tau$.

It follows immediately that $\varphi$ is satisfied in
$\mathfrak{K}$. It should be also clear that $\mathfrak{K}$
satisfies \FSA{}. This proves (i). The existence of $R$-maximal
points in sets of the form $A_{w,\boldsymbol{x},\tau}$ (where
$w\in W$, $\boldsymbol{x}\in V$ and $\tau$ is a spatio-temporal
term) follows from a result of Fine \citeyear{Fine74b}; see also
\cite[Theorem 10.36]{Chagrov&Z97}.

%Here is a sketch of the proof. Consider the family
%
%\begin{equation*}
%\mathcal{X} = \{ X \subseteq A_{w,\boldsymbol{x},\tau} \mid
%R\cap(X\times X) \text{ is a linear order with smallest element }
%\boldsymbol{x} \}.
%\end{equation*}
%
%Let $C$ be a $\subseteq$-maximal set in $\mathcal{X}$ (i.e., for
%every $C'\in \mathcal{X}$, $C\subseteq C'$ implies $C'=C$); its
%existence can be readily proved with the help of Zorn's lemma. Now
%take the set
%
%\begin{equation*}
%\boldsymbol{y}_0 = \{ A \subseteq U \mid \exists \boldsymbol{z}
%\in C\; \forall \boldsymbol{z'} \in C\ (\boldsymbol{z} R
%\boldsymbol{z'} \to A \in \boldsymbol{z'})\}.
%\end{equation*}
%
%This set is not empty, since
%$\mathfrak{U}(\tau,w)\in\boldsymbol{y}_0$, and clearly
%$\boldsymbol{y}_0$ has the finite intersection property. Hence we
%can find an ultrafilter $\boldsymbol{y}$ containing
%$\boldsymbol{y}_0$. Then it is easy to see, using the definition
%of $R$, that
%
%\begin{equation}\label{eq:PST:maximal}
%\forall \boldsymbol{z}\in C\ \boldsymbol{z} R \boldsymbol{y}.
%\end{equation}
%
%We claim that $\boldsymbol{y}$ is $R$-maximal in
%$A_{w,\boldsymbol{x},\tau}$. Indeed, take some $\boldsymbol{y'}\in
%A_{w,\boldsymbol{x},\tau}$ such that $\boldsymbol{y} R
%\boldsymbol{y'}$. If $\boldsymbol{y'}\in C$ than $\boldsymbol{y'}
%R \boldsymbol{y}$ holds by~\eqref{eq:PST:maximal}. If
%$\boldsymbol{y'}\notin C$ then, by the $\subseteq$-maximality of
%$C$ in $\mathcal{X}$, $C\cup\{\boldsymbol{y'}\}$ is not linearly
%ordered by $R$. Since by~\eqref{eq:PST:maximal} and
%$\boldsymbol{y} R \boldsymbol{y'}$, we have $\boldsymbol{z} R
%\boldsymbol{y'}$ for all $\boldsymbol{z}\in C$, there exists a
%$\boldsymbol{z'}\in C$ such that $\boldsymbol{y'} R
%\boldsymbol{z'}$, and so, again by~\eqref{eq:PST:maximal},
%$\boldsymbol{y'} R \boldsymbol{y}$ as required.

\smallskip
(ii) The construction is the same as in (i). First we show by
induction that, for every spatio-temporal term $\tau$ of
$\PTL\circ\SFU$, $(\mathfrak{K},\auf
w,\boldsymbol{x}\zu)\models\tau$ iff $\mathfrak{U}(\tau,w)\in
\boldsymbol{x}$. This time, however, instead of $\until$ and
$\since$ we need the inductive step for $\nxt$.

{\em Case} $\tau = \nxt\tau'$. We have $(\mathfrak{K},\auf
w,\boldsymbol{x}\zu)\models \nxt\tau'$ iff there exists an immediate
successor $w'$ of $w$ such that $(\mathfrak{K},\auf
w',\boldsymbol{x}\zu) \models\tau'$ iff, by IH, there is an
immediate successor $w'$ of $w$ such that $\mathfrak{U}(\tau',w')\in
\boldsymbol{x}$. It remains to recall that $\mathfrak{U}(\nxt
\tau',w)= \mathfrak{U}(\tau',w')$ whenever $w'$ is the immediate
successor of $w$ and $\mathfrak{U}(\nxt \tau',w)=\emptyset$ whenever
$w$ has no immediate successor.

The remaining part of the proof is the same as in (i).
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{proof}[\proofof{Proposition~\ref{proposition:fsa}}]
The implication ($\Leftarrow$) follows immediately from the
definition.

($\Rightarrow$) Suppose that a $\PTL\times\SFU$-formula $\varphi$
is satisfied in a tt-model with \FSA{} and a flow of time
$\mathfrak{F} = \auf W,<\zu$. Then, by Lemma~\ref{simula1}~(i),
$\varphi$ is satisfiable in an Aleksandrov tt-model $\mathfrak{M}
= \auf\mathfrak{F},\mathfrak{G},\mathfrak{V}\zu$ with \FSA{} and
based on a quasi-order $\mathfrak{G}=\auf V,R\zu$. In view of
\FSA{}, for every $\tau\in\textit{term}\,\varphi$, there are
finitely many sets $A_1,\dots,A_k\subseteq V$ such that $\{
\mathfrak{V}(\tau,w) \mid w \in W\}  = \{A_1,\dots, A_k\}$.
Therefore, there are finitely many time points $w_1,\dots,w_m\in
W$ such that, for every $w\in W$, there is $w_i$, $1 \leq i \leq
m$, with $\mathfrak{V}(\tau,w) = \mathfrak{V}(\tau,w_i)$ for
\emph{all} $\tau\in\textit{term}\,\varphi$. Now we use the Lemmon
filtration \cite<see, e.g.,>{Chagrov&Z97} to construct a tt-model
based on a \emph{finite} Aleksandrov topological space. First,
define an equivalence relation $\sim$ on $V$ by taking $x \sim y$
if
%
\begin{equation*}
(\mathfrak{M},\auf w_i,x\zu)\models\tau\quad \text{iff}\quad
(\mathfrak{M},\auf w_i,y\zu)\models\tau, %
\quad\text{for all } i,\ 1 \leq i \leq m, \text{ and } \tau\in\textit{term}\,\varphi.  %
\end{equation*}
%
Denote by $[x]$ the equivalence class of $x\in V$. The set
$V/_\sim$ of pairwise distinct equivalence classes is clearly
finite. Define a binary relation $S$ on $V/_\sim$ by taking
$[x]S[y]$ if
%
\begin{equation*}
(\mathfrak{M},\auf w_i,y\zu)\models\IB\tau\quad\text{whenever}\quad (\mathfrak{M},\auf w_i,x\zu)\models\IB\tau, %
\quad\text{for all } i,\ 1 \leq i \leq m, \text{ and } \tau\in\textit{term}\,\varphi.  %
\end{equation*}
%
Clearly,  $S$ is well-defined, reflexive and transitive, and so
$\mathfrak{G}' = \auf V/_\sim, S\zu$ is a finite quasi-order. Let
$\mathfrak{V}'(p,w) = \{[x] \mid x\in\mathfrak{V}(p,w)\}$, for
every spatial variable $p$ and every $w\in W$.

Consider the tt-model $\mathfrak{M}' =
\auf\mathfrak{F},\mathfrak{G}',\mathfrak{V}'\zu$. First we show
that for all $\tau\in\textit{term}\,\varphi$, $x\in V$ and $w\in
W$,
%
\begin{equation*}
(\mathfrak{M},\auf w,x\zu)\models\tau\qquad\text{iff}\qquad(\mathfrak{M}',\auf w,[x]\zu)\models\tau.%
\end{equation*}
%
The basis of induction follows from the definition of
$\mathfrak{V}'$, the cases of  intersection and  complement are
trivial, and those of temporal operators follow by IH.

Suppose that $(\mathfrak{M},\auf w,x\zu)\models\IB\tau$ and
$[x]S[y]$. Then there is a moment $w_i$ such that
$(\mathfrak{M},\auf w,z\zu)\models\tau$ iff $(\mathfrak{M},\auf
w_i,z\zu)\models\tau$, for all $\tau\in\textit{term}\,\varphi$ and
$z\in V$. By the definition of $S$, we have $(\mathfrak{M},\auf
w_i,y\zu)\models\tau$, and so $(\mathfrak{M},\auf
w,y\zu)\models\tau$. Finally, by IH, $(\mathfrak{M}',\auf
w,[y]\zu)\models\tau$, and since $y$ was arbitrary, we obtain
$(\mathfrak{M}',\auf w,[x]\zu)\models\IB\tau$.

Conversely, let $(\mathfrak{M}',\auf w,[x]\zu)\models\IB\tau$ and
$xRy$. Then $[x]S[y]$, and so $(\mathfrak{M}',\auf
w,[y]\zu)\models\tau$, from which, by IH, $(\mathfrak{M},\auf
w,y\zu)\models\tau$. Thus, $(\mathfrak{M},\auf
w,x\zu)\models\IB\tau$.

Finally, by a straightforward induction on the structure  of
$\varphi$, one can show that
%
\begin{equation*}
(\mathfrak{M},w)\models\psi\qquad\text{iff}\qquad(\mathfrak{M}',w)\models\psi,%
\end{equation*}
%
for all $\psi\in\textit{sub}\,\varphi$ and  $w\in W$. It follows
that $\varphi$ is satisfied in $\mathfrak{M}'$.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection[Temporalisations of S4u]{Temporalisations of $\SFU$}\label{appendix:temporalisation}

\begin{lemma}\label{PST0:bigmodel}
Let $\Gamma$ be a finite set of  $\SFU$-formulas. Then there is a
finite quasi-order $\mathfrak{G}$ such that every satisfiable
subset $\Phi\subseteq\Gamma$ is satisfied in some Aleksandrov
model based on $\mathfrak{G}$.
\end{lemma}
%
\begin{proof}
For every satisfiable $\Phi\subseteq\Gamma$, fix a model based on
a finite quasi-order $\mathfrak{G}_\Phi=\auf V_\Phi,R_\Phi\zu$ and
satisfying $\Phi$. Let $n= \max\{|V_\Phi| :\Phi\subseteq \Gamma,\
\Phi \mbox{ is satisfiable}\}$ and let $\mathfrak{G}$ be the
disjoint union of $n$ full $n$-ary (transitive) trees of depth $n$
whose nodes are clusters of cardinality $n$. It is not difficult
to see that every $\mathfrak{G}_\Phi$ is a p-morphic image of
$\mathfrak{G}$. Therefore, every satisfiable $\Phi\subseteq
\Gamma$ is satisfied in an Aleksandrov model based on
$\mathfrak{G}$.
\end{proof}

\begin{proof}[\proofof{Theorem \ref{PST0:pspace:text}}]
PSPACE-hardness follows from Theorem~\ref{s4u:pspace} or
\ref{complextempPSP}. We show the matching upper bound.

Let $\varphi$ be a $\PTL[\SFU]$-formula. Since $\varphi$ is a
$\PTL\circ\SFU$-formula, by Lemma~\ref{simula1} (ii), it is
satisfiable in a tt-model iff it is satisfiable in an Aleksandrov
tt-model based on the same flow of time. With every (spatial)
subformula $\Ubox\tau$ of $\varphi$ we associate a fresh
propositional variable $\textp{p}_\tau$ and denote by
$\varphi^\ast$ the $\PTL$-formula that results from $\varphi$ by
replacing all its subformulas of the form $\Ubox\tau$ with
$\textp{p}_\tau$. We claim that $\varphi$ is satisfiable in an
Aleksandrov tt-model over a flow of time $\mathfrak{F} = \auf
W,<\zu$ iff
%
\begin{itemize}
\item there exists a temporal model
$\mathfrak{N}=\auf \mathfrak{F},{\mathfrak U}\zu$ satisfying
$\varphi^{\ast}$ and,
%
\item for every $w\in W$, the set
$\Phi_w = \{ \Ubox\tau \mid (\mathfrak{N},w)\models \textp{p}_\tau
\} \cup \{ \neg\Ubox\tau \mid (\mathfrak{N},w)\models \neg
\textp{p}_\tau \}$ of spatial formulas is satisfiable.
\end{itemize}
%
The implication $(\Rightarrow)$ is obvious. Conversely, suppose
that we have a temporal model $\mathfrak{N}$ satisfying the
conditions above. Let $\Gamma = \bigcup_{w\in W} \Phi_w$. By
Lemma~\ref{PST0:bigmodel}, there is a finite quasi-order
$\mathfrak{G}$ such that, for every $w\in W$, we have
$\auf\mathfrak{G},\mathfrak{V}_w\zu \models \Phi_w$ for some
valuation $\mathfrak{V}_w$. It should be clear that $\varphi$ is
satisfied in the Aleksandrov tt-model $\auf\mathfrak{F},
\mathfrak{G},\mathfrak{V}\zu$, where $\mathfrak{V}(p,w) =
\mathfrak{V}_w(p)$, for every spatial variable $p$ and every $w\in
W$.

Now, to devise a decision procedure for $\PTL[\SFU]$ which uses
polynomial space in the length of the input formula, one can take
the corresponding nondeterministic PSPACE algorithm for $\PTL$
\cite{Sistla&Clarke85,Reynolds99,Reynolds03} and modify it as
follows. The algorithm constructs a `pure' temporal model
$\mathfrak{N} = \auf\mathfrak{F},\mathfrak{U} \zu$ for
$\varphi^\ast$ and every time it produces a state for a time
instant $w\in W$, it additionally checks whether the set $\Phi_w$
of spatial formulas is satisfiable. By Theorem \ref{s4u:pspace},
this extra test can also be performed by a PSPACE algorithm, which
does not increase the complexity of the `combined' algorithm.
\end{proof}


\begin{proof}[\proofof{Theorem~\ref{getlost}}]
The proof is essentially the same as that of
Theorem~\ref{PST0:pspace:text}, but now nondeterministic
polynomial-time algorithms for the component logics are available.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection[Undecidability of PTLoS4u and PTLxS4u]{Undecidability of $\PTL\circ\SFU$ and $\PTL_\Box\times\SFU$}\label{appendix:product}

Note that although our spatio-temporal languages contain no
propositional variables, we still can simulate them: for a spatial
variable $p$, formula $\Ubox p$ can be regarded as a proposition.
Thus, in what follows by a \emph{propositional variable}
$\textp{p}$ we mean the formula $\Ubox p$, for a spatial variable
$p$ (note the different typefaces used to denote propositional and
spatial variables).

\begin{proof}[\proofof{Theorem~\ref{PST1undecidable}}]
The proof is by reduction of the undecidable Post's
\citeyear{Post46} correspondence problem or PCP, for short.  It is
formulated as follows.  Given a finite alphabet $A$ and a finite
set $P$ of pairs $\auf v_1,u_1\zu,\dots,\auf v_k,u_k\zu$ of
nonempty finite words
%
\begin{equation*}
v_i  = \auf b^i_1,\dots,b^i_{l_i}\zu,\qquad %
u_i  = \auf c^i_1,\dots,c^i_{r_i}\zu\qquad (i=1,\dots,k)
\end{equation*}
%
over $A$, an instance of PCP, decide whether there exist an $N\geq
1$ and a sequence $i_{1},\dots,i_{N}$ of indices such that
%
\begin{equation}\label{eq:PCP}
v_{i_{1}}\!\ast\dots\ast v_{i_{N}} = u_{i_{1}}\!\ast\dots\ast u_{i_{N}}, %
\end{equation}
%
where $\ast$ is the concatenation operation. We will construct
(using only future-time temporal operators) a
$\PTL\circ\SFU$-formula $\varphi_{A,P}$ such that
%
\begin{itemize}
\item[(i)] the length of $\varphi_{A,P}$ is a polynomial function in the
size of both $A$ and $P$;
%
\item[(ii)] if $\varphi_{A,P}$ is satisfiable in a tt-model based on
$\auf\mathbb{N},<\zu$ then there exist an $N\geq 1$ and a sequence
$i_1,\dots,i_N$ of indices such that \eqref{eq:PCP} holds;
%
\item[(iii)] if there exist an $N\geq 1$ and a
sequence $i_1,\dots,i_N$ of indices such that \eqref{eq:PCP} holds
then $\varphi_{A,P}$ is satisfiable in a tt-model with \FSA{} and
based on $\auf\mathbb{N},<\zu$;
%
\item[(iv)] $\varphi_{A,P}$ is satisfiable in a tt-model based on
$\auf\mathbb{N},<\zu$ iff $\varphi_{A,P}$ is satisfiable in a
tt-model based on a finite flow of time.
\end{itemize}
%
The case of $\auf\mathbb{Z},<\zu$ follows immediately. By
Lemma~\ref{simula1} (ii), it suffices to consider only Aleksandrov
tt-models for $\varphi_{A,P}$.

We build $\varphi_{A,P}$ using spatial variables $\textv{left}_a$
and $\textv{right}_a$ ($a\in A$), $\textv{left}$, $\textv{right}$
and $\textv{stripe}$, as well as propositional variables
$\textp{pair}_i$, for every pair $\auf v_i,u_i\zu$, $1\le i\le k$,
and $\textp{range}$.

The variable $\textp{range}$ is required to `relativise' temporal
operators $\Box_F$ and $\Diamond_F$ in order to ensure that we can
construct a model based on a finite flow of time. The variable
$\textv{stripe}$ is used to introduce a new `strict closure'
operator in Aleksandrov spaces by taking, for every
spatio-temporal term $\tau$,
%
\begin{equation*}
\Dvs\tau = \big(\textv{stripe} \imp \CB (\overline{\textv{stripe}} \sqcap \CB\tau)\big) \sqcap %
           \big(\overline{\textv{stripe}} \imp \CB(\textv{stripe} \sqcap \CB\tau)\big).
\end{equation*}
%
Denote by $\Dvs^n$ a sequence of $n$ operators $\Dvs$. Other
abbreviations we need are $\tau_1\equi\tau_2$ which stands for
$(\tau_1\imp\tau_2)\sqcap(\tau_2\imp\tau_1)$ and $\Box_F^+\varphi$
which replaces $\varphi\land\Box_F\varphi$.

The formula $\varphi_{A,P}$ is defined as the conjunction
%
\begin{equation*}
\varphi_{A,P} = \varphi_{\textit{range}} \;\land\;
\varphi_{\textit{stripe}}\;\land\; \varphi_{\textit{pair}}
\;\land\; \varphi_{\textit{eq}} \;\land\; \varphi_{\textit{left}}
\;\land\; \varphi_{\textit{right}},
\end{equation*}
%
where
%
\begin{align*}
\varphi_{\textit{range}} &= \textp{range} \land \Diamond_F\neg\textp{range} \land \Box_F(\neg\textp{range} \to \Box_F\neg\textp{range}),\\
%
\varphi_{\textit{pair}} &= \Box_F^+ \Big(\Diamond_F\textp{range} \to %
\bigvee_{1\le i\le k}\textp{pair}_i\;\land\!\!\! %
\bigwedge_{1\le i < j\le k} \!\!\!\neg(\textp{pair}_i\land\textp{pair}_j)\Big),\\
%
\varphi_{\textit{stripe}} &= \Box_F^+ \bigl(\Diamond_F\textp{range} \to \Ubox(\textv{stripe} \equi \nxt\textv{stripe})\bigr),\\
%
\varphi_{\textit{eq}} &= \Diamond_F \Bigl(\textp{range} \land %
\bigwedge_{a\in A}  \Ubox(\textv{left}_{a}\equi \textv{right}_{a})\Bigr), %
\end{align*}
%
$\varphi_{\textit{left}}$ is the conjunction of
\eqref{eq:pcp2}--\eqref{eq:twocirc}, for all $i$ with $1\le i\le
k$,
%
\begin{align}
\label{eq:pcp2}%
& \bigwedge_{\begin{subarray}{c}a\ne b\\ a,b\in A\end{subarray}}%
 \!\!\!\!\Box_F^+\neg\Udiamond\bigl(\textv{left}_a\sqcap\textv{left}_b\bigr)%
 \;\wedge\; \Box_F^+\Ubox\bigl(\textv{left} \equi  \bigsqcup_{a\in A} \textv{left}_{a}\bigr),\\
%
\label{eq:pcp2b}%
& \bigwedge_{a\in A} \Box_F^+\bigl(\textp{pair}_i\to\Ubox(\textv{left}_a \imp \nxt \textv{left}_a)\bigr),\\
%
\label{eq:pcp1}%
& \Ubox\overline{\textv{left}} \;\land\; \Box_F^+\Ubox(\overline{\textv{left}} \imp \overline{\Dvs\textv{left}}),\\
%
\label{eq:onecirc}%
& \Box_F^+\bigl(\textp{pair}_i\to\Ubox(\overline{\textv{left}} \imp \nxt\overline{\Dvs^{l_i}\textv{left}})\bigr),\\
%
\label{eq:fourcirc}%
& \Box_F^+\bigl(\textp{pair}_i \to \bigwedge_{j < l_i}\nxt\Ubox
((\Dvs^j \textv{left} \sqcap \overline{\Dvs^{j+1}\textv{left}})
\imp \textv{left}_{b^i_{l_i-j}})\bigr),\\
%
\label{eq:one+circ}%
& \textp{pair}_i \to \nxt\Udiamond\tau^{\textit{left}}_i,\\
%
\label{eq:twocirc}  %
& \Box_F \bigl(\textp{pair}_i \to %
\Ubox((\textv{left} \sqcap \overline{\Dvs\textv{left}}) \imp\nxt\Dvs \tau^{\textit{left}}_i)\bigr),%
\end{align}
%
where
%
\begin{equation*}
\tau^{\textit{left}}_i = \textv{left}_{b^i_1}\sqcap \Dvs
\bigl(\textv{left}_{b^i_2} \sqcap \Dvs(\textv{left}_{b^i_3}\sqcap
\dots \sqcap\Dvs \textv{left}_{b^i_{l_i}})\dots\bigr)
\end{equation*}
%
(remember that $l_i$ is the length of the word $v_i$). The
conjunct $\varphi_{\textit{right}}$ is defined by replacing in
$\varphi_{\textit{left}}$ all occurrences of $\textv{left}$ with
$\textv{right}$, $\textv{left}_a$ with $\textv{right}_a$ (for
$a\in A$), $l_i$ with $r_i$ and $\tau^{\textit{left}}_i$ with
$\tau^{\textit{right}}_i$, which is defined similarly. (Note that
$\textp{pair}_i$ occurs in both $\varphi_{\textit{left}}$ and
$\varphi_{\textit{right}}$.)


\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{.075cm}
\begin{picture}(140,110)(0,-10)
%\put(0,0){\line(1,0){160}}\put(0,0){\line(0,1){170}}

\put(4,-5) {
%
\multiput(15,15)(0,25){4}{%
\multiput(0,0)(18,0){5}{\circle{1.5}}%
\multiput(0,15)(18,0){5}{\circle{1.5}}%
}
\multiput(15,115)(18,0){5}{\circle{1.5}}%

\put(12,6){$\textp{pair}_{i_1}$}%
\put(30,6){$\textp{pair}_{i_2}$}%
\put(48,6){$\textp{pair}_{i_3}$}%
\put(66,6){$\textp{pair}_{i_4}$}%

\multiput(33,0)(18,0){4}{%
\multiput(0,15)(0,15){2}{\circle*{1.5}}%
%\put(-2,17){$\textv{left}_{b_1}$}%
%\put(-2,32){$\textv{left}_{b_{n_1}}$}%
}%
\multiput(51,25)(18,0){3}{%
\multiput(0,15)(0,15){2}{\circle*{1.5}}%
%\put(-2,17){$\textv{left}_{b_{n_1 + 1}}$}%
%\put(-2,32){$\textv{left}_{b_{n_2}}$}%
}
\multiput(69,50)(18,0){2}{%
\multiput(0,15)(0,15){2}{\circle*{1.5}}%
%\put(-2,17){$\textv{left}_{b_{n_2 + 1}}$}%
%\put(-2,32){$\textv{left}_{b_{n_3}}$}%
}
\multiput(87,75)(18,0){1}{%
\multiput(0,15)(0,15){2}{\circle*{1.5}}%
%\put(-2,17){$\textv{left}_{b_{n_3 + 1}}$}%
%\put(-2,32){$\textv{left}_{b_{n_4}}$}%
}

\multiput(0, 0)(0, 25){4}{%
%\multiput(14.5, 21)(18, 0){5}{\tiny\vdots}
\put(5, 20){\vdots}} %

\put(0, 75){\multiput(14.5, 21)(18, 0){4}{\tiny\vdots}} %
\put(0, 50){\multiput(14.5, 21)(18, 0){3}{\tiny\vdots}} %
\put(0, 25){\multiput(14.5, 21)(18, 0){2}{\tiny\vdots}} %
\put(0, 0){\multiput(14.5, 21)(18, 0){1}{\tiny\vdots}} %


%\multiput(33,22.5)(18,0){4}{\oval(3,19)}%
%\multiput(51,47.5)(18,0){3}{\oval(3,19)}%
%\multiput(69,72.5)(18,0){2}{\oval(3,19)}%
%\multiput(87,97.5)(18,0){1}{\oval(3,19)}%

{%
\linethickness{0.8pt}
\multiput(33,15)(18,0){4}{\line(0,1){15}}%
\multiput(51,40)(18,0){3}{\line(0,1){15}}%
\multiput(69,65)(18,0){2}{\line(0,1){15}}%
\multiput(87,90)(18,0){1}{\line(0,1){15}}%
}%

%{%
%\linethickness{0.1pt}
%\put(33,15){\vector(1,0){61}}%
%\put(33,30){\vector(1,0){61}}%
%
%\put(51,40){\vector(1,0){43}}%
%\put(51,55){\vector(1,0){43}}%
%
%\put(69,65){\vector(1,0){25}}%
%\put(69,80){\vector(1,0){25}}%
%
%\put(87,90){\vector(1,0){7}}%
%\put(87,105){\vector(1,0){7}}%
%}%


\put(4,0){
\put(95,14){$b_1^{i_1}$}%
\put(96,21){\tiny$\vdots$}
\put(95,29){$b_{l_{i_1}}^{i_1}$}%
}%
%
\put(4,25){
\put(95,14){$b_1^{i_2}$}%
\put(96,21){\tiny$\vdots$}
\put(95,29){$b_{l_{i_2}}^{i_2}$}}%
%
\put(4,50){
\put(95,14){$b_1^{i_3}$}%
\put(96,21){\tiny$\vdots$}
\put(95,29){$b_{l_{i_3}}^{i_3}$}}%
%
\put(4,75){
\put(95,14){$b_1^{i_4}$}%
\put(96,21){\tiny$\vdots$}
\put(95,29){$b_{l_{i_4}}^{i_4}$}}%

\put(17,21){$\left\}\rule{0pt}{21pt} l_{i_1}\right.$}%
\put(35,46){$\left\}\rule{0pt}{21pt} l_{i_2}\right.$}%
\put(53,71){$\left\}\rule{0pt}{21pt} l_{i_3}\right.$}%
\put(71,96){$\left\}\rule{0pt}{21pt} l_{i_4}\right.$}%

\put(36,22){$v_{i_1}$}%
\put(54,47){$v_{i_2}$}%
\put(72,72){$v_{i_3}$}%
\put(90,97){$v_{i_4}$}%

\put(33,30){\circle*{2.5}}%
\put(51,55){\circle*{2.5}}%
\put(69,80){\circle*{2.5}}%
\put(87,105){\circle*{2.5}}%

\put(0, -1){%
\put(0, 15){\makebox[30pt][c]{$y_1$}}%
\put(0, 30){\makebox[30pt][c]{$y_{n_1}$}}%
\put(0, 40){\makebox[30pt][c]{$y_{n_1 + 1}$}}%
\put(0, 55){\makebox[30pt][c]{$y_{n_2}$}}%
\put(0, 65){\makebox[30pt][c]{$y_{n_2 + 1}$}}%
\put(0, 80){\makebox[30pt][c]{$y_{n_3}$}}%
\put(0, 90){\makebox[30pt][c]{$y_{n_3 + 1}$}}%
\put(0, 105){\makebox[30pt][c]{$y_{n_4}$}}%
}


\put(14, 0){$0$}%
\put(32, 0){$1$}%
\put(50, 0){$2$}%
\put(68, 0){$3$}%
\put(86, 0){$4$}%
\put(100, 0){\dots}%

\put(12, -2){\line(1,0){78}}
\put(12,-2){\line(0,1){2}}\put(90,-2){\line(0,1){2}}
\put(45,-6){$\textp{range}$}

{%
\linethickness{0.1pt}%
\multiput(32,10)(4.5,0){17}{\line(1,0){2}}%
\multiput(32,35)(4.5,0){17}{\line(1,0){2}}%
\multiput(50,60)(4.5,0){13}{\line(1,0){2}}%
\multiput(68,85)(4.5,0){9}{\line(1,0){2}}%
\multiput(86,110)(4.5,0){5}{\line(1,0){2}}%
}%
%

\put(0,13){%
\put(120,100){\put(0,1.1){\circle{1.5}}\quad = $\overline{\textv{left}}$}%
\put(120,94){\put(0,1.1){\circle*{1.5}}\quad = $\textv{left}$}%
\put(120,88){\put(0,1.1){\circle*{2.5}}\quad = $\textv{left}\sqcap\overline{\Dvs\textv{left}}$}%
\put(117,86){\framebox(42,19){}}
}%
%
}
\end{picture}
\end{center}
%
\caption{Model satisfying $\varphi_{\textit{left}}$, for $N = 4$.}
%
\end{figure}


Let us prove that $\varphi_{A,P}$ is as required. Suppose that
$(\mathfrak{M}, 0)\models\varphi_{A,P}$, for an Aleksandrov
tt-model $\mathfrak{M} = \auf\auf\mathbb{N},<\zu, \mathfrak{G},
\mathfrak{V}\zu$ with $\mathfrak{G} = \auf V,R\zu$. Since
$(\mathfrak{M},0)\models\varphi_{\textit{eq}}$, we can find an
$N$, $1\le N<\omega$, such that
%
\begin{equation}\label{eq:lrua}
(\mathfrak{M},N)\models \textp{range}\land \bigwedge_{a\in A}
\Ubox(\textv{left}_a\equi \textv{right}_{a}).
\end{equation}
%
In view of $\varphi_{\textit{range}}$, we have
$(\mathfrak{M},j)\models \textp{range}$ for all $j$, $0\le j\le
N$. Let $i_1,\dots,i_N$ be the sequence of indices such that, for
$1\leq j \leq N$, we have $(\mathfrak{M},
j-1)\models\textp{pair}_{i_j}$ ($\varphi_{\textit{pair}}$ ensures
that there is a unique sequence of this sort).  We claim that
\eqref{eq:PCP} holds for this sequence.

Since $\varphi_{\textit{stripe}}$ holds in $\mathfrak{M}$ at 0, we
have, for every $y\in V$, $(\mathfrak{M},\auf 0,y\zu)\models
\textv{stripe}$ iff $(\mathfrak{M},\auf j,y\zu)\models
\textv{stripe}$ for all $j$, $0 \leq j \leq N$. Denote by $R_s$
the transitive binary relation on $V$ defined by taking $x R_s y$
if there is $z\in V$ such that $xRzRy$ and $(\mathfrak{M},\auf
0,x\zu)\models \textv{stripe}$ holds iff $(\mathfrak{M},\auf
0,z\zu)\not\models \textv{stripe}$. Then we clearly have that, for
every $j$, $0 \leq j \leq N$, and every $x\in V$,
%
\begin{equation*}
(\mathfrak{M},\auf j,x\zu)\models\Dvs\tau \qquad\text{iff}\qquad
\text{there is } y\in V \text{ such that } x R_s y \text{ and }
(\mathfrak{M},\auf j, y\zu)\models\tau.
\end{equation*}



Call a sequence $\auf y_1,\dots, y_l\zu$ of (not necessarily
distinct) points from $V$ an \emph{$R_s$-path in
$\mathfrak{V}(\textv{left}, j)$ of length} $l$ if $y_1,\dots,
y_l\in \mathfrak{V}(\textv{left}, j)$ and $y_1 R_s y_2 R_s\dots
R_s y_l$. For every sequence $z_1, \dots, z_l$ of points from
$\mathfrak{V}(\textv{left}, j)$ we define
%
\begin{equation*}
\textit{leftword}_j(z_1,\dots,z_l) = \auf a_1,\dots, a_l\zu,
\end{equation*}
%
where the $a_i$ are the (uniquely determined by \eqref{eq:pcp2})
symbols from $A$ with $(\mathfrak{M},\auf j,z_i\zu)\models
\textv{left}_{a_i}$.

We will show now that, for every $j$, $1\leq j\leq N$, the
following holds:
%
\begin{itemize}
\item[(a)] there exists an $R_s$-path $\auf y_1,\dots, y_{n_j}\zu$
in $\mathfrak{V}(\textv{left}, j)$ of length
$n_j=l_{i_1}+\dots+l_{i_j}$
%
such that
%
\begin{equation*}
\textit{leftword}_j(y_{1},\dots,y_{n_j}) =
v_{i_1}\!\ast\dots\!\ast v_{i_j};
\end{equation*}
%
\item[(b)] every $R_s$-path in $\mathfrak{V}(\textv{left}, j)$ is of
length $\leq n_j$;
%
\item[(c)] for every $R_s$-path $\auf y_1,\dots, y_{n_j}\zu$ in
$\mathfrak{V}(\textv{left}, j)$, we have
%
\begin{equation*}
\textit{leftword}_{j}(y_1,\dots,y_{n_j}) =
v_{i_1}\!\ast\dots\!\ast v_{i_j}.
\end{equation*}
%
\end{itemize}
%
Indeed, for $j=1$, we have (a) by $(\mathfrak{M}, 0) \models
\textp{pair}_{i_1}$ and \eqref{eq:one+circ}, (b) by
\eqref{eq:pcp1} and \eqref{eq:onecirc}, and (c) by
\eqref{eq:fourcirc}. Now assume inductively that (a)--(c) hold for
some $j$, $1\leq j<N$. Let $\auf y_1,\dots,y_{n_j}\zu$ be a
maximal $R_s$-path in $\mathfrak{V}(\textv{left}, j)$. First, by
\eqref{eq:pcp2b}, $y_1,\dots,y_{n_j}\in\mathfrak{V}(\textv{left},
j+1)$.  Second, since $(\mathfrak{M},\auf
j,y_{n_j}\zu)\models\textv{left} \sqcap \overline{\Dvs
\textv{left}}$ and $(\mathfrak{M}, j)\models
\textp{pair}_{i_{j+1}}$, \eqref{eq:twocirc} now implies that there
exist $y_{n_j+1},\dots,y_{n_j+l_{i_{j+1}}}$ such that
$\bigl\langle y_1,\dots,y_{n_j+l_{i_{j+1}}}\bigr\rangle$ is an
$R_s$-path in $\mathfrak{V}(\textv{left}, j+1)$, as required in
(a). For (b) and (c), observe first that for every $R_s$-path
$\auf y_1,\dots,y_{l}\zu$ in $\mathfrak{V}(\textv{left}, j+1)$,
$\bigl\langle y_1,\dots,y_{l-l_{i_{j+1}}}\bigr\rangle$ is an
$R_s$-path in $\mathfrak{V}(\textv{left}, j)$, by
\eqref{eq:onecirc}.  So $l\leq n_{j+1}$ must hold. If $l=n_{j+1}$
then $\textit{leftword}_{j}(y_{1},\dots,y_{l-l_{i_{j+1}}}) =
v_{i_1}\!\ast\dots\!\ast v_{i_j}$ by the induction hypothesis, and
so $\textit{leftword}_{j+1}(y_{1},\dots,y_{l-l_{i_{j+1}}}) =
v_{i_1}\!\ast\dots\!\ast v_{i_j}$ by \eqref{eq:pcp2b}. On the
other hand,
$\textit{leftword}_{j+1}(y_{l-l_{i_{j+1}}+1},\dots,y_{l}) =
v_{i_{j+1}}$ by \eqref{eq:fourcirc}, and so we have
$\textit{leftword}_{j+1}(y_{1},\dots,y_{l}) =
v_{i_1}\!\ast\dots\!\ast v_{i_j}\!\ast v_{i_{j+1}}$, as required.

We can repeat the argument above for the `right side' as well. For
every sequence $z_1,\dots, z_l$ of points from
$\mathfrak{V}(\textv{right}, j)$, define
%
\begin{equation*}
\textit{rightword}_j(z_1,\dots,z_l) = \auf a_1,\dots, a_l\zu,
\end{equation*}
%
where the $a_i$ are the uniquely determined elements of $A$ such
that $(\mathfrak{M},\auf j,z_i\zu)\models \textv{right}_{a_i}$. We
then have, for every $1\leq j\leq N$:
%
\begin{itemize}
\item[(a$'$)] there is an $R_s$-path $\auf y_1,\dots,
y_{m_j}\zu$ in $\mathfrak{V}(\textv{right}, j)$ of length
$m_j=r_{i_1}+\dots+r_{i_j}$ such that
%
\begin{equation*}
\textit{rightword}_{j}(y_1,\dots,y_{m_j}) =
u_{i_1}\!\ast\dots\!\ast u_{i_j};
\end{equation*}
%
\item[(b$'$)] every $R_s$-path in $\mathfrak{V}(\textv{right}, j)$ is
of length $\leq m_j$;
%
\item[(c$'$)] for every $R_s$-path $\auf y_{1},\dots, y_{m_j}\zu$ in
$\mathfrak{V}(\textv{right}, j)$, we have
%
\begin{equation*}
\textit{rightword}_{j}(y_1,\dots,y_{m_j}) =
u_{i_1}\!\ast\dots\!\ast u_{i_j}.
\end{equation*}
%
\end{itemize}

Now, by \eqref{eq:pcp2} and \eqref{eq:lrua}, we have
$\mathfrak{V}(\textv{left}, N)= \mathfrak{V}(\textv{right}, N)$.
By (a), there exists an $R_s$-path $\auf y_1,\dots,y_l\zu$ in
$\mathfrak{V}(\textv{left}, N)$ such that $l=n_N$ and
$\textit{leftword}_N(y_1,\dots,y_l) = v_{i_1}\!\ast\dots \!\ast
v_{i_N}$. By (b$'$), we have $n_N\leq m_N$. Similarly, using
(a$'$) and (b), we obtain $m_N\leq n_N$, from which $n_N=m_N$.
Hence, by (c$'$), $\textit{rightword}_N(y_{1},\dots,y_{l}) =
u_{i_{1}}\!\ast\dots \!\ast u_{i_{N}}$. Since, by \eqref{eq:lrua},
%
\begin{equation*}
\textit{leftword}_N(y_1,\dots,y_l)=
\textit{rightword}_N(y_1,\dots,y_l),
\end{equation*}
%
we finally obtain $v_{i_1}\!\ast\dots \!\ast v_{i_N} =
u_{i_1}\!\ast\dots \!\ast u_{i_N}$, as required.

\smallskip
Conversely, suppose there is an $N\geq 1$ and a sequence
$i_1,\dots,i_N$ for which \eqref{eq:PCP} holds. We will show that
$\varphi_{A,P}$ is satisfiable in an Aleksandrov tt-model
$\mathfrak{M} = \auf \auf \mathbb{N},< \zu, \auf
\mathbb{N},\leq\zu, \mathfrak{V} \zu$ with \FSA{}. Let
$n_j=l_{i_1}+\dots +l_{i_j}$ and $m_j=r_{i_1}+\dots +r_{i_j}$ for
every $j$, $1\leq j\le N$. By our assumption, $n_N=m_N$ and we
have
%
\begin{equation*}
v_{i_{1}}\!\ast\dots\!\ast v_{i_N} = \auf a_1,\dots,a_{n_N}\zu =
u_{i_{1}}\!\ast\dots\!\ast u_{i_N}.
\end{equation*}
%
Define a valuation $\mathfrak{V}$ by taking
%
\begin{itemize}
\item $\mathfrak{V}(\textp{range}, j)$ is true iff $0 \le j\le N$,
\item $\mathfrak{V}(\textv{stripe}, j) = \{ 2m \mid m<\omega,\ 0 \le j\le N \}$,
\item $\mathfrak{V}(\textp{pair}_i, j - 1)$ is true iff  $i= i_j$ and $1 \le j \le N$,%
\item $\mathfrak{V}(\textv{left}_a, j) = \{ k \mid 1 \leq k \leq n_j,\,a_k = a\}$ for  $a \in A$ and $1 \le j\le N$,
\item $\mathfrak{V}(\textv{right}_a, j) = \{ k \mid 1 \leq k \leq m_j,\,a_k = a\}$ for  $a \in A$ and $1 \le j\le N$,
\item $\mathfrak{V}(\textv{left},j) =\bigcup\limits_{a\in A} \mathfrak{V}(\textv{left}_a,j)$\quad and\quad%
  $\mathfrak{V}(\textv{right},j) =\bigcup\limits_{a\in A} \mathfrak{V}(\textv{right}_a,j)$. %
\end{itemize}
%
One can easily check that under this valuation we have
$(\mathfrak{M}, 0)\models\varphi_{A,P}$ and $\mathfrak{M}$
satisfies \FSA{}. It is also readily seen that $\varphi_{A,P}$ is
satisfiable in a tt-model based on $\auf\mathbb{N},<\zu$ iff it is
satisfiable in a tt-model based on a finite flow of time.
\end{proof}


\begin{proof}[\proofof{Theorem~\ref{PSTBundecidable}}]
We show this by modifying formulas from the proof of
Theorem~\ref{PST1undecidable}. First, we replace
$\varphi_{\textit{stripe}}$ with
%
\begin{equation*}
\Box_F^+\Ubox(\textv{stripe}\imp\Box_F\textv{stripe})\ \land\ \Box_F^+\Ubox(\overline{\textv{stripe}}\imp\Box_F\overline{\textv{stripe}}). %
\end{equation*}
%
Then, $\varphi_{\textit{left}}$ is the conjunction of
\eqref{eq:pcp2:2}--\eqref{eq:twocirc:2}, for all $i$ with $1\le
i\le k$,
%
\begin{align}
\tag{\ref{eq:pcp2}$'$}\label{eq:pcp2:2}%
& \bigwedge_{\begin{subarray}{c}a\ne b\\ a,b\in A\end{subarray}}%
 \!\!\!\!\Box_F^+\neg\Udiamond\bigl(\textv{left}_a\sqcap\textv{left}_b\bigr)%
 \;\wedge\; \Box_F^+\Ubox\bigl(\textv{left} \equi  \bigsqcup_{a\in A} \textv{left}_{a}\bigr),\\
%
\tag{\ref{eq:pcp2b}$'$}\label{eq:pcp2b:2}%
& \bigwedge_{a\in A} \Box_F^+\bigl(\textp{pair}_i \to\Ubox(\textv{left}_a \imp \Box_F \textv{left}_a)\bigr),\\
%
\tag{\ref{eq:pcp1}$'$}\label{eq:pcp1:2}%
& \Ubox\overline{\textv{left}} \;\land\; \Box_F^+\Ubox(\overline{\textv{left}} \imp \overline{\Dvs\textv{left}}),\\
%
\tag{\ref{eq:onecirc}$'$}\label{eq:onecirc:2}%
& \Box_F^+\bigl(\textp{pair}_i\to\Ubox(\overline{\textv{left}} \imp \Diamond_F\overline{\Dvs^{l_i}\textv{left}})\bigr),\\
%
\tag{\ref{eq:fourcirc}$'$}\label{eq:fourcirc:2}%
& \Box_F^+\bigl(\textp{pair}_i \to \bigwedge_{j <
l_i}\Ubox\bigl((\overline{\textv{left}}\sqcap\Box_F\textv{left})
\imp \Box_F((\Dvs^j \textv{left} \sqcap
\overline{\Dvs^{j+1}\textv{left}})
\imp \textv{left}_{b^i_{l_i-j}})\bigr)\bigr),\\
%
\tag{\ref{eq:one+circ}$'$}\label{eq:one+circ:2}%
& \textp{pair}_i \to \Box_F\Udiamond\tau^{\textit{left}}_i,\\
%
\tag{\ref{eq:twocirc}$'$}\label{eq:twocirc:2}  %
& \Box_F \bigl(\textp{pair}_i \to %
\Ubox((\textv{left} \sqcap \overline{\Dvs\textv{left}}) \imp\Box_F\Dvs \tau^{\textit{left}}_i)\bigr),%
\end{align}
%
where the $\tau^{\textit{left}}_i$ are defined exactly as in the
proof of Theorem~\ref{PST1undecidable}. Formula
$\varphi_{\textit{right}}$ is modified in a similar way.
\end{proof}

\begin{remark}
In fact, the set of PCP instances without solutions is not
recursively enumerable and therefore, the proofs above show that
the sets of $\PTL\circ\SFU$ and $\PTL_\Box\times\SFU$-formulas
which are true in all models based on $\auf\mathbb{N},<\zu$, $\auf
\mathbb{Z},<\zu$ or finite flows of time are not recursively
enumerable either. Therefore, these logics are not recursively
axiomatisable.
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\section[Spatio-Temporal Logics Based on RC]{Spatio-Temporal Logics Based on $\RC$}\label{appendix:RC}%

In this appendix we establish lower and upper complexity bounds
for a wide range of decidable spatio-temporal combinations and, in
particular, prove
Theorems~\ref{st1m:pspace}--\ref{theorem:brcc8:expspace:box:dense}.
We begin with a straightforward generalisation of
Lemma~\ref{s4u:simula2} to the spatio-temporal case:
%
\begin{lemma}\label{rc:simula2}
\textup{(i)} If a $\PTL\times\RC$-formula $\varphi$ is satisfiable
in a tt-model with \FSA{} and based on a flow of time
$\mathfrak{F}$ then $\varphi$ is satisfiable in an Aleksandrov
tt-model based on $\mathfrak{F}$ and a finite disjoint union of
finite brooms.

\textup{(ii)} If a $\PTL\circ\RC$-formula $\varphi$ is satisfiable
in a tt-model based on a flow of time $\mathfrak{F}$ then
$\varphi$ is satisfiable in an Aleksandrov tt-model based on
$\mathfrak{F}$ and a \textup{(}possibly infinite\textup{)}
disjoint union of $\omega$-brooms.
\end{lemma}
%
\begin{proof}
(i) By Lemma~\ref{simula1} (i), $\varphi$ is satisfiable in an
Aleksandrov tt-model based on $\mathfrak{F}$ and a finite
quasi-order $\mathfrak{G}$. The rest of the proof is similar to
that of Lemma~\ref{s4u:simula2}. Further details are left to the
reader.

\smallskip

(ii) By Lemma~\ref{simula1} (ii), $\varphi$ is satisfiable in an
Aleksandrov tt-model based on $\mathfrak{F}$ and a quasi-order
$\mathfrak{G} = \auf V,R\zu$. The rest of the proof again is
similar to that of Lemma~\ref{s4u:simula2}. We only note that
although $\mathfrak{G}$ can be infinite, still for every $x\in V$
there is a $y\in V_0$ such that $xRy$. This is guaranteed by the
condition that the set $A_{w,x,\top}$ has a maximal point.
\end{proof}

Observe that Aleksandrov spaces are essentially infinite in
case~(ii) of Lemma~\ref{rc:simula2} and a generalisation of
Lemma~\ref{s4u:simula3} does not go through. First, we can easily
enforce a topological space to be infinite using the
$\PTL\circ\RCCE$ formula
%
\begin{equation*}
%\Box_F^+\Udiamond(\overline{\reg{p}} \sqcap \reg{\nxt p}) \ \land\ %
%\Box_F^+\Ubox(\reg{p} \imp \reg{\nxt p}).
\Box_F^+\mathsf{NTTP}(p, \nxt p).
\end{equation*}
%
Moreover, the formula
%
\begin{equation*}
 \Udiamond(\reg{p} \sqcap \overline{\IB\reg{p}})  \quad \land\quad %
 \Box_F^+\Ubox(\reg{p} \imp \reg{\nxt p}) \quad \land\quad %
 \Box_F^+\Ubox\bigl(\bigl(\reg{p} \sqcap
\overline{\IB\reg{p}}\bigr) \imp \bigl(\overline{\IB\reg{\nxt p}}
\sqcap \reg{\overline{p}\sqcap\nxt
p}\bigr)\bigr)  %
\end{equation*}
%
is satisfied in an Aleksandrov tt-model based on a single
$\omega$-broom, but cannot be satisfied in an Aleksandrov tt-model
based on a union of $n$-brooms for any finite $n$.

On the other hand, Aleksandrov tt-models based on disjoint unions
of $n$-brooms, where $n$ is bounded by the width of the formula,
are enough for spatio-temporal logics based on $\RCm$. Recall that
spatial terms $\tau$ of $\PTL\times\RCm$ (and $\PTL\circ\RCm$) are
defined as follows
%
\begin{align*}%\label{eq:rcm:rclosed}
\delta \quad::=\quad & \varrho \quad|\quad \overline{\sigma},\\
%\label{eq:rcm:ropen}%
\sigma \quad::=\quad & \IB\varrho \quad|\quad \overline{\delta}
\quad|\quad \sigma_1\sqcap\sigma_2,\\
%\label{eq:rcm:tau}%
\tau \quad::=\quad &\overline{\delta_1\sqcap\dots\sqcap\delta_m}
\quad|\quad \overline{\delta\sqcap\sigma} \quad|\quad
\overline{\sigma},
\end{align*}
%
where the $\varrho$ are spatio-temporal Boolean region terms of
$\PTL\times\BRCCE$ (and $\PTL\circ\BRCCE$, respectively). It is
not hard to see that, for every tt-model $\mathfrak{M}=\auf
\mathfrak{F}, \mathfrak{T}, \mathfrak{V}\zu$ with
$\mathfrak{F}=\auf W,<\zu$, $\mathfrak{T} = \auf U,\mathbb{I}\zu$
and every $w\in W$, we have
%
\begin{equation}\label{nadoelo}
\mathfrak{V}(\delta,w) =
\mathbb{C}\mathbb{I}\mathfrak{V}(\delta,w) \qquad \text{and}
\qquad \mathfrak{V}(\sigma,w) =
\mathbb{I}\mathbb{C}\mathfrak{V}(\sigma,w),
\end{equation}
%
i.e., the $\delta$ are always interpreted by regular closed sets,
whereas the $\sigma$ by regular open ones.

We define\label{def:width} the \emph{width} $w(\varphi)$ of a
$\PTL\times\RCm$-formula $\varphi$ as the maximal number $m$ of
conjuncts in its subformulas of the form
$\Ubox(\overline{\delta_1\sqcap\dots\sqcap\delta_m})$, if such
subformulas exist, and put $w(\varphi)=1$ otherwise.

\begin{lemma}\label{rc:simula3}
\textup{(i)} If a $\PTL\times\RCm$-formula $\varphi$ is
satisfiable in a tt-model with \FSA{} and based on a flow of time
$\mathfrak{F}$ then $\varphi$ is satisfiable in an Aleksandrov
tt-model based on $\mathfrak{F}$ and a finite disjoint union of
$w(\varphi)$-brooms.

\textup{(ii)} If a $\PTL\circ\RCm$-formula $\varphi$ is
satisfiable in a tt-model based on a flow of time $\mathfrak{F}$
then $\varphi$ is satisfiable in an Aleksandrov tt-model based on
$\mathfrak{F}$ and a \textup{(}possibly infinite\textup{)}
disjoint union of $w(\varphi)$-brooms.
%
%If a
%$\PTL\times\RCm$-formula $\varphi$ is satisfied in an Aleksandrov
%tt-model $\mathfrak{M}$ based on a flow of time $\mathfrak{F}$ and
%a partial order of depth 1 then $\varphi$ is satisfied in an
%Aleksandrov tt-model $\mathfrak{M}'$ based on $\mathfrak{F}$ and a
%partial order of depth $1$ and width $\leq w(\varphi)$. Moreover,
%if $\mathfrak{M}$ satisfies \FSA{} then $\mathfrak{M}'$ satisfies
%\FSA{} as well.
\end{lemma}
%
\begin{proof}
By Lemma~\ref{rc:simula2}, we may assume that $\varphi$ is
satisfied in an Aleksandrov tt-model $\mathfrak{M}=\auf
\mathfrak{F}, \mathfrak{G}, \mathfrak{V}\zu$, where $\mathfrak F=
\auf W,<\zu$ and $\mathfrak{G}= \auf V,R\zu$ is a disjoint union
of brooms (in (i), the union and the brooms are finite). Without
loss of generality we may assume that $\varphi$ is composed (using
temporal operators and the Booleans) from formulas of the set
$\Sigma_\varphi =
\{\Udiamond\tau_1,\dots,\Udiamond\tau_n\}$,\!\footnote{%
We treat {\normalsize $\Udiamond$} as primitive and {\normalsize
$\Ubox$} as an abbreviation.
}%
\ %
where every $\tau_i$ has one of the following forms
%
\begin{equation}\label{eq:rcm:term}
\delta_1 \sqcap \dots \sqcap \delta_m, \qquad %
\delta \sqcap \sigma \qquad\text{or}\qquad %
\sigma,%
\end{equation}
%
and the $\delta_i$, $\delta$ and $\sigma$ are as defined above.

For every $\Udiamond\tau\in \Sigma_\varphi$ and every $w\in W$
with $(\mathfrak{M},w)\models\Udiamond\tau$, we fix a point
$x_{\tau,w}\in V$ such that $(\mathfrak{M},\auf
w,x_{\tau,w}\zu)\models\tau$. We may assume that the $x_{\tau,w}$
are pairwise distinct and that the roots of all the brooms are the
points of the form $x_{\tau,w}$ for some $w\in W$ and
$\Udiamond\tau\in\Sigma_\varphi$. Therefore, $\mathfrak{G}$ is a
disjoint union of brooms $\mathfrak{b}_{\tau,w}$, for
$\Udiamond\tau\in\Sigma_\varphi$ and $w\in W$.

Let us construct a model $\mathfrak{M}'=\auf \mathfrak{F},
\mathfrak{G}', \mathfrak{V}'\zu$ as follows. Given a broom
$\mathfrak{b}_{\tau,w}$, we delete some of its leaves depending on
the form of $\tau$. Three cases are possible:

\smallskip

\emph{Case $\tau = \delta_1\sqcap\dots\sqcap\delta_m$}: take $m$
leaves  $y_1, \dots, y_m$  of $\mathfrak{b}_{\tau,w}$ such that
$(\mathfrak{M},\auf w,y_i\zu)\models\delta_i$ and $x_{\tau,w}Ry_i$
for $i = 1,\dots, m$ and remove all leaves different from
$y_1,\dots,y_m$.

\emph{Case $\tau = \delta\sqcap\sigma$}: take a leaf $y$ of
$\mathfrak{b}_{\tau,w}$ such that $(\mathfrak{M},\auf
w,y\zu)\models\delta$ and $x_{\tau,w}Ry$ and remove all other
leaves. Note that, by~\eqref{nadoelo}, we have $(\mathfrak{M},\auf
w,y\zu)\models\sigma$, and therefore $(\mathfrak{M},\auf
w,y\zu)\models\tau$.

\emph{Case $\tau = \sigma$}: take a leaf $y$ of
$\mathfrak{b}_{\tau,w}$ such that $x_{\tau,w}Ry$ and remove all
other leaves. By~\eqref{nadoelo}, we have $(\mathfrak{M},\auf
w,y\zu)\models\tau$.

\smallskip

Denote by $\mathfrak{b}'_{\tau,w}$ the resulting broom. Clearly,
it is a $w(\varphi)$-broom. Let $\mathfrak{G}'= \auf V',R'\zu$ be
the disjoint union of all $\mathfrak{b}_{\tau,w}$, for
$\Udiamond\tau\in\Sigma_\varphi$ and $w\in W$. It should be clear
that $\mathfrak{G}'$ is as required. Finally, we define
$\mathfrak{V}'$ by taking for every spatial variable $p$, every
$w\in W$ and every $x\in V'$,
%
\begin{equation*}
x \in \mathfrak{V}'(p,w) \quad \text{iff}\quad \text{there is }
y\in V' \text{ of depth } 0 \text{ such that }
 xR'y \text{ and } y \in \mathfrak{V}(p,w).
\end{equation*}
%

To show that $\varphi$ is satisfied in $\mathfrak{M}'$, we first
prove that, for all $w\in W$ and all
$\Udiamond\tau\in\Sigma_\varphi$,
%
\begin{equation*}
(\mathfrak{M}',w)\models\Udiamond\tau\qquad\text{iff}\qquad
(\mathfrak{M},w)\models \Udiamond\tau.
\end{equation*}
%
It is readily proved by induction that we have
$(\mathfrak{M}',\auf w,x\zu) \models \tau$ iff $(\mathfrak{M},\auf
w, x\zu)\models\tau$, for all points $x\in V'$ of depth $0$, all
$w\in W$ and \emph{all spatio-temporal terms} $\tau$.

Then, by the construction, we also have that, for all formulas
$\Udiamond\tau\in\Sigma_\varphi$ and all $w\in W$,
$(\mathfrak{M},w)\models\Udiamond\tau$ implies
$(\mathfrak{M}',w)\models\Udiamond\tau$. So it remains to show
that $(\mathfrak{M},w)\models\neg\Udiamond\tau$ implies
$(\mathfrak{M}',w)\models\neg\Udiamond\tau$ for all
$\Udiamond\tau\in\Sigma_\varphi$ and all $w\in W$. Suppose that we
have $(\mathfrak{M}',\auf w,x\zu)\models\tau$ and
$(\mathfrak{M},w)\models\neg\Udiamond\tau$. Consider three
possible cases for $\tau$:

\smallskip

\emph{Case $\tau=\delta_1\sqcap\dots\sqcap\delta_m$}. Then, for
every $i$, $1 \leq i \leq m$, there is $y_i\in V'$ of depth 0 such
that $xR'y_i$ and $(\mathfrak{M}',\auf w,y_i\zu)\models\delta_i$.
But then $(\mathfrak{M},\auf w,y_i\zu)\models\delta_i$ and, by
\eqref{nadoelo}, $(\mathfrak{M},\auf w,x\zu)\models\delta_i$.
Therefore, $(\mathfrak{M},\auf w,x\zu)\models\tau$, contrary to
$(\mathfrak{M},w)\models\neg\Udiamond\tau$.

\emph{Case $\tau=\delta\sqcap\sigma$.} Then there is $y\in V'$ of
depth 0 such that $xR'y$, $(\mathfrak{M}',\auf
w,y\zu)\models\delta$ and, by \eqref{nadoelo},
$(\mathfrak{M}',\auf w,y\zu)\models\sigma$. Thus
$(\mathfrak{M}',\auf w,y\zu)\models\tau$. But then
$(\mathfrak{M},\auf w,y\zu)\models\tau$, contrary to
$(\mathfrak{M},w)\models\neg\Udiamond\tau$.

\emph{Case $\tau=\sigma$.} Then there is $y\in V'$ of depth 0 with
$xR'y$ and, by \eqref{nadoelo}, $(\mathfrak{M}',\auf
w,y\zu)\models\tau$. But then $(\mathfrak{M},\auf
w,y\zu)\models\tau$, contrary to
$(\mathfrak{M},w)\models\neg\Udiamond\tau$.

\smallskip

Now, by a straightforward induction we can easily show that, for
all $w\in W$ and all formulas $\psi$ built from $\Sigma_\varphi$
using the temporal operators and the Booleans,
%
\begin{equation*}
(\mathfrak{M}',w)\models\psi\qquad\text{iff}\qquad
(\mathfrak{M},w)\models\psi.
\end{equation*}
%
It follows that $\varphi$ is satisfied in $\mathfrak{M}'$.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection[Lower Complexity Bounds (I)]{Lower Complexity Bounds (I)}\label{appendix:RC:lowerI}

\begin{proof}[\proofof{Theorem~\ref{theorem:rc:2expspace}, lower bound}]
The proof is by reduction of an arbitrary problem in 2EXPSPACE to
the satisfiability problem of $\PTL\circ\RC$. Let $\boldsymbol{A}$
be a (single-tape, deterministic) Turing machine such that
$\boldsymbol{A}$ halts on every input (accepting or rejecting it),
and $\boldsymbol{A}$ uses $\leq 2^{2^{f(n)}}$ cells of the tape on
any input of length $n$, for some polynomial $f$. Given any such
Turing machine $\boldsymbol{A}$ and an input $x$ for it, we will
construct a $\PTL\circ\RC$-formula $\varphi_{\boldsymbol{A},x}$
(using only future-time temporal operators) such that
%
\begin{itemize}
\item[(i)] the length of $\varphi_{\boldsymbol{A},x}$ is polynomial
in the size of $\boldsymbol{A}$ and $x$;
%
\item[(ii)] if $\varphi_{\boldsymbol{A},x}$ is satisfiable in a tt-model based on $\auf\mathbb{N},<\zu$ then
$\boldsymbol{A}$ accepts $x$; and
%
\item[(iii)] if $\boldsymbol{A}$ accepts $x$ then
$\varphi_{\boldsymbol{A},x}$ is satisfiable in a tt-model with
\FSA{} and based on $\auf\mathbb{N},<\zu$.
\end{itemize}
%
The case of $\auf\mathbb{Z},<\zu$ as a flow of time (with or
without \FSA{}) follows immediately. The case of finite flows of
time can be proved by relativising the temporal operators of
$\varphi_{\boldsymbol{A},x}$ (say, by a propositional variable
$\textp{range}$ as in the proof of Theorem~\ref{PST1undecidable}
in Appendix~\ref{appendix:product} and in the proof of the lower
bound of Theorem~\ref{st1:expspace} below): we can obtain a
formula $\varphi'_{\boldsymbol{A},x}$ such that
$\varphi'_{\boldsymbol{A},x}$ is satisfiable in an Aleksandrov
tt-model based on $\auf\mathbb{N},<\zu$ iff it is satisfiable in
an Aleksandrov tt-model based on the same quasi-order but on a
finite flow of time. So this way all the lower bound results of
this theorem follow.

Given a Turing machine $\boldsymbol{A}$, polynomial $f$, and input
$x=\langle x_1,\dots,x_n\rangle$ as above, let $d=f(n)$,
%
\begin{equation*}
\textit{exp}(1,d)=d\cdot 2^d\quad\text{and}\quad
\textit{exp}(2,d)=\textit{exp}(1,d)\cdot 2^{\textit{exp}(1,d)}.
\end{equation*}
%
Then we have
%
\begin{equation}\label{dok}
2^{2^{f(n)}}\leq \textit{exp}(2,d).
\end{equation}
%
Our plan is as follows. First, we will show that `yardsticks' of
length $\textit{exp}(2,d)$ (similar to those used by Stockmeyer,
\citeyearR{Stockmeyer74} or Halpern and Vardi,
\citeyearR{Halpern&Vardi89}) can be encoded by
$\PTL\circ\RC$-formulas of length polynomial in $d$. These
yardsticks will be used to define a temporal operator
$\nxt^{\textit{exp}(2,d)}$. Then, using this operator, we will
encode the computation of $\boldsymbol{A}$ on input $x$.

By Lemma~\ref{rc:simula2} (ii), if a $\PTL\circ\RC$-formula
$\varphi_{\boldsymbol{A},x}$ is satisfied in a tt-model based on a
flow of time $\auf \mathbb{N},<\zu$, then it is satisfied in an
Aleksandrov tt-model $\mathfrak{M}=\auf\auf\mathbb{N},<\zu,
\mathfrak{G},\mathfrak{U}\zu$, where $\mathfrak{G} = \auf V,R\zu$
is a disjoint union of $\omega$-brooms. Take such a model
$\mathfrak{M}$ and suppose that the
$\PTL\circ\RC$-formula%
%
\footnote{Recall that $\tau_1\equi\tau_2$ stands for
$(\tau_1\imp\tau_2)\sqcap(\tau_2\imp\tau_1)$. We assume that
$\sqcap$ and $\sqcup$ bind stronger than $\equi$.}
%
\begin{equation}\label{eq:rc:1}
\Box_F^+\Ubox\bigl(\reg{\textv{aux}} \equi \reg{\nxt
\textv{aux}}\bigr)
\end{equation}
%
is true in $\mathfrak{M}$ at moment $0$. Since region
$\reg{\textv{aux}}$ does not change over time, we can divide all
points in $V$ into three disjoint sets: \emph{external},
\emph{boundary} and \emph{internal} points with respect to
$\reg{\textv{aux}}$---i.e., those satisfying
%
\begin{equation*}
\textit{ep}(\textv{aux})=\overline{\reg{\textv{aux}}},\qquad
\textit{bp}(\textv{aux})=\reg{\textv{aux}} \sqcap
\overline{\IB\reg{\textv{aux}}}\qquad\text{and}\qquad
\IB{\reg{\textv{aux}}},
\end{equation*}
%
respectively. Note that every boundary point has a non-boundary
$R$-successor, so boundary points can only be of depth 1. In what
follows we simply speak about external and boundary points not
mentioning `with respect to $\reg{\textv{aux}}$.'

We define the `$\nxt^{\textit{exp}(2,d)}$ operator' by a
$\PTL\circ\RC$-formula of length polynomial in $d$ as follows:
%(see Fig.~\ref{fig:yardststicks:overview}):
%
\begin{itemize}
\item[(a)] First, we `encode' yardsticks of length
$d$. We will use different formulas for yardsticks on external
points and for yardsticks on boundary points.
%
\item[(b)] Then, with the help of $d$-yardsticks,
we `encode' yardsticks of length $\textit{exp}(1,d)$. We will
again use different formulas for external and boundary points.
%
\item[(c)] Next, with the help of $\textit{exp}(1,d)$-yardsticks
on both boundary and external points, we `encode' yardsticks of
length $\textit{exp}(2,d)$ on boundary points.
%
\item[(d)] Finally, with the help of $\textit{exp}(2,d)$-yardsticks
on boundary points, we define a polyno\-mial-length
`$\nxt^{\textit{exp}(2,d)}$ operator' applicable to propositional
variables.
\end{itemize}

\emph{Step} (a). Suppose that \eqref{eq:rc:1} and the following
formula hold in $\mathfrak{M}$ at $0$:
%
\begin{equation}\label{eq:rc:2}
\Box_F^+\Ubox \bigl(\textit{bp}(\textv{aux}) \imp
\delta_{0,d}\bigr)\;\land\; \Box_F^+\Ubox
\bigl(\textit{ep}(\textv{aux}) \imp
\delta^{\textit{ext}}_{0,d}\bigr)
\end{equation}
%
where
%
\begin{equation*}
\delta_{0,d} =
%\Bigl(\bigsqcup_{j=0}^{d-1}\reg{\nxt^j p_0}\Bigr)\;\sqcap\; %
\bigl(\reg{\textv{delim}_0} \equi \reg{\nxt^d \textv{delim}_0}\bigr)\;\sqcap\;%
\Bigl(\reg{\textv{delim}_0} \imp \bigsqcap_{j=1}^{d-1}\overline{\reg{\nxt^j \textv{delim}_0}}\Bigr),%
\end{equation*}
%
and $\delta^{\textit{ext}}_{0,d}$ results from $\delta_{0,d}$ by
replacing each occurrence of $\textv{delim}_0$ with
$\textv{ext\_delim}_0$.

Take a boundary point $z$. Suppose that $v\in\mathbb{N}$ is such
that $(\mathfrak{M},\auf v,x\zu)\models \reg{\textv{delim}_0}$. By
$\delta_{0,d}$, for every time moment $w\geq v$,
%
\begin{equation*}
(\mathfrak{M},\auf w,z\zu)\models
\reg{\textv{delim}_0}\qquad\text{iff}\qquad w \equiv v \pmod{d},
\end{equation*}
%
that is, on $z$, $\reg{\textv{delim}_0}$ holds once in every $d$
time instants, starting from $v$. By the second conjunct of
\eqref{eq:rc:2}, external points of $\reg{\textv{aux}}$ behave
similarly with respect to $\reg{\textv{ext\_delim}_0}$.

\smallskip

\emph{Step} (b). To encode yardsticks of length
$\textit{exp}(1,d)$, recall first that every number $a<2^d$ can be
represented in binary by a sequence $a_0\dots a_{d-1}$ of bits. We
will `mark' the bits of binary numbers by a region term
$\reg{\textv{bit}_1}$ as follows. Given a boundary point $z$ and a
time moment $v$ such that $(\mathfrak M,\langle
v,z\rangle)\models\reg{\textv{delim}_0}$, we say that an interval
$[w,w + d-1]$, for some $w = v+j\cdot d$, $j\in\mathbb N$,
\emph{encodes a number $a<2^d$ on} $z$, if for every $i<d$,
%
\begin{equation*}
(\mathfrak M,\langle w+i,z\rangle)\models\reg{\textv{bit}_1} \qquad\text{iff}\qquad a_i=1.%
\end{equation*}
%
Recall that the binary representation $b_0\dots b_{d-1}$ is the
successor of $a_0\dots a_{d-1}$ modulo $2^d$ if the following
holds: for all $i$,  $0 \leq i < d$, we have $a_i=b_i$ iff $a_j
=0$, for some $j$, $i < j < d$. We will use the $d$-intervals
starting from $v$ to encode $<2^d$ numbers in such a way that
consecutive intervals encode consecutive (modulo $2^d$) numbers,
starting from $0$.

So, suppose that \eqref{eq:rc:1}, \eqref{eq:rc:2} and the
following formula hold in $\mathfrak{M}$ at $0$:
%
\begin{equation}\label{eq:rc:3}
\Box_F^+\Ubox\bigl(\textit{bp}(\textv{aux})\;\imp\;
\bigl(\gamma_{1,d}\;\sqcap\;\delta_{1,d}\bigr)\bigr)\ \land\
\Box_F^+\Ubox\bigl(\textit{ep}(\textv{aux})\;\imp\;
\bigl(\gamma^{\textit{ext}}_{1,d}\;\sqcap\;\delta^{\textit{ext}}_{1,d}\bigr)\bigr),
\end{equation}
%
where
%
\begin{align*}
\gamma_{1,d} \quad = \quad & \bigl(\reg{\textv{lwr}_1} \ \equi\ \reg{\nxt\textv{delim}_0} \sqcup \bigl(\reg{\nxt\textv{bit}_1}\sqcap\reg{\nxt\textv{lwr}_1}\bigr)\bigr)\ \sqcap\\ %
%
& \bigl(\reg{\textv{zr}_1} \ \equi\ \overline{\reg{\textv{bit}_1}} \sqcap \bigl(\reg{\nxt \textv{delim}_0} \sqcup \reg{\nxt\textv{zr}_1}\bigr)\bigr)\;\sqcap\; %
%
\bigl(\reg{\textv{delim}_1} \ \equi\  \reg{\textv{delim}_0} \sqcap \reg{\textv{zr}_1}\bigr),\\[3pt] %
%
\delta_{1,d} \quad = \quad &  \bigl(\overline{\reg{\textv{lwr}_1}}\ \equi\ \bigl(\reg{\textv{bit}_1} \equi \reg{\nxt^d \textv{bit}_1}\bigr)\bigr),%
\end{align*}
%
and both $\gamma^{\textit{ext}}_{1,d}$ and
$\delta^{\textit{ext}}_{1,d}$ result from $\gamma_{1,d}$ and
$\delta_{1,d}$, respectively, by attaching prefix $\textv{ext\_}$
to all of their spatial variables (save $\textv{aux}$).

Take a boundary point $z$. Suppose that $v\in\mathbb{N}$ is such
that $(\mathfrak{M},\langle
v,z\rangle)\models\reg{\textv{delim}_1}$. Then, by the last
conjunct of $\gamma_{1,d}$, we have $(\mathfrak{M},\langle
v,z\rangle)\models\reg{\textv{delim}_0}$. Since, by (a),
$\reg{\textv{delim}_0}$ holds once in every $d$ time instants on
$z$, $\reg{\textv{delim}_0}$ `marks' the starting moment of each
$d$-interval. Then, by the first conjunct of $\gamma_{1,d}$, for
every $i$, $0 \leq i < d$, we have%
%
\footnote{%
Since we cannot apply the $\until$ operator to form
spatio-temporal terms, auxiliary regions are used instead:
$\reg{\textv{lwr}_1}\;\equiv\; \reg{\nxt\textv{delim}_0} \sqcup
\bigl(\reg{\nxt\textv{bit}_1}\sqcap\reg{\nxt\textv{lwr}_1}\bigr)$
ensures that $\reg{\textv{lwr}_1}$ behaves as
$\reg{\textv{bit}_1}\until\reg{\textv{delim}_0}$. This equality
term can indeed be regarded as a fixed point characterisation of
the $\until$ operator. Note also that we do not need to require
(as we should do for the fixed point characterisation)
$\reg{\textv{lwr}_1} \imp \Diamond_F\reg{\textv{delim}_0}$ to be
true because the eventuality is already enforced by
$\delta_{0,d}$.
}%
%
\begin{equation*}
(\mathfrak{M},\auf
v+i,z\zu)\models\reg{\textv{lwr}_1}\qquad\text{iff}\qquad
(\mathfrak{M},\auf v+j,z\zu)\models\reg{\textv{bit}_1},\ \text{for
all } j,\ i < j < d.
\end{equation*}
%
Therefore, $\delta_{1,d}$ says that consecutive $<2^d$ numbers
(starting with $0$) are encoded by consecutive $d$-intervals
(starting from $v$). Similarly to the first conjunct of
$\gamma_{1,d}$, its second conjunct ensures that, for every $i$,
$0 \leq i < d$,
%
\begin{equation*}
(\mathfrak{M},\auf
v+i,z\zu)\models\reg{\textv{zr}_1}\qquad\text{iff}\qquad
(\mathfrak{M},\auf
v+j,z\zu)\models\overline{\reg{\textv{bit}_1}},\ \text{for all }
j,\ i \leq j < d.
\end{equation*}
%
So, by the last conjunct of $\gamma_{1,d}$,
$\reg{\textv{delim}_1}$ holds on $z$ once in every
$\textit{exp}(1,d)=d\cdot 2^d$ time instants, starting from $v$.
By the second conjunct of \eqref{eq:rc:3}, external points behave
similarly with respect to $\reg{\textv{ext\_delim}_1}$.

\smallskip

\emph{Step} (c). Now we construct yardsticks of length
$\textit{exp}(2,d)$, using the $\textit{exp}(1,d)$-yardsticks
constructed in (b). Suppose that \eqref{eq:rc:1}--\eqref{eq:rc:3}
and the following formulas hold in $\mathfrak{M}$ at $0$:
%
\begin{align}
\label{eq:rc:5}
& \Box_F^+\Ubox\bigl(\textit{bp}(\textv{aux})\;\imp\;\reg{\textv{ext\_delim}_1}\bigr)\ \land\  %
  \Box_F^+\Ubox\bigl(\reg{\textv{ext\_delim}_1}\ \imp\ (\textit{ep}(\textv{aux})\sqcup\textit{bp}(\textv{aux}))\bigr),\\
%
%\label{eq:rc:5b} & \Box_F^+\Ubox\bigl(\textit{bp}(z) \imp
%(\reg{\textv{bit}_2} \equi \IB\reg{\textv{bit}_2})\bigr),\\
%
\label{eq:rc:5c} & \Box_F^+\Ubox\bigl(\textit{ep}(\textv{aux})\ \imp\ \eta_{1,d}(\textv{bit}_2)\bigr),\\ %
%
\label{eq:rc:5d} &
\Box_F^+\Ubox\bigl(\textit{bp}(\textv{aux})\ \imp\ \gamma_{2,d} \;\sqcap\; \delta_{2,d}\bigr), %
\end{align}
%
where $\gamma_{2,d}$ is defined similarly to $\gamma_{1,d}$ and
%
\begin{align*}
\eta_{1,d}(\textv{bit}_2) \quad = \quad & \bigl(\reg{\textv{jm}_1\textv{\_bit}_2} \ \equi\ \\ & \hspace*{3em} \bigl(\reg{\nxt\textv{ext\_delim}_1}\sqcap \reg{\nxt\textv{bit}_2}\bigr) \ \sqcup\ \bigl(\overline{\reg{\nxt\textv{ext\_delim}_1}} \sqcap \reg{\nxt\textv{jm}_1\textv{\_bit}_2}\bigr)\bigr), \\
%
\delta_{2,d} \quad = \quad & \bigl(\reg{\textv{bit}_2} \equi \IB\reg{\textv{bit}_2}\bigr) \ \sqcap\ \bigl(\overline{\reg{\textv{lwr}_2}}\ \equi\ \bigl(\reg{\textv{bit}_2} \equi \mathbf{J}_{1,d}\,\textv{bit}_2\bigr)\bigr),\\[2pt] %
%
\mathbf{J}_{1,d}\,\textv{bit}_2 \quad = \quad & \IB\reg{(\overline{\textv{aux}} \sqcap \textv{ext\_delim}_1) \;\imp\; \textv{jm}_1\textv{\_bit}_2}. %
\end{align*}
%
Take a boundary point $z$. Suppose $v$ is a time moment such that
$(\mathfrak{M},\langle v,z\rangle)\models\reg{\textv{delim}_2}$.
Then, by the last conjunct of $\gamma_{2,d}$,
$(\mathfrak{M},\langle v,z\rangle)\models\reg{\textv{delim}_1}$.
We know from (b) that $\reg{\textv{delim}_1}$ holds on $z$ once in
every $\textit{exp}(1,d)$ time instants starting from $v$. So, by
$\delta_{2,d}$ and the first conjunct of $\gamma_{2,d}$ we intend
to express that consecutive $<2^{exp(1,d)}$ numbers (starting with
$0$) are encoded by consecutive $\textit{exp}(1,d)$-intervals
starting from $v$. If we could do this then, by the last conjunct
of $\gamma_{2,d}$, $\reg{\textv{delim}_2}$ would hold on $z$ once
in every $\textit{exp}(2,d)$ time instants starting from $v$. The
only problem (and the only difference from step (b)) is that to
`mark' the bits of $<2^{\textit{exp}(1,d)}$ binary numbers by a
term $\reg{\textv{bit}_2}$, we need to show that the (polynomial
length) term $\mathbf{J}_{1,d}\,\textv{bit}_2$ actually defines
`$\nxt^{\textit{exp}(1,d)}\reg{\textv{bit}_2}$' in the sense that,
for every $w \geq v$,
%
\begin{equation}\label{longys}
(\mathfrak{M},\langle w, z\rangle)\models
\mathbf{J}_{1,d}\,\textv{bit}_2 \qquad\text{iff}\qquad
(\mathfrak{M},\bigl\langle
w+\textit{exp}(1,d),z\bigr\rangle)\models \reg{\textv{bit}_2}.
\end{equation}
%
Suppose first that $(\mathfrak{M},\langle w, z\rangle)\models
\mathbf{J}_{1,d}\,\textv{bit}_2$. Then, by \eqref{eq:rc:5}, there
is an external $R$-successor $y_w$ of $z$ (of depth 0) such that
$(\mathfrak{M},\langle w,
y_w\rangle)\models\reg{\textv{ext\_delim}_1}$, and so
$(\mathfrak{M},\langle w, y_w\rangle)\models
\reg{\textv{jm}_1\textv{\_bit}_2}$. On the other hand, it is not
hard to see that if $(\mathfrak{M},\langle w, z\rangle)\not\models
\mathbf{J}_{1,d}\,\textv{bit}_2$, then there is an external
$R$-successor $y'_w$ of $z$ (of depth $0$) such that $(\mathfrak
M,\langle w, y'_w\rangle)\models \reg{\textv{ext\_delim}_1}$ but
$(\mathfrak M,\langle w, y'_w\rangle)\models
\overline{\reg{\textv{jm}_1\textv{\_bit}_2}}$.

In both cases, it is readily checked that if $(\mathfrak M,\langle
w, y\rangle)\models \reg{\textv{ext\_delim}_1}$, for some external
point $y$, then, by \eqref{eq:rc:5c},
%
\begin{equation*}
(\mathfrak M,\langle w, y\rangle)\models \reg{\textv{jm}_1\textv{\_bit}_2}\qquad\text{iff}\qquad %
(\mathfrak{M},\bigl\langle
w+\textit{exp}(1,d),y\bigr\rangle)\models \reg{\textv{bit}_2}.
\end{equation*}
%
Now \eqref{longys} follows by the first conjunct of
$\delta_{2,d}$.

\smallskip

\emph{Step} (d). We are now in a position to define a
polynomial-length `$\nxt^{\textit{exp}(2,d)}$ operator' $\jump{}$
applicable to propositional variables. Recall that a propositional
variable $\textp{p}$ stands for spatial formula $\Ubox\textv{p}$,
where $\textv{p}$ is a spatial variable associated with
$\textp{p}$. Now, for every propositional variable $\textp{p}$ we
intend to apply the new operator to, we introduce a fresh spatial
variable $\textv{jm}_2\textv{\_p}$. Suppose that
\eqref{eq:rc:1}--\eqref{eq:rc:5d} and the following formulas hold
in $\mathfrak{M}$ at $0$:
%
\begin{align}
\label{eq:rc:6}
& \Box_F^+\Udiamond\bigl(\textit{bp}(\textv{aux})\; \sqcap\; \reg{\textv{delim}_2}\bigr),\\
%
\label{eq:rc:6b}
& \Box_F^+\bigl(\Ubox\reg{\textv{p}} \leftrightarrow \neg\Ubox\overline{\reg{\textv{p}}}\bigr)\ \land\ \Box_F^+\Ubox\; \eta_{2,d}(\textv{p}), %\\
\end{align}
%
where $\eta_{2,d}(\textv{p})$ is obtained by replacing
$\textv{bit}_2$, $\textv{jm}_1\textv{\_bit}_2$ and
$\textv{ext\_delim}_1$ in $\eta_{1,d}(\textv{bit}_2)$ with
$\textv{p}$, $\textv{jm}_2\textv{\_p}$ and $\textv{delim}_2$,
respectively. Let
%
\begin{equation*}
\jump{\textp{p}} =
\Ubox\bigl((\textit{bp}(\textv{aux})\sqcap\reg{\textv{delim}_2})
\ \imp\ \reg{\textv{jm}_2\textv{\_p}}\bigr). %
\end{equation*}
%
We claim that, for every time moment $w$ and every propositional
variable $\textp{p}$,
%
\begin{equation}\label{eq:jumpok}
(\mathfrak{M},w)\models\jump{\textp{p}}\qquad\text{iff}\qquad
(\mathfrak{M},w + \textit{exp}(2,d))\models\textp{p}.
\end{equation}
%
Suppose first that $(\mathfrak{M},w)\models\jump{\textp{p}}$.
Then, by \eqref{eq:rc:6}, there is a boundary point $z$ such that
$(\mathfrak{M},\langle w,z\rangle)\models\reg{\textv{delim}_2}$,
and therefore $(\mathfrak{M},\langle w,z\rangle)\models
\reg{\textv{jm}_2\textv{\_p}}$. On the other hand, if
$(\mathfrak{M},w)\not\models\jump{\textp{p}}$, then there is a
boundary point $z'$ with $(\mathfrak{M},\langle
w,z'\rangle)\models\reg{\textv{delim}_2}$ but
$(\mathfrak{M},\langle w,z'\rangle)\models
\overline{\reg{\textv{jm}_2\textv{\_p}}}$. In both cases, it is
readily checked that if $(\mathfrak M,\langle w, z\rangle)\models
\reg{\textv{delim}_2}$, for some boundary point $z$, then, by the
second conjunct of~\eqref{eq:rc:6b},
%
\begin{equation*}
(\mathfrak{M},\langle w, z\rangle)\models \reg{\textv{jm}_2\textv{\_p}}\qquad\text{iff}\qquad %
(\mathfrak{M},\bigl\langle w+\textit{exp}(2,d),z\bigr\rangle)\models \reg{\textv{p}}. %
\end{equation*}
%
Now~\eqref{eq:jumpok} follows by the first conjunct
of~\eqref{eq:rc:6b}.


Finally, we are in a position to define the $\PTL\circ\RC$-formula
$\varphi_{\boldsymbol{A},x}$ that encodes the computation of
Turing machine $\boldsymbol{A}$ on input $x$. Let $A$ be the tape
alphabet (with the blank symbol $b\in A$) and $S$ the set of
states (with two halt states $s_{\textit{yes}}$ and
$s_{\textit{no}}$ in $S$) of $\boldsymbol{A}$. We use the symbol
$\pounds\notin A$ to mark the left end of the tape. We know that
the space used by $\boldsymbol A$ on input $x=\langle
x_1,\dots,x_n\rangle$ is $\leq 2^{2^{f(n)}}$, which is $\leq
\textit{exp}(2,d)$ by \eqref{dok}. So we can represent each
configuration of the computation of $\boldsymbol A$ on $x$ as a
finite word
%
\begin{equation*}
\langle \pounds, a_1,\dots, a_{i-1},\auf s, a_i\zu,a_{i+1},\dots,
a_m,b,\dots, b\rangle
\end{equation*}
%
of length $\textit{exp}(2,d)$, where $a_1,\dots,a_m\in A$ and
$\auf s, a_i\zu\in S\times A$ represents the current state and the
active cell. The transition function $\delta$ of $\boldsymbol{A}$
takes triples of the form $\auf a_i,\auf s,a_j\zu, a_k\zu$ (for
$a_i\in A\cup \{\pounds\}$, $a_j,a_k\in A$, $s\in
S-\{s_{\textit{yes}},s_{\textit{no}}\}$) to similar triples. For
instance,
%
\begin{equation*}
\delta(a_i,\auf s,a_j\zu, a_k)=\langle a_i, a_j,\langle
s',a_k\rangle\rangle
\end{equation*}
%
means that, when being in state $s$ and reading symbol $a_j$, the
new state should be $s'$ and the head should move one cell to the
right. We also assume that, for all $a_i\in A\cup\{\pounds\}$ and
$a_j,a_k\in A$, we have $\delta(a_i,\auf
s_{\textit{yes}},a_j\zu,a_k) = \langle a_i, a_j, a_k\rangle$ and
$\delta(a_i,\auf s_{\textit{no}},a_j\zu,a_k) = \langle a_i, a_j,
a_k\rangle$ meaning that the head is removed after
$\boldsymbol{A}$ is being halted.

Now, for every $\alpha\in A\cup\{\pounds\}\cup(S\times A)$, we
introduce a fresh propositional variable $\textp{p}_\alpha$. Let
$\varphi_{\boldsymbol{A},x}$ be the conjunction of
\eqref{eq:rc:1}--\eqref{eq:rc:5d}, \eqref{eq:rc:6} and an instance
of~\eqref{eq:rc:6b}, for each $\textp{p}_\alpha$, as well as the
following formulas:
%
\begin{align}
\label{eq:rc:tmfirst}
& \bigwedge_{\begin{subarray}{c}\alpha,\beta\in A\cup\{\pounds\}\cup S\times A\\\alpha\ne \beta\end{subarray}} \hspace*{-2em}\Box_F^+\bigl(\neg \textp{p}_\alpha \lor \neg \textp{p}_{\beta}\bigr),\\
%
& \textp{p}_\pounds \land \nxt(\textp{p}_{\auf s_0,x_1\zu} \land\nxt(\textp{p}_{x_2}\land\nxt(\cdots \land \nxt(\textp{p}_{x_n}\land \textp{p}_b \until \textp{p}_\pounds)\cdots))),\\[8pt]  %
%
& \Box_F^+\Bigl(\nxt\textp{af\_head}\ \leftrightarrow \hspace*{-1em}\bigvee_{\auf s,a\zu \in S \times A}  \hspace*{-1em}\textp{p}_{\auf s,a\zu}\Bigr), %\\
%\end{align}
\\
%\begin{align}
%
& \bigwedge_{\begin{subarray}{c}\delta(\alpha,\beta,\gamma) =\\
\auf\alpha',\beta',\gamma'\zu\end{subarray}}\hspace*{-1em}\Box_F^+
\Bigl(\nxt\nxt\textp{af\_head} \to%\\  & \hspace*{6cm} %
  \bigl(\textp{p}_\alpha \to \jump{\textp{p}_{\alpha'}}\bigr)\land%\\ \nonumber  & \hspace*{5cm} %
  \nxt\bigl(\textp{p}_\beta \to \jump{\textp{p}_{\beta'}}\bigr)\land%\\  & \hspace*{6cm} %
  \nxt\nxt\bigl(\textp{p}_\gamma \to \jump{\textp{p}_{\gamma'}}\bigr)\Bigr), \\
%
\label{eq:rc:tmlast}
& \bigwedge_{a\in A\cup\{\pounds\}}\Box_F^+\Bigl(\neg\bigl(\nxt\nxt\textp{af\_head} \lor \nxt\textp{af\_head} \lor \textp{af\_head}\bigr) \to \bigl({\textp{p}_a} \to \jump{\textp{p}_a}\bigr)\Bigr),\\
\label{eq:rc:tmstop} & \Diamond_F\bigvee_{a\in A}\textp{p}_{\auf
s_{\textit{yes}},a\zu}
\;\land\; \neg\Diamond_F\bigvee_{a\in A}\textp{p}_{\auf s_{\textit{no}},a\zu}. %
\end{align}
%
Suppose first that $\varphi_{\boldsymbol{A},x}$ holds in
$\mathfrak{M}$ at time moment $0$. By
\eqref{eq:rc:tmfirst}--\eqref{eq:rc:tmlast} and~\eqref{eq:jumpok},
the consecutive configurations of the computation of
$\boldsymbol{A}$ on input $x$ are properly `encoded' along the
time axis. (For instance, $\textp{p}_{\pounds}$ holds once in
every $\textit{exp}(2,d)$ time moments.) Finally,
\eqref{eq:rc:tmstop} says that $\boldsymbol A$ accepts input $x$.

Conversely, suppose that $\boldsymbol A$ accepts input $x$. We
will define an Aleksandrov tt-model
$\mathfrak{M}=\auf\auf\mathbb{N},<\zu,\mathfrak{G},\mathfrak{U}\zu$
with \FSA{} that satisfies $\varphi_{\boldsymbol{A},x}$. Let the
partial order $\mathfrak{G} = \auf V,R\zu$ be a disjoint union of
$\textit{exp}(2,d)$ many $(\textit{exp}(1,d)+1)$-brooms
%
(see Fig.~\ref{fig:yardststicks:overview}):%
%
%
\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{.050cm}
\begin{picture}(290,75)(-20,-20)
%
%\put(-20,0){\line(1,0){290}}
%\put(0,-25){\line(0,1){80}}
%
\multiput(-15,0)(165,0){2}{%
%
\put(50,10){\circle*{2}}\put(50,10){\circle{3.5}}%
\put(110,30){\circle*{2}}%
\put(52,10.5){\vector(3,1){56}}%
\put(50,30){\circle{2.5}}%
\put(50,12){\vector(0,1){16}}%
\put(48.5,11.8){\vector(-1,1){16.5}}%
\put(30.5,30){\circle{2.5}}%
\put(51.5,11.8){\vector(1,1){16.5}}%
\put(69.5,30){\circle{2.5}}%
\put(48,11.2){\vector(-2,1){35.5}}%
\put(10.5,30){\circle{2.5}}%
\put(52,11.2){\vector(2,1){35.5}}%
\put(89.5,30){\circle{2.5}}%
%
\put(50,30){\oval(90,8)}%
%\put(19,41){\vector(-2,-1){12}}%
\put(60,25){\oval(130,60)}%
\put(30,9){\vector(1,0){15}}
%
%\put(50,0){\line(2,1){50}}%
%\put(50,0){\line(-3,1){70}}%
%\put(-20,25){\line(0,-1){1.5}}
%
\put(102,17){\scriptsize $\IB \reg{\textv{aux}}$}%
\put(2,17){\scriptsize $\textit{ep}(\textv{aux})$}%
\put(38,0){\scriptsize $\textit{bp}(\textv{aux})$}%
%
}%
%
\put(5,0){%
\put(0,40){\small $\textit{exp}(1,d)$ many $l_0^j$}%
\put(81,36){\small $l_0^{\textit{exp}(1,d)}$}%
\put(4,7){\small $r_0$}%
}%
%
\put(160,40){\small $\textit{exp}(1,d)$ many $l_{\textit{exp}(2,d)-1}^j$}%
\put(246,36){\small $l_{\textit{exp}(2,d)-1}^{\textit{exp}(1,d)}$}%
\put(148,7){\small $r_{\textit{exp}(2,d)-1}$}%
%
\put(120,25){\LARGE \dots}%
%
\put(-20,-10){$\underbrace{\hspace*{38em}}_{\displaystyle\textit{exp}(2,d) \text{ many } (\textit{exp}(1,d)+1)\text{-brooms}}$} %
%
\end{picture}
\end{center}
\caption{Structure of
yardsticks.}\label{fig:yardststicks:overview}
\end{figure}
%
%
%
\begin{align*}
& V=\{r_i\mid i<\textit{exp}(2,d)\}\cup
\{l_i^j\mid i<\textit{exp}(2,d),\;j\leq\textit{exp}(1,d)\},\\
& zRy\qquad\text{iff}\qquad z=y\quad\text{or}\quad z=r_i,\
y=l_i^j, \text{ for some }i,j.
\end{align*}
%
Suppose that the number of steps in the computation of
$\boldsymbol A$ on $x$ is $m$. Then $\mathfrak M$ will have a
prefix of length $N=m\cdot\textit{exp}(2,d)$ after which the final
configuration (without a halting state) repeats to infinity. For
 $w\in\mathbb{N}$, let
%
\begin{equation*}
\mathfrak{U}(w,\textv{aux})=\{l_i^{\textit{exp}(1,d)}\mid
  i<\textit{exp}(2,d)\}.
\end{equation*}
%
Then it is easy to see that the boundary points are the $r_i$, and
the external points are the $l_i^j$, for $i<\textit{exp}(1,d)$.
%
Now put, for every $w\in \mathbb N$,
%
\begin{align*}
& \mathfrak{U}(w,\textv{delim}_2)=\{l_i^{\textit{exp}(1,d)} \mid i\equiv w \pmod{\textit{exp}(2,d)}\},\\
& \mathfrak{U}(w,\textv{delim}_1)=\{l_i^{\textit{exp}(1,d)} \mid i\equiv w \pmod{\textit{exp}(1,d)}\},\\
& \mathfrak{U}(w,\textv{delim}_0)=\{l_i^{\textit{exp}(1,d)} \mid i\equiv w \pmod{d}\},\\
& \mathfrak{U}(w,\textv{ext\_delim}_1)=\{l_i^{v} \mid v\equiv w \pmod{\textit{exp}(1,d)},\ i<\textit{exp}(2,d)\},\\
& \mathfrak{U}(w,\textv{ext\_delim}_0)=\{l_i^{v} \mid v\equiv w
\pmod{d},\ i<\textit{exp}(2,d)\}.
\end{align*}
%
The valuations for the other variables should be clear. We then
have
%
\begin{align*}
& (\mathfrak{M},\auf w,z\zu)\models \reg{\textv{delim}_2}
&&\quad\text{iff}\quad z=r_i\text{ or }z=l_i^{\textit{exp}(1,d)}
\mbox{ for some }i\equiv w \pmod{\textit{exp}(2,d)},\\
%
& (\mathfrak{M},\auf w,z\zu)\models \reg{\textv{delim}_1}
&&\quad\text{iff}\quad z=r_i \text{ or } z=l_i^{\textit{exp}(1,d)}
\mbox{ for some }i\equiv w \pmod{\textit{exp}(1,d)},\\
%
& (\mathfrak{M},\auf w,z\zu)\models \reg{\textv{ext\_delim}_1}
&&\quad\text{iff}\quad z=r_i \text{ or } z=l_i^v
\mbox{ for some }v\equiv w \pmod{\textit{exp}(2,d)}\\
&&& \hspace*{7cm} \text{and } i<\textit{exp}(2,d),
\end{align*}
%
and so on, as required. It is not hard to see that $\mathfrak{M}$
satisfies \FSA{} and
$(\mathfrak{M},0)\models\varphi_{\boldsymbol{A},x}$.
\end{proof}


\begin{proof}[\proofof{Theorem~\ref{st1:expspace}, lower bound}]
The proof is by reduction of the \emph{$2^n$-corridor tiling
problem} which is known to be
EXPSPACE-complete~\cite{Chlebus86,EmdeBoas97}. The problem can be
formulated as follows: given an instance $\mathcal{T} = \auf
T,t_0,t_1,n\zu$, where $T$ is a finite set of tile types, $t_0,
t_1\in T$ and $n>0$, decide whether there is an $m\in\mathbb{N}$
such that $\mathcal{T}$ \emph{tiles} the $m\times 2^n$-grid (or
corridor) in such a way that $t_0$ is placed onto $\auf 0, 0\zu$,
$t_1$  onto $\auf m - 1, 0\zu$, and the top and bottom sides of
the corridor are of some fixed colour, say, $\textit{white}$.

Suppose $\mathcal{T} = \auf T,t_0,t_1,n\zu$ is given. Our aim is
to construct (using only future-time temporal operators) a
$\PTL\circ\BRCCE$ formula $\varphi_{\mathcal{T}}$ such that
%
\begin{itemize}
\item[(i)] the length of $\varphi_{\mathcal{T}}$ is a polynomial
function of $|T|$ and $n$;
%
\item[(ii)] if $\varphi_{\mathcal{T}}$ is satisfiable in a tt-model based on $\auf\mathbb{N},<\zu$ then
there is $m\in\mathbb{N}$ such that $\mathcal{T}$ tiles the
$m\times 2^n$-corridor;
%
\item[(iii)] if there is $m\in\mathbb{N}$ such that $\mathcal{T}$ tiles the $m\times
2^n$-corridor, then $\varphi_{\mathcal{T}}$ is satisfiable in a
tt-model with \FSA{} and based on $\auf\mathbb{N},<\zu$;
%
\item[(iv)]  $\varphi_{\mathcal{T}}$ is satisfiable in a tt-model
based on $\auf\mathbb{N},<\zu$ iff it is satisfiable in a tt-model
based on a finite flow of time.
\end{itemize}
%
The case of $\auf\mathbb{Z},<\zu$ follows immediately.

Recall that, by Lemma~\ref{rc:simula3} (ii), if
$\varphi_{\mathcal{T}}$ is satisfied in a tt-model then it is
satisfied in an Aleksandrov tt-model $\mathfrak{M} = \auf
\auf\mathbb{N},<\zu, \mathfrak{G},\mathfrak{V}\zu$, where
$\mathfrak{G} = \auf V,R\zu$ is a disjoint union of
$\omega$-brooms. To explain the meaning of
$\varphi_{\mathcal{T}}$'s subformulas, we assume that such a model
$\mathfrak M$ is given. Throughout the proof we use only a
restricted subset of $\RCCE$ predicates: for spatio-temporal terms
$\tau_1$ and $\tau_2$ constructed from spatial variables using
only the complement, the intersection and the next-time operator
$\nxt$, we need $\EQ(\tau_1,\tau_2)$ as well as two abbreviations
$\PT(\tau_1, \tau_2) = \textsf{EQ}(\tau_1,\tau_2) \lor
\textsf{TPP}(\tau_1,\tau_2) \lor \textsf{NTPP}(\tau_1,\tau_2)$ and
$\NE\tau_1 = \neg\textsf{DC}(\tau_1, \tau_1)$ standing for
`$\tau_1$ is a part of $\tau_2$' and `$\tau_1$ is nonempty,'
respectively. Clearly, this language forms a subset of
$\PTL\circ\BRCCE$ (in fact, as we show in
Remark~\ref{remark:rcc8:expspace} below, the proof goes through
for an even more restricted subset of the langauge).

Our first step in the construction of $\varphi_{\mathcal{T}}$
(which will contain, among many others, spatial variables
$\textv{t}$ for all $t\in T$) is to write down formulas forcing a
sequence $y_0, y_1, \dots, y_{m\cdot 2^n - 1}$ of distinct points
(of depth 0) from $V$, for some $m\in\mathbb{N}$, such that, for
each $i < m\cdot 2^n$, $(\mathfrak{M}, \auf i,y_i\zu)\models
\textv{t}$ for a unique tile type $t\in T$. If $i = k\cdot 2^n +
j$, for some $k< m$, $j < 2^n$, then we will use $y_i$ (at time
$i$) to encode the pair $\auf k, j\zu$ of the $m\times 2^n$-grid.
Thus, the up neighbour $\auf k, j + 1\zu$ of $\auf k, j\zu$ will
be coded by the point $y_{i + 1}$ at time $i + 1$, while its right
neighbour $\auf k + 1, j\zu$ by $y_{i + 2^n}$ at moment $i + 2^n$
(see Fig.~\ref{figure:brcc8:slices}).

Let $\textp{q}_0, \dots, \textp{q}_{n-1}$ be pairwise distinct
propositional variables and
%
\begin{equation*}
\sigma_j = \textp{q}_0^{d_0}\land \dots \land
\textp{q}_{n-1}^{d_{n-1}},
\end{equation*}
%
where $d_{n-1}\dots d_0$ is the binary representation of $j <
2^n$, $\textp{q}_i^0 = \neg \textp{q}_i$ and $\textp{q}_i^1 =
\textp{q}_i$, for each $i$. Suppose that the formula
%
\begin{equation}\label{eq:brcc8:corridor:count1}
\bigl(\textp{count} \land \sigma_0 \bigr) \;\land\; %
\bigl(\textp{count} \until (\sigma_0 \land \Box_F^+ \neg \textp{count}) \bigr) \;\land\; %
\Box_F^+\bigl(\textp{count} \to \chi\bigr) %
\end{equation}
%
is true in $\mathfrak{M}$ at $0$, where $\textp{count}$ is a fresh
propositional variable and $\chi$ is the following `counting'
formula (the length of which is polynomial in $n$)
%
\begin{equation*}
\chi = \bigwedge_{k < n}%
\Bigl(\Bigl(\bigwedge_{i < k} \textp{q}_i \land \neg \textp{q}_k\Bigr) \to %
\Bigl(\bigwedge_{i < k} \nxt\neg \textp{q}_i \land \nxt \textp{q}_k \land \bigwedge_{k < i < n}(\textp{q}_i \leftrightarrow \nxt \textp{q}_i)\Bigr)\Bigr)\;\land\;%
\bigl(\sigma_{2^n-1} \to \nxt\sigma_0\bigr).
\end{equation*}
%
Then there is an $m\in\mathbb{N}$ such that $\textp{count}$ is
true before moment $(m + 1)\cdot 2^n$ and false starting from $(m
+ 1)\cdot 2^n$. The sequence $\sigma_0, \sigma_1, \dots,
\sigma_{2^n-1}$ is repeated $m+1$ times along the time-line, i.e.,
while $\textp{count}$ is true. Let
%
\begin{equation*}
\textp{range} = \Diamond_F(\textp{count} \land \sigma_0).
\end{equation*}
%
Clearly, $\textp{range}$ is true before moment $m\cdot 2^n$ and
then always false (see Fig.~\ref{figure:brcc8:sigmas}).

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{.070cm}
\begin{picture}(200,28)(5,0)
%
\put(15,13){\vector(1,0){180}}%
\multiput(15,7)(40,0){5}{\multiput(0,0)(0,5){5}{\line(0,1){2}}}%

\put(15,23){\line(1,0){160}}%
%\put(18,12){\line(0,1){11}}%
%\put(175,12){\line(0,1){11}}%
\put(90,25){\scriptsize $\textp{count}$}%

\put(15,16){\line(1,0){120}}%
%\put(18,12){\line(0,1){4}}%
%\put(135,12){\line(0,1){4}}%
\put(70,18){\scriptsize $\textp{range}$}%

{\thicklines\multiput(20,12)(10,0){17}{\line(0,1){2}}}%

\put(0, -4){%
\multiput(18.7,4)(10,0){17}{$\uparrow$}%
\multiput(16.8,-2)(10,0){17}{\framebox(6.4,11){ }}%
\put(19,11){${}_{0}$} \put(18,-0.5){\small $\sigma_0$}
\put(29,11){${}_{1}$} \put(28,-0.5){\small $\sigma_1$}
\put(39,11){${}_{2}$} \put(38,-0.5){\small $\sigma_2$}
\put(49,11){${}_{3}$} \put(48,-0.5){\small $\sigma_3$}
\put(59,11){${}_{4}$} \put(58,-0.5){\small $\sigma_0$}
\put(69,11){${}_{5}$} \put(68,-0.5){\small $\sigma_1$}
\put(79,11){${}_{6}$} \put(78,-0.5){\small $\sigma_2$}
\put(89,11){${}_{7}$} \put(88,-0.5){\small $\sigma_3$}
\put(99,11){${}_{8}$} \put(98,-0.5){\small $\sigma_0$}
\put(109,11){${}_{9}$} \put(108,-0.5){\small $\sigma_1$}
\put(118,11){${}_{{10}}$} \put(118,-0.5){\small $\sigma_2$}
\put(128,11){${}_{{11}}$} \put(128,-0.5){\small $\sigma_3$}
\put(138,11){${}_{{12}}$} \put(138,-0.5){\small $\sigma_0$}
\put(148,11){${}_{{13}}$} \put(148,-0.5){\small $\sigma_1$}
\put(158,11){${}_{{14}}$} \put(158,-0.5){\small $\sigma_2$}
\put(168,11){${}_{{15}}$} \put(168,-0.5){\small $\sigma_3$}
\put(178,11){${}_{{16}}$} \put(178,-0.5){\small $\sigma_0$}
%
\multiput(188,11)(2,0){3}{.}
}%

\put(14,30){\small $0$}%
\put(130,30){\small $m\cdot 2^n$}%
\put(165,30){\small $(m+1)\cdot 2^n$}
\end{picture}
\end{center}
\caption{Counting formulas for $m=3$ and
$n=2$.}\label{figure:brcc8:sigmas}
\end{figure}

Let $\textv{equ}$, $\textv{p}_0,\dots, \textv{p}_{n-1}$ and
$\textv{e}_0,\dots, \textv{e}_{n-1}$ be fresh distinct spatial
variables, and
%
\begin{equation*}
\pi_j = \textv{p}_0^{d_0}\sqcap\dots\sqcap
\textv{p}_{n-1}^{d_{n-1}},
\end{equation*}
%
where $d_{n-1}\dots d_0$ is the binary representation of $j <
2^n$, $\textv{p}_i^0 = \overline{\textv{p}_i}$ and $\textv{p}_i^1
= \textv{p}_i$, for each $i$. Suppose
that~\eqref{eq:brcc8:corridor:count1} and
%
\begin{equation}\label{eq:brcc8:corridor:equ}%
\Box_F^+ \EQ\bigl(\textv{equ},\; \bigsqcap_{i < n}\textv{e}_i\bigr) \;\land\;%
\bigwedge_{i < n}\Box_F^+\bigl(\textp{q}_i \leftrightarrow \EQ(\textv{e}_i,\; \textv{p}_i)\bigr) \;\land\;%
\bigwedge_{i < n} \Box_F^+ \EQ\bigl(\textv{p}_i, \nxt \textv{p}_i\bigr) %
\end{equation}
%
are true in $\mathfrak{M}$ at 0. Then, by the first two conjuncts
of~\eqref{eq:brcc8:corridor:equ}, for all $i\in\mathbb{N}$ and
$y\in V$ of depth 0, there is $j < 2^n$ such that
$(\mathfrak{M},\auf i,y\zu)\models\textv{equ}$ iff $(\mathfrak{M},
i)\models\sigma_j$ and $(\mathfrak{M},\auf i,y\zu)\models\pi_j$.
By the last conjunct of~\eqref{eq:brcc8:corridor:equ}, we then
have
%
\begin{align}\tag{$\ast$}\label{eq:brcc8:corridor:equ:con}%
& (\mathfrak{M},\auf i,y\zu)\models\textv{equ}\quad\text{iff}\quad
\exists j < 2^n\; \Bigl( (\mathfrak{M}, i)\models\sigma_j\quad
\text{and}\quad (\mathfrak{M},\auf k,y\zu)\models\pi_j, \text{ for
all } k\in\mathbb{N}\Bigr).
\end{align}


We can generate the required sequence of points $y_i$ using the
formulas:
%
\begin{align}\label{eq:brcc8:corridor:sequence}%
& \textp{range}\; \land\; \Box_F^+ (\textp{range} \to \NE\textv{tile}),\\
%
\label{eq:brcc8:corridor:tiledef}%
& \Box_F^+ \EQ\Bigl(\textv{tile},\; \textv{equ} \sqcap
\bigl(\bigsqcup_{t\in T} \textv{t}\bigr) \sqcap%
\bigsqcap_{t\in T} \textv{no\_t\_in\_future}\Bigr),\\
%
\label{eq:brcc8:corridor:no:mark:in:future}%
& \bigwedge_{t\in T}\Box_F^+ \PT\bigl(\textv{no\_t\_in\_future},\;
\nxt\overline{\textv{t}}\sqcap
\nxt\textv{no\_t\_in\_future}\bigr),%
\end{align}
%
where $\textv{tile}$ and the $\textv{no\_t\_in\_future}$ (for all
$t\in T$) are fresh spatial variables. Indeed, suppose the
conjunction of
\eqref{eq:brcc8:corridor:count1}--\eqref{eq:brcc8:corridor:no:mark:in:future}
holds at time~$0$ in $\mathfrak{M}$. Then, by the first conjunct
of~\eqref{eq:brcc8:corridor:sequence} and
\eqref{eq:brcc8:corridor:count1},
$(\mathfrak{M},0)\models\textp{range}\land\sigma_0$ and, by the
second conjunct of~\eqref{eq:brcc8:corridor:sequence},
$(\mathfrak{M},\auf 0,y_0\zu)\models\textv{tile}$ for some $y_0\in
V$. We may assume that $y_0$ is of depth 0. Then, by
\eqref{eq:brcc8:corridor:tiledef}, we have
%
\begin{itemize}
\item[(a$_0$)] $(\mathfrak{M}, \auf 0,y_0\zu)\models \textv{equ}$, and,
by \eqref{eq:brcc8:corridor:equ:con}, $(\mathfrak{M}, \auf
k,y_0\zu) \models\pi_0$ for all $k\in\mathbb{N}$;
%
\item[(b$_0$)] for all $t\in T$, $(\mathfrak{M},\auf 0,y_0\zu)\models\textv{no\_t\_in\_future}$
and, by \eqref{eq:brcc8:corridor:no:mark:in:future},
$(\mathfrak{M}, \auf k,y_0\zu)\models\overline{\textv{t}}$ for all
$k> 0$.
\end{itemize}
%
Next, by \eqref{eq:brcc8:corridor:count1}, we have $(\mathfrak{M},
1) \models \textp{range}\land\sigma_1$ and,
by~\eqref{eq:brcc8:corridor:sequence}, there is $y_1\in V$ (again,
of depth 0) such that $(\mathfrak{M}, \auf
1,y_1\zu)\models\textv{tile}$. In particular, we have:
%
\begin{itemize}
\item[(a$_1$)] $(\mathfrak{M}, \auf 1,y_1\zu)\models \textv{equ}$, and,
by \eqref{eq:brcc8:corridor:equ:con}, $(\mathfrak{M}, \auf
k,y_1\zu) \models\pi_1$ for all $k\in\mathbb{N}$;
%
\item[(b$_1$)] for all $t\in T$, $(\mathfrak{M},\auf 1,y_1\zu)\models\textv{no\_t\_in\_future}$
and, by \eqref{eq:brcc8:corridor:no:mark:in:future},
$(\mathfrak{M}, \auf k,y_1\zu)\models\overline{\textv{t}}$ for all
$k> 1$.
\end{itemize}
%
By (b$_0$), $(\mathfrak{M}, \auf
1,y_0\zu)\models\overline{\textv{t}}$, for all $t\in T$, and thus
$y_1\ne y_0$. Now we consider $y_1$ at moment~$1$ and use the same
argument to find a point $y_2\in V$ (which is different from $y_1$
by (b$_1$)), and so forth; see Fig.~\ref{figure:brcc8:slices}.
This gives us points $y_0, y_1, \dots, y_{m\cdot 2^n -1}$ (of
depth 0) from $V$ we need.

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{.070cm}
\begin{picture}(200,158)(-7,-5)

%\put(0,0){\line(1,0){193}} \put(-7,0){\line(0,1){150}}

\put(0,132){$V$}
%\multiput(140,135)(10,0){2}{\vector(0,-1){4}}
%
\put(30,0){%
\put(120,145){\put(2.5,0){\circle{1.5}}\put(5,0){${}_{=\;\overline{\textv{t}}\;\; \text{ for all }\;\; t\in T}$}}%
\put(120,140){\put(-2.5,0){\circle*{1}}\put(2.5,0){\circle*{1}}\put(0,0){\oval(7,2.1)}\put(5,0){${}_{=\;\textv{no\_equ\_U\_tile}}$}}%
%
\put(115,137.5){\framebox(45,10){\ }}
}%

%\multiput(24,19)(10,10){11}{\line(1,1){5}}

\multiput(30,15)(10,0){12}{\circle{1.5}}
\put(20,25){\circle*{1}}%
\multiput(40,25)(10,0){11}{\circle{1.5}}
\multiput(20,35)(10,0){2}{\circle*{1}}
\multiput(50,35)(10,0){10}{\circle{1.5}}
\multiput(20,45)(10,0){3}{\circle*{1}}
\multiput(60,45)(10,0){9}{\circle{1.5}}
\multiput(20,55)(10,0){4}{\circle*{1}}
\multiput(70,55)(10,0){8}{\circle{1.5}}
\multiput(20,65)(10,0){5}{\circle*{1}}
\multiput(80,65)(10,0){7}{\circle{1.5}}
\multiput(20,75)(10,0){6}{\circle*{1}}
\multiput(90,75)(10,0){6}{\circle{1.5}}
\multiput(20,85)(10,0){7}{\circle*{1}}
\multiput(100,85)(10,0){5}{\circle{1.5}}
\multiput(20,95)(10,0){8}{\circle*{1}}
\multiput(110,95)(10,0){4}{\circle{1.5}}
\multiput(20,105)(10,0){9}{\circle*{1}}
\multiput(120,105)(10,0){3}{\circle{1.5}}
\multiput(20,115)(10,0){10}{\circle*{1}}
\multiput(130,115)(10,0){2}{\circle{1.5}}
\multiput(20,125)(10,0){11}{\circle*{1}}
\multiput(140,125)(10,0){1}{\circle{1.5}}
\multiput(20,15)(10,10){12}{\circle*{1.5}}

%\multiput(20,55)(10,10){8}{\multiput(0,0)(10,0){4}{\put(-1.18,-1.17){$*$}}}
%\multiput(20,45)(10,0){3}{\put(-1.18,-1.17){$*$}}%
%\multiput(20,35)(10,0){2}{\put(-1.18,-1.17){$*$}}%
%\multiput(20,25)(10,0){1}{\put(-1.18,-1.17){$*$}}%

\multiput(20,55)(10,10){8}{\put(15,0){\oval(32,2.1)}}
\put(20,45){\put(10,0){\oval(22,2.1)}}
\put(20,35){\put(5,0){\oval(12,2.1)}}
\put(20,25){\put(0,0){\oval(2.1,2.1)}}


\put(18,133){\line(1,0){117}}%
\put(18,132){\line(0,1){2}}%
\put(55,132){\line(0,1){2}}%
\put(95,132){\line(0,1){2}}%
\put(135,132){\line(0,1){2}}%
\put(70,135){\scriptsize $\textp{range}$}%

\put(-7,0){%
\put(0,14){\framebox{\small $\pi_0\to$}} \put(17,15){${}_{y_0}$}
\put(0,24){\framebox{\small $\pi_1\to$}} \put(17,25){${}_{y_1}$}
\put(0,34){\framebox{\small $\pi_2\to$}} \put(17,35){${}_{y_2}$}
\put(0,44){\framebox{\small $\pi_3\to$}} \put(17,45){${}_{y_3}$}
\put(0,54){\framebox{\small $\pi_0\to$}} \put(17,55){${}_{y_4}$}
\put(0,64){\framebox{\small $\pi_1\to$}} \put(17,65){${}_{y_5}$}
\put(0,74){\framebox{\small $\pi_2\to$}} \put(17,75){${}_{y_6}$}
\put(0,84){\framebox{\small $\pi_3\to$}} \put(17,85){${}_{y_7}$}
\put(0,94){\framebox{\small $\pi_0\to$}} \put(17,95){${}_{y_8}$}
\put(0,104){\framebox{\small $\pi_1\to$}} \put(17,105){${}_{y_{9}}$}%
\put(0,114){\framebox{\small $\pi_2\to$}} \put(17,115){${}_{y_{10}}$} %
\put(0,124){\framebox{\small $\pi_3\to$}} \put(17,125){${}_{y_{11}}$}%
}%

\put(0, -5){%
\multiput(18.7,4)(10,0){13}{$\uparrow$}%
\multiput(16.8,-2)(10,0){13}{\framebox(6.4,11){ }}%
\put(19,11){${}_{0}$} \put(18,-0.5){\small $\sigma_0$}
\put(29,11){${}_{1}$} \put(28,-0.5){\small $\sigma_1$}
\put(39,11){${}_{2}$} \put(38,-0.5){\small $\sigma_2$}
\put(49,11){${}_{3}$} \put(48,-0.5){\small $\sigma_3$}
\put(59,11){${}_{4}$} \put(58,-0.5){\small $\sigma_0$}
\put(69,11){${}_{5}$} \put(68,-0.5){\small $\sigma_1$}
\put(79,11){${}_{6}$} \put(78,-0.5){\small $\sigma_2$}
\put(89,11){${}_{7}$} \put(88,-0.5){\small $\sigma_3$}
\put(99,11){${}_{8}$} \put(98,-0.5){\small $\sigma_0$}
\put(109,11){${}_{9}$} \put(108,-0.5){\small $\sigma_1$}
\put(118,11){${}_{{10}}$} \put(118,-0.5){\small $\sigma_2$}
\put(128,11){${}_{{11}}$} \put(128,-0.5){\small $\sigma_3$}
\put(138,11){${}_{{12}}$} \put(138,-0.5){\small $\sigma_0$}
%\put(148,11){${}_{{13}}$} \put(148,1.5){$\sigma_1$}

\multiput(148,11.5)(2,0){3}{.}
}%


%\put(21,18){${}_{\sf equ}$} %\put(21,15){${}_{\sf tile}$}

\multiput(17.5,17)(10,10){12}{${}_{\textv{equ}}$}
\multiput(17.6,12.1)(10,10){12}{${}_{\textv{tile}}$}
\multiput(19,27.5)(10,10){11}{${}_{\textv{up}}$}
\multiput(17.5,57.5)(10,10){8}{${}_{\textv{equ}}$}
\multiput(17,52.0)(10,10){8}{${}_{\textv{right}}$}
\multiput(17.5,97)(10,10){4}{${}_{\textv{equ}}$}

%\put(59,17){${}_{\sf right}$}
\multiput(57.5,17.5)(10,10){9}{${}_{\textv{equ}}$}
%\put(59,20){${}_{\sf equ}$}
%\multiput(69,22.5)(10,10){9}{${}_{\sf right}$}
%\put(99,17){${}_{\sf right}$}
\multiput(97.5,17.5)(10,10){5}{${}_{\textv{equ}}$}
%\put(99,20){${}_{\sf equ}$}
%\multiput(109,23)(10,10){5}{${}_{\sf right}$}
\multiput(137.5,17.5)(10,10){1}{${}_{\textv{equ}}$}
%
{%
\linethickness{0.1pt}
\multiput(15,10)(10,10){12}{\put(0,0){\framebox(9.9,9.9){}}}
%
\multiput(27,10)(5,0){27}{\line(1,0){2}}%
\multiput(67,50)(5,0){19}{\line(1,0){2}}%
\multiput(107,90)(5,0){13}{\line(1,0){2}}%
\multiput(137,130)(5,0){9}{\line(1,0){2}}%
}%


\put(-10,0){%
%
\multiput(170,10)(10,40){3}{\put(0,0){\line(0,1){40}}\put(10,0){\line(0,1){40}}\multiput(0,0)(0,10){5}{\line(1,0){10}}}

{%
\linethickness{0.1pt}
%
\multiput(190,11)(0,5){8}{\line(0,1){2}}%
\multiput(200,11)(0,5){8}{\line(0,1){2}}%
\multiput(180,10)(0,10){5}{\multiput(2,0)(5,0){4}{\line(1,0){2}}}
}%

\put(175,60){\vector(0,1){15}}%
\put(185,100){\vector(0,1){15}}

%\put(174,13){$t_0$}\put(194,13){$t_1$}\put(194,93){$t_1$}

\put(185,25){\circle{2}}\put(186,25){\vector(1,0){4.5}}\put(185,26){\vector(0,1){7}}%
\put(191.2,25){${}_{\textv{right}}$}\put(183,35){${}_{\textv{up}}$}%
%
\put(185,65){\circle*{2}}\put(185,66){\vector(0,1){7}}%
\put(191.2,105){${}_{\textv{right}}$}\put(183,75){${}_{\textv{up}}$}%
%\put(186,65){\line(1,0){9}}\put(195,65){\vector(0,1){38}}
\put(186,74){\oval(18,18)[rb]}\put(195,74){\vector(0,1){29}}

\put(169,8){$\underbrace{\hspace*{2.2cm}}$}%
\put(170,0){\small $3\times 2^2$ corridor}
}%

\end{picture}
\end{center}
\caption{Satisfying $\varphi_{\mathcal{T}}$, $n = 2$, in a
tt-model based on space with $3\cdot 2^2$
points.}\label{figure:brcc8:slices}
\end{figure}


Our next aim is to write down formulas that could serve as
pointers to the up and right neighbours of a given pair in the
corridor (at this moment we do not bother about its top border).
Consider the formulas
%
\begin{align}
& \Box_F^+ \EQ\bigl(\textv{up},\; \nxt\textv{tile}\bigr), \\
%
& \Box_F^+ \EQ\bigl(\textv{right},\; \textv{equ} \sqcap \textv{no\_equ\_U\_tile} \bigr), \\
%
\label{eq:brcc8:corridor:rightfp}%
& \Box_F^+ \EQ\bigl(\textv{no\_equ\_U\_tile},\;
\nxt\textv{tile}\sqcup\nxt\bigl(\overline{\textv{equ}} \sqcap
\textv{no\_equ\_U\_tile}\bigr)\bigr),
\end{align}
%
where $\textv{up}$, $\textv{right}$ and $\textv{no\_equ\_U\_tile}$
are fresh spatial variables. We claim that, for all \mbox{$i,j<
m\cdot 2^n$,}
%
\begin{itemize}
\item $(\mathfrak{M},\auf
i,y_j\zu)\models\textv{up}$ \quad iff\quad $j = i + 1$;
%
\item $(\mathfrak{M}, \auf i,y_j\zu)\models\textv{right}$
\quad iff\quad $j = i + 2^n$.
\end{itemize}
%
The former is obvious. Let us prove the latter. To show that
$(\mathfrak{M}, \auf i,y_j\zu)\models\textv{right}$, for $j = i +
2^n$, we first observe that $(\mathfrak{M}, \auf
j,y_j\zu)\models\textv{equ}$ and $(\mathfrak{M}, \auf
i,y_j\zu)\models\textv{equ}$ by~\eqref{eq:brcc8:corridor:equ:con}.
It follows from $(\mathfrak{M}, \auf j,y_j\zu)\models\textv{tile}$
by \eqref{eq:brcc8:corridor:rightfp} that $(\mathfrak{M}, \auf j -
1,y_j\zu)\models\textv{no\_equ\_U\_tile}$. Then, applying
\eqref{eq:brcc8:corridor:rightfp} (from right to left)
sufficiently many times, we obtain $(\mathfrak{M}, \auf
i,y_j\zu)\models\textv{no\_equ\_U\_tile}$, $(\mathfrak{M}, \auf i
- 1,y_j\zu)\not\models\textv{no\_equ\_U\_tile}$, and so
$(\mathfrak{M}, \auf i,y_j\zu)\models\textv{right}$.

Conversely, suppose that $(\mathfrak{M}, \auf
i,y_j\zu)\models\textv{right}$ for some $y_j$. Then
$(\mathfrak{M}, \auf i,y_j\zu)\models\textv{equ}$ and,
by~\eqref{eq:brcc8:corridor:equ:con} (note that $i + 2^n <
(m+1)\cdot 2^n$, and so $\textp{count}$ is still true at $i +
2^n$),
%
\begin{align}\tag{$\ast\ast$}\label{eq:brcc8:corridor:rightfp:proof}
& (\mathfrak{M}, \auf i + 2^n,y_j\zu)\models\textv{equ}.
\end{align}
%
We have $(\mathfrak{M}, \auf
i,y_j\zu)\models\textv{no\_equ\_U\_tile}$. Then
applying~\eqref{eq:brcc8:corridor:rightfp} (from left to right)
sufficiently many times we arrive at $(\mathfrak{M}, \auf i + 2^n
- 1,y_j\zu)\models\textv{no\_equ\_U\_tile}$ which together
with~\eqref{eq:brcc8:corridor:rightfp:proof} gives $(\mathfrak{M},
\auf i + 2^n,y_j\zu)\models\textv{tile}$. But then $j = i + 2^n$.

It should be noted that at every time point the extension of
$\textv{no\_equ\_U\_tile}$ coincides with the extension of the
term $\overline{\textv{equ}} \until \textv{tile}$ on elements of
the sequence $y_0,\dots,y_{m\cdot2^n}$, and
that~\eqref{eq:brcc8:corridor:rightfp} is indeed the fixed point
characterisation of this $\until$ operator.

Finally, the formulas below ensure that every point of the
$m\times 2^n$-corridor is covered by at most one tile, $\auf 0,
0\zu$ is covered by $t_0$, $\auf m - 1, 0\zu$  by $t_1$, the top
and bottom sides are $\textit{white}$ and that the colours on
adjacent edges of adjacent tiles match:
%
\begin{align}\label{eq:brcc8:corridor:tileinit}
&  \bigwedge_{\begin{subarray}{c}t, t'\in T\\ t\ne t'\end{subarray}} %
  \Box_F^+ \neg\NE (\textv{t}\sqcap \textv{t}'), \\
%
&  \PT(\textv{tile},\; \textv{t}_0)\;\land\; \Box_F^+\Bigl(\bigl(\sigma_0\land\textp{range}\bigr)\land\neg\Diamond_F\bigl(\sigma_0 \land \textp{range}\bigr) \to %
  \PT\bigl(\textv{tile},\; \textv{t}_1\bigr)\Bigr),  \\
%
&  \Box_F^+ \Bigl(\sigma_{2^n - 1}\to
  \hspace*{-1.2em}\bigvee_{\begin{subarray}{c}t\in T\\ \textit{up}(t) = \textit{white}\end{subarray}}\hspace*{-1.2em} %
  \PT\bigl(\textv{tile},\;\textv{t}\bigr)\Bigr), \\
%
&  \Box_F^+ \Bigl(\sigma_0\to
  \hspace*{-1.5em}\bigvee_{\begin{subarray}{c}t\in T\\ \textit{down}(t) = \textit{white}\end{subarray}}\hspace*{-1.5em} %
  \PT\bigl(\textv{tile},\;\textv{t}\bigr)\Bigr), \\
%
& \hspace*{-2em} \bigwedge_{\begin{subarray}{c}t,t'\in T\\ \textit{up}(t)\ne \textit{down}(t')\end{subarray}}\hspace*{-1em} %
  \Box_F^+ \Bigl(\neg\sigma_{2^n - 1}\ \land\ \NE\textv{t}\ \to\
  \PT\bigl(\textv{up},\; \textv{no\_t$'\!\!$\_in\_future}\bigr)\Bigr),  \\
%
& \label{eq:brcc8:corridor:last}%
  \hspace*{-2em}\bigwedge_{\begin{subarray}{c}t,t'\in T\\ \textit{right}(t)\ne \textit{left}(t')\end{subarray}}\hspace*{-1.5em} %
   \Box_F^+ \Bigl(\NE\textv{t}\ \to\
  \PT\bigl(\textv{right},\; \textv{no\_t$'\!\!$\_in\_future}\bigr)\Bigr). %
\end{align}
%

Let $\varphi_{\mathcal{T}}$ be the conjunction
of~\eqref{eq:brcc8:corridor:count1}--\eqref{eq:brcc8:corridor:last}.
Suppose that $\varphi_{\mathcal{T}}$ holds at $0$ in
$\mathfrak{M}$. Then there is $m\in\mathbb{N}$ such that
$(\mathfrak{M}, m\cdot 2^n - 1)\models\textp{range}$ and, for
every $i \geq m\cdot 2^n$, $(\mathfrak{M},
i)\models\neg\textp{range}$. Then we define a map
$\boldsymbol{\tau}\colon m\times 2^n \to T$ by taking
%
\begin{equation*}
\boldsymbol{\tau}(k, j) = t \qquad\text{iff}\qquad (\mathfrak{M},
\auf i, y_i\zu) \models \textv{t} \text{ and } i = k\cdot 2^n + j.
\end{equation*}
%
We leave it to the reader to check that $\boldsymbol{\tau}$ is a
tiling of $m\times 2^n$, as required.

\smallskip

For the other direction, suppose that there is a tiling
$\boldsymbol{\tau}$ of the $m\times 2^n$-corridor by
$\mathcal{T}$, for some $m>0$. Then $\varphi_{\mathcal{T}}$ is
satisfied in the Aleksandrov tt-model $\mathfrak{M} = \auf \auf
\mathbb{N}, <\zu, \auf V,R\zu, \mathfrak{V}\zu$, where $V = \{
y_0, \dots, y_{m\cdot 2^n - 1}\}$, $R$ is the minimal reflexive
relation on $V$,
%
\begin{equation*}
\mathfrak{V}(\textv{t},i) = \{ y_i \in V \mid \boldsymbol{\tau}(k,
j) = t \textrm{ and } i = k\cdot 2^n + j \},
\end{equation*}
%
and the other variables of $\varphi_{\mathcal{T}}$ are interpreted
as shown in Fig.~\ref{figure:brcc8:slices}. Clearly,
$\mathfrak{M}$ satisfies \FSA{}. Moreover, $\varphi_{\mathcal{T}}$
is satisfiable in tt-models over finite flows of time iff it is
satisfiable in tt-models over $\auf\mathbb{N},<\zu$. Details are
left to the reader.
\end{proof}

\begin{remark}\label{remark:rcc8:expspace}
It may be of interest to note that the language used in the proof
above is rather limited. In fact, it is enough to extend the
PSPACE-complete logic $\PTL\circ\RCCE$ with predicates of the form
$\EQ(\varrho_1, \varrho_2\sqcup\varrho_3)$ (where the $\varrho_i$
are \emph{atomic} spatio-temporal region terms) to make it
EXPSPACE-hard. To show this, we transform the $\PTL\circ\BRCCE$
formula $\varphi_{\mathcal{T}}$ constructed above in the following
way. First, we take a fresh spatial variable $\textv{u}$ (denoting
`the universe') and add to $\varphi_{\mathcal{T}}$ the conjunct
$\Box_F^+\EQ(\textv{u}, \nxt \textv{u})$. Next, for every
spatio-temporal Boolean region term $\varrho$ of
$\varphi_{\mathcal{T}}$, we introduce a spatial variable
$\textv{neg\_}\varrho$ (`the complement of $\varrho$ with respect
to $\textv{u}$'), add to $\varphi_{\mathcal{T}}$ conjuncts
$\Box_F^+\EQ(\textv{u}, \varrho \sqcup \textv{neg\_}\varrho) \land
\Box_F^+\textsf{DC}(\varrho, \textv{neg\_}\varrho)$, and replace
every occurrence of $\overline{\varrho}$ in the resulting formula
with $\textv{neg\_}\varrho$. Finally, for every spatio-temporal
term of the form $\varrho_1\sqcap\varrho_2$, we introduce a fresh
spatial variable $\varrho_1\textv{\_and\_}\varrho_2$, add the
conjuncts
%
\begin{equation*}
\Box_F^+\PT(\varrho_1\textv{\_and\_}\varrho_2,\; \varrho_1) \;\land\; %
\Box_F^+\PT(\varrho_1\textv{\_and\_}\varrho_2,\; \varrho_2) \;\land\; %
\Box_F^+\PT(\varrho_1,\;\textv{neg\_}\varrho_2 \sqcup \varrho_1\textv{\_and\_}\varrho_2)   %
\end{equation*}
%
and replace occurrences of $\varrho_1\sqcap\varrho_2$ with
$\varrho_1\textv{\_and\_}\varrho_2$. One can readily see that (i)
the length of the resulting formula $\varphi'_{\mathcal{T}}$ is
linear in the length of $\varphi_{\mathcal{T}}$ and (ii)
$\varphi'_{\mathcal{T}}$ is satisfiable in a tt-model based on
$\auf\mathbb{N},<\zu$ (with \FSA{}) iff $\varphi_{\mathcal{T}}$ is
satisfiable in a tt-model based on $\auf\mathbb{N},<\zu$ (with
\FSA{}).
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection[Upper Complexity Bounds (I): Quasimodels for PTLxRC]{Upper Complexity Bounds (I): Quasimodels for $\PTL\times\RC$}\label{appendix:quasimodels}

In this appendix we define quasimodels for $\PTL\times\RC$ in the
spirit of the paper \cite{Hodkinsonetal00} in order to establish
the upper complexity bounds of Theorems~\ref{theorem:rc:2expspace}
and \ref{theorem:rc:2expspace:full}. We remind the reader that
spatio-temporal terms of $\PTL\times\RC$ are of the form:
%
\begin{align*}
\tau \quad ::= \quad & \varrho \quad|\quad \IB\varrho \quad|\quad
\overline{\tau}\quad|\quad \tau_1\sqcap\tau_2,\\ %
%
\varrho \quad ::= \quad & \CB\IB p \quad|\quad
\CB\IB\overline{\varrho} \quad|\quad
\CB\IB(\varrho_1\sqcap\varrho_2)\quad|\quad
\CB\IB(\varrho_1\until\varrho_2) \quad|\quad
\CB\IB(\varrho_1\since\varrho_2),
\end{align*}
%
and that $\PTL\circ\RC$ forms a sublanguage of
$\PTL\times\RC$---it differs from the latter only in the
definition of spatio-temporal \emph{region} terms:
%
\begin{equation*}
\varrho \quad ::= \quad \CB\IB p \quad|\quad
\CB\IB\overline{\varrho} \quad|\quad
\CB\IB(\varrho_1\sqcap\varrho_2)\quad|\quad \CB\IB\nxt\varrho. %
\end{equation*}

Let $\varphi$ be a $\PTL\times\RC$-formula. Recall from
p.~\pageref{def:subformulas:subterms} that by
$\textit{sub}\,\varphi$ we denote the set of all subformulas of
$\varphi$ and by $\textit{term}\,\varphi$ the set of all its
spatio-temporal terms including those of the form $\tau$ and
$\varrho$. A \emph{type} $\boldsymbol{t}$ for $\varphi$ is a
subset of $\textit{term}\,\varphi$ such that
%
\begin{itemize}\itemsep=0pt
\item for every
$\tau_1\sqcap\tau_2\in\textit{term}\,\varphi$,\qquad
$\tau_1\sqcap\tau_2\in\boldsymbol{t}$\quad iff\quad
$\tau_1\in\boldsymbol{t}$ and $\tau_2\in\boldsymbol{t}$;
%
\item for every
$\overline{\tau}\in\textit{term}\,\varphi$,\qquad
$\overline{\tau}\in\boldsymbol{t}$\quad iff\quad
$\tau\notin\boldsymbol{t}$.
\end{itemize}
%
Clearly, the number $\flat(\varphi)$ of different types for
$\varphi$ is bounded by $2^{|\textit{term}\,\varphi|}$.

A \emph{broom type} $\mathfrak{b}$ for $\varphi$ is a pair $\auf
\auf T,\leq\zu,\boldsymbol{t}\zu$, where $\auf T,\leq\zu$ is a
broom (with $T^0$ being its leaves) and $\boldsymbol{t}$ a
labelling function associating with each $x\in T$ a type
$\boldsymbol{t}(x)$ for $\varphi$ such that the following
conditions hold:
%
\begin{description}
\item[\textbf{(bt0)}] $\boldsymbol{t}(x)\ne\boldsymbol{t}(y)$,
for each pair of distinct points $x,y\in T^0$;
%
\item[\textbf{(bt1)}] for every $x\in T^0$,
\begin{itemize}\itemsep=0pt
\item for every
$\CB\IB(\varrho_1\sqcap\varrho_2)\in\textit{term}\,\varphi$,\quad
$\CB\IB(\varrho_1\sqcap\varrho_2)\in\boldsymbol{t}(x)$\quad
iff\quad $\varrho_1\in\boldsymbol{t}(x)$ and
$\varrho_2\in\boldsymbol{t}(x)$,
%
\item and for every
$\CB\IB\overline{\varrho}\in\textit{term}\,\varphi$,\quad
$\CB\IB\overline{\varrho}\in\boldsymbol{t}(x)$\quad iff\quad
$\varrho\notin\boldsymbol{t}(x)$;
\end{itemize}
%
\item[\textbf{(bt2)}] for every
$\IB\varrho\in\textit{term}\,\varphi$,\qquad
$\IB\varrho\in\boldsymbol{t}(x)$ \quad iff\quad
$\varrho\in\boldsymbol{t}(y)$ for every $y\in T$, $x \leq y$;
%
\item[\textbf{(bt3)}] for
every $\varrho\in\textit{term}\,\varphi$,\qquad
$\varrho\in\boldsymbol{t}(x)$ \quad iff\quad $\exists y\in T^0$
with $x \leq y$ and $\varrho\in\boldsymbol{t}(y)$.
\end{description}
%
Broom types $\mathfrak{b}_1 = \auf\auf
T_1,\leq_1\zu,\boldsymbol{t}_1\zu$ and $\mathfrak{b}_2 = \auf\auf
T_2,\leq_2\zu,\boldsymbol{t}_2\zu$ for $\varphi$ are said to be
\emph{isomorphic} if
%
\begin{itemize}
\item for every $x_1\in T_1^0$, there is $x_2\in T_2^0$ such that
$\boldsymbol{t}_1(x_1) = \boldsymbol{t}_2(x_2)$ and
%
\item for every $x_2\in T_2^0$, there is $x_1\in T_1^0$ such
that $\boldsymbol{t}_1(x_1) = \boldsymbol{t}_2(x_2)$.
\end{itemize}
%
Clearly, given two isomorphic broom types $\mathfrak{b}_1$ and
$\mathfrak{b}_2$, we also have $\boldsymbol{t}_1(r_1) =
\boldsymbol{t}_2(r_2)$, where $r_1$ and $r_2$ are the roots of
$\mathfrak{b}_1$ and $\mathfrak{b}_2$, respectively.

A \emph{quasistate} for $\varphi$ is a pair $\auf \boldsymbol{s},
\mathfrak{m}\zu$, where $\boldsymbol{s}$ is a Boolean-saturated
subset of $\textit{sub}\,\varphi$ and $\mathfrak{m}$ a disjoint
union $\auf\auf T,\leq\zu,\boldsymbol{t}\zu$ of broom types
$\mathfrak{b}_1,\dots,\mathfrak{b}_n$ for $\varphi$ such that the
following conditions hold:
%
\begin{description}
\item[\textbf{(qs0)}] $\mathfrak{b}_i$ and $\mathfrak{b}_j$ are not isomorphic,
for $i \ne j$;
%
\item[\textbf{(qs1)}] for every
$\Ubox\tau\in\textit{sub}\,\varphi$,\quad
$\Ubox\tau\in\boldsymbol{s}$\quad iff\quad
$\tau\in\boldsymbol{t}(x)$\ \ for every $x\in T$.
\end{description}
%
Clearly, the number $\sharp(\varphi)$ of quasistates for $\varphi$
is bounded by $2^{2^{\flat(\varphi)}}\cdot
2^{|\textit{sub}\,\varphi|}$.

Fix a flow of time $\mathfrak{F} = \auf W,<\zu$. A \emph{basic
structure} for $\varphi$ is a pair $\auf\mathfrak{F},
\boldsymbol{q}\zu$, where $\boldsymbol{q}$ is a function
associating with each $w\in W$ a quasistate $\boldsymbol{q}(w) =
\auf \boldsymbol{s}_w, \mathfrak{m}_w\zu$ for $\varphi$ such that,
for each $w\in W$,
%
\begin{itemize}\itemsep=0pt
\item for every $\psi_1\until\psi_2\in\textit{sub}\,\varphi$,
$\psi_1\until\psi_2\in \boldsymbol{s}_w$ iff there is $v
> w$ such that $\psi_2\in \boldsymbol{s}_v$ and $\psi_1\in
\boldsymbol{s}_u$ for all $u\in(w,v)$;
%
\item for every $\psi_1\since\psi_2\in\textit{sub}\,\varphi$,
$\psi_1\since\psi_2\in \boldsymbol{s}_w$ iff there is $v < w$ such
that $\psi_2\in \boldsymbol{s}_v$ and $\psi_1\in \boldsymbol{s}_u$
for all $u\in(v,w)$.
\end{itemize}
%
Let $\auf\mathfrak{F},\boldsymbol{q}\zu$ be a basic structure for
$\varphi$, where $\boldsymbol{q}(w) =
\auf\boldsymbol{s}_w,\mathfrak{m}_w\zu$ and
$\mathfrak{m}_w=\auf\auf T_w,\leq_w\zu,\boldsymbol{t}_w\zu$ for
$w\in W$. Denote by $T^0_w$ the set of all leaves in $\auf
T_w,\leq_w\zu$ and by $T^1_w$ the set of all roots of brooms in
it. A \emph{$1$-run through} $\auf\mathfrak{F},\boldsymbol{q}\zu$
is a function $r$ giving for each $w\in W$ a point $r(w)\in
T^1_w$; a \emph{coherent and saturated $0$-run through}
$\auf\mathfrak{F},\boldsymbol{q}\zu$ is a function $r$ giving for
each $w\in W$ a point $r(w)\in T^0_w$ such that the following
conditions hold:
%
\begin{itemize}\itemsep=0pt
\item for every $\CB\IB(\varrho_1\until\varrho_2)
\in\textit{term}\,\varphi$, $\CB\IB(\varrho_1\until\varrho_2)\in
\boldsymbol{t}_w(r(w))$ iff there is $v > w$ such that
$\varrho_2\in \boldsymbol{t}_v(r(v))$ and $\varrho_1\in
\boldsymbol{t}_u(r(u))$ for all $u\in(w,v)$;
%
\item for every $\CB\IB(\varrho_1\since\varrho_2)
\in\textit{term}\,\varphi$, $\CB\IB(\varrho_1\since\varrho_2)\in
\boldsymbol{t}_w(r(w))$ iff there is $v < w$ such that
$\varrho_2\in \boldsymbol{t}_v(r(v))$ and $\varrho_1\in
\boldsymbol{t}_u(r(u))$ for all $u\in(v,w)$.
\end{itemize}
%
Say that a quadruple $\mathfrak{Q} = \auf\mathfrak{F},
\boldsymbol{q}, \mathfrak{R}, \vartriangleleft\zu$ is a
\emph{quasimodel for $\varphi$ based on $\mathfrak{F}$} if
$\auf\mathfrak{F},\boldsymbol{q}\zu$ is a basic structure for
$\varphi$, $\mathfrak{R} = \mathfrak{R}^0\cup\mathfrak{R}^1$, with
$\mathfrak{R}^1$ being a set of 1-runs and $\mathfrak{R}^0$ a set
of coherent and saturated 0-runs through
$\auf\mathfrak{F},\boldsymbol{q}\zu$, and $\vartriangleleft$ the
reflexive closure of a subset of
$\mathfrak{R}^1\times\mathfrak{R}^0$ such that
%
\begin{description}
\item[\textbf{(qm2)}] $\exists w_0\in W$
$\varphi\in\boldsymbol{s}_{w_0}$;
%
\item[\textbf{(qm3)}] for every $w\in W$ and every
$x\in T_w$, there is $r\in\mathfrak{R}$ with $r(w)=x$;
%
\item[\textbf{(qm4)}] for all $r,r'\in\mathfrak{R}$, if $r\vartriangleleft
r'$ then $r(w)\leq_w r'(w)$ for all $w\in W$;
%
\item[\textbf{(qm5)}] for all $r\in \mathfrak{R}$, $w\in W$ and
$x\in T^0_w$, if $r(w)\leq_w x$ then there is
$r'\in\mathfrak{R}^0$ such that $r'(w) = x$ and $r
\vartriangleleft r'$.
\end{description}
%
A quasimodel $\mathfrak{Q}$ is said to be \emph{finitary} if the
set $\mathfrak{R}$ of runs is finite.


\begin{lemma}\label{lemma:rc:quasimodel}
A $\PTL\times\RC$-formula $\varphi$ is satisfiable in an
Aleksandrov tt-model based on a flow of time $\mathfrak{F}$ and a
\textup{(}finite\textup{)} disjoint union of
\textup{(}finite\textup{)} brooms iff there is a
\textup{(}finitary\textup{)} quasimodel for $\varphi$ based on
$\mathfrak{F}$.
\end{lemma}
%
\begin{proof}
$(\Leftarrow)$ Let $\varphi$ be a $\PTL\times\RC$-formula and
$\mathfrak{Q} =
\auf\mathfrak{F},\boldsymbol{q},\mathfrak{R},\vartriangleleft\zu$
a quasimodel for $\varphi$, where $\mathfrak{F}= \auf W, <\zu$ and
$\boldsymbol{q}(w) = \auf \boldsymbol{s}_w,\auf\auf
T_w,\leq_w\zu,\boldsymbol{t}_w\zu\zu$ for $w\in W$. We construct
an Aleksandrov tt-model $\mathfrak{M}=\auf
\mathfrak{F},\mathfrak{G},\mathfrak{V}\zu$ by taking
$\mathfrak{G}=\auf\mathfrak{R},\vartriangleleft\zu$ and, for each
spatial variable $p$ and $w\in W$,
%
\begin{equation*}
\mathfrak{V}(p,w) = \{ r \mid \CB\IB p \in
\boldsymbol{t}_w(r(w))\}.
\end{equation*}
%
Clearly, if $\mathfrak{Q}$ is finitary then $\mathfrak{G}$ is
finite. Thus, it remains to prove that $\varphi$ is satisfied in
$\mathfrak{M}$.

First, we show by induction on the construction of a \emph{region}
term $\varrho\in\textit{term}\,\varphi$ that, for every $w\in W$
and every $r\in\mathfrak{R}$,
%
\begin{equation}\label{eq:rc:quasimodel:region}
(\mathfrak{M},\auf w,r\zu)\models \varrho \qquad\text{iff}\qquad
\varrho\in\boldsymbol{t}_w(r(w)).
\end{equation}
%
The \emph{basis} of induction: $\varrho = \CB\IB p$. Let
$(\mathfrak{M},\auf w,r\zu)\models \varrho$. Then there is
$r'\in\mathfrak{R}$ such that $r \vartriangleleft r'$ and
$(\mathfrak{M},\auf w,r'\zu)\models \IB p$. By \textbf{(qm4)},
$r(w) \leq_w r'(w)$. Take any $y\in T^0_w$, $r'(w) \leq_w y$. By
\textbf{(qm5)}, there is a run $r''\in\mathfrak{R}^0$ such that
$r'\vartriangleleft r''$ and $r''(w) = y$. Then
$(\mathfrak{M},\auf w,r''\zu)\models p$ and, by the definition of
$\mathfrak{V}$, $\CB\IB p\in\boldsymbol{t}_w(r''(w))$ and, by
\textbf{(bt3)}, $\varrho\in\boldsymbol{t}_w(r(w))$.

Conversely, if $\varrho\in\boldsymbol{t}_w(r(w))$ then, by
\textbf{(bt3)}, there is $y\in T_w^0$ with $r(w) \leq_w y$ and
$\varrho\in\boldsymbol{t}_w(y)$. By \textbf{(qm5)}, there is
$r''\in\mathfrak{R}^0$, $r\vartriangleleft r''$, such that $r''(w)
= y$. Then $\CB\IB p\in\boldsymbol{t}_w(r''(w))$ and, by the
definition of $\mathfrak{V}$, $(\mathfrak{M},\auf w,r''\zu)\models
p$. Therefore, $(\mathfrak{M},\auf w,r\zu)\models \varrho$.

The induction \emph{steps} for $\varrho =
\CB\IB\overline{\varrho_1}$, $\CB\IB(\varrho_1\sqcap\varrho_2)$,
$\CB\IB(\varrho_1\until\varrho_2)$ and
$\CB\IB(\varrho_1\since\varrho_2)$ are similar, but instead of the
definition of $\mathfrak{V}$, we use $\textbf{(bt1)}$ for the
cases of the Booleans and coherence and saturatedness of $r''$ for
the cases of temporal operators.

Next, we extend~\eqref{eq:rc:quasimodel:region} to arbitrary
spatio-temporal terms $\tau\in\textit{term}\,\varphi$.

\emph{Case} $\tau=\IB\varrho$. Suppose that $(\mathfrak{M},\auf
w,r\zu)\models\IB\varrho$. Take any $y \in T_w$, $r(w) \leq_w y$.
If $y\in T^0_w$ then, by \textbf{(qm5)}, there is
$r'\in\mathfrak{R}^0$ such that $r \vartriangleleft r'$ and $r'(w)
= y$. If $y\notin T^0_w$ then clearly $y = r(w)$ and take $r'= r$.
We have $(\mathfrak{M},\auf w,r'\zu)\models\varrho$, which, by IH,
implies $\varrho\in\boldsymbol{t}_w(r'(w))$. Therefore,
$\varrho\in\boldsymbol{t}_w(y)$ for every $y \geq_w r(w)$ and, by
\textbf{(bt2)}, $\IB\varrho\in\boldsymbol{t}_w(r(w))$.

Conversely, if $\IB\varrho\in\boldsymbol{t}_w(r(w))$ then, by
\textbf{(bt2)}, $\varrho\in\boldsymbol{t}_w(y)$, for every $y
\geq_w r(w)$. Take any run $r'\in\mathfrak{R}$ such that $r
\vartriangleleft r'$. By \textbf{(qm4)}, $r(w) \leq_w r'(w)$, and
so $\varrho\in\boldsymbol{t}_w(r'(w))$, from which, by IH,
$(\mathfrak{M},\auf w,r'\zu)\models\varrho$. Hence,
$(\mathfrak{M},\auf w,r\zu)\models\IB\varrho$.


\emph{Cases} $\tau = \tau_1\sqcap\tau_2$ and $\overline{\tau_1}$
follow from IH by the definition of type.

\smallskip
%
Finally, we show by induction on the construction of
$\psi\in\textit{sub}\,\varphi$ that, for every $w\in W$,
%
\begin{equation}\label{eq:ptlrc:correctness}
(\mathfrak{M},w)\models \psi \qquad\text{iff}\qquad
\psi\in\boldsymbol{s}_w.
\end{equation}

\emph{Case} $\psi=\Ubox\tau$. Suppose
$(\mathfrak{M},w)\models\Ubox\tau$. Take any $x\in T_w$. By
\textbf{(qm3)}, there is $r\in\mathfrak{R}$ such that $r(w) = x$.
Then $(\mathfrak{M},\auf w,r\zu)\models\tau$ and, by IH,
$\tau\in\boldsymbol{t}_w(r(w))$. Therefore, by~\textbf{(qs1)},
$\Ubox\tau\in\boldsymbol{s}_w$. Conversely, let
$\Ubox\tau\in\boldsymbol{s}_w$. Take any run $r\in\mathfrak{R}$.
By \textbf{(qs1)}, we have $\tau\in\boldsymbol{t}_w(r(w))$, from
which, by IH, $(\mathfrak{M},\auf w,r\zu)\models\tau$. Hence,
$(\mathfrak{M}, w)\models\Ubox\tau$.

\emph{Cases} $\psi = \psi_1\land\psi_2$ and $\neg\psi_1$ follow
from IH by the Boolean-saturatedness of the $\boldsymbol{s}_w$.

It follows from \eqref{eq:ptlrc:correctness} and \textbf{(qm2)}
that $\varphi$ is satisfiable in $\mathfrak{M}$.

\medskip
%
($\Rightarrow$) Let $\varphi$ be a $\PTL\times\RC$-formula and
suppose that $\varphi$ is satisfied in an Aleksandrov tt-model
$\mathfrak{M} = \auf\mathfrak{F},\mathfrak{G},\mathfrak{V}\zu$,
where $\mathfrak{F}=\auf W,<\zu$ and $\mathfrak{G} = \auf
\Delta,\leq\zu$ is a disjoint union of brooms.

Denote by $\Delta^0$ and $\Delta^1$ the leaves and the roots of
brooms in $\mathfrak{G}$, respectively. With every pair $\auf
w,x\zu\in W\times\Delta$ we associate the type
%
\begin{equation*}
\boldsymbol{t}(w,x) = \{\tau\in\textit{term}\,\varphi \mid
(\mathfrak{M},\auf w,x\zu)\models \tau\}.
\end{equation*}
%
Fix a $w\in W$ and define a binary relation on $\Delta$ as
follows. For $x,x'\in\Delta^0$, let $x \sim_w x'$ iff
$\boldsymbol{t}(w,x) = \boldsymbol{t}(w,x')$ and, for $z,z'\in
\Delta^1$, let $z\sim_w z'$ iff the brooms generated by $z$ and
$z'$ are isomorphic, i.e.,
%
\begin{align*}
& \forall x\in\Delta^0\, (z \leq x \;\to\; \exists x'\in \Delta^0\, (z' \leq x' \;\land\; x\sim_w x'))\land{} \\
& \forall x'\in\Delta^0\, (z' \leq x' \;\to\; \exists x\in \Delta^0\, (z \leq x \;\land\; x\sim_w x')).% \\
\end{align*}
%
Clearly, $\sim_w$ is an equivalence relation on $\Delta$. Denote
by $[x]_w$ the $\sim_w$-equivalence class of $x$ and define a map
$f_w$ by taking, for each $x\in \Delta$,
%
\begin{equation*}
f_w(x) = \begin{cases}[x]_w, & x\in\Delta^1,\\
\auf [z]_w, [x]_w\zu, & x\in\Delta^0 \quad\text{and}\quad z\in\Delta^1\quad\text{such that }z\leq x.\end{cases} %
\end{equation*}
%
Since $\mathfrak{G}$ is a disjoint union of brooms, $f_w$ is
well-defined. Now put
%
\begin{align*}
& T_w = \{f_w(x) \mid x\in\Delta\},\\
& u \leq_{w} v \quad\text{iff}\quad \exists x,y\in \Delta \quad \text{such that}\ x \leq y,\  u = f_w(x)\ \text{ and } v = f_w(y),\\
%& f_w(x) \leq_{w,z} f_w(y) \quad\text{iff}\quad x \leq y, \qquad \text{for } x,y\in\Delta |_z \\
& \boldsymbol{t}_w(f_w(x))  = \boldsymbol{t}(w,x),\quad \text{for } x\in\Delta. %
\end{align*}
%
By definition of $f_w$, $\auf T_w,\leq_w\zu$ is a union of brooms
and $\boldsymbol{t}_{w}$ is well-defined. Consider the structure
$\auf \boldsymbol{s}_w,\mathfrak{m}_w\zu$, where
%
\begin{equation*}
\mathfrak{m}_w = \auf\auf T_w,\leq_w\zu,\boldsymbol{t}_w\zu%
\qquad\text{and}\qquad%
\boldsymbol{s}_w = \{\psi\in\textit{sub}\,\varphi \mid (\mathfrak{M},w)\models\psi\}. %
\end{equation*}
%
It is readily seen that for each of the brooms of $\mathfrak{m}_w$
we have \textbf{(bt0)} and that $\mathfrak{m}_w$ satisfies
\textbf{(qs0)}. Moreover, as $f_w$ is a p-morphism from
$\auf\Delta,\leq\zu$ onto $\auf T_w,\leq_w\zu$, we also have
\textbf{(bt1)}--\textbf{(bt3)} and \textbf{(qs1)}. So, by taking
$\boldsymbol{q}(w) = \auf \boldsymbol{s}_w,\mathfrak{m}_w\zu$ for
each $w\in W$ we obtain a basic structure
$\auf\mathfrak{F},\boldsymbol{q}\zu$ for $\varphi$ satisfying
\textbf{(qm2)}.

It remains to define appropriate runs through
$\auf\mathfrak{F},\boldsymbol{q}\zu$. For $k=0,1$, let
$\mathfrak{R}^k$ be the set of all maps $r\colon w \mapsto f_w(x)$
for $x\in \Delta^k$. Clearly, $\mathfrak{R}^1$ and
$\mathfrak{R}^0$ are sets of 1- and coherent and saturated 0-runs,
respectively. Put $\mathfrak{R} =
\mathfrak{R}^0\cup\mathfrak{R}^1$ and for $r,r'\in\mathfrak{R}$,
$r\vartriangleleft r'$ iff $r(w) \leq_w r'(w)$ for all $w\in W$.
Then \textbf{(qm4)} holds by definition. Let $v\in W$ and $y \in
T_v$. Then there is $x\in\Delta$ such that $f_v(x) = y$. Clearly,
$\mathfrak{R}$ contains the run $r\colon w \mapsto f_w(x)$, which
proves \textbf{(qm3)}. Finally, let $r\in\mathfrak{R}$, $v\in W$
and $y \in T^0_v$ be such that $r(v) \leq_v y$. There are some
$z,x\in\Delta$ such that $f_w(z) = r(w)$, for every $w\in W$, and
$f_v(x) = y$. We clearly have $z\leq x$ and $x\in\Delta^0$. Then
take the run $r'\colon w\mapsto f_w(x)$. By definition, $r
\vartriangleleft r'$, which proves \textbf{(qm5)}. Thus,
$\mathfrak{Q} =
\auf\mathfrak{F},\boldsymbol{q},\mathfrak{R},\vartriangleleft\zu$
is a quasimodel for $\varphi$. Note that if $\mathfrak{G}$ is
finite then $\mathfrak{R}$ is finite as well and therefore,
$\mathfrak{Q}$ is finitary.
\end{proof}


We are now in a position to establish the upper complexity bounds
of the satisfiability problem for $\PTL\times\RC$- and
$\PTL\circ\RC$-formulas in tt-models based on
$\auf\mathbb{N},<\zu$, $\auf\mathbb{Z},<\zu$ or arbitrary finite
flows of time.

\begin{proof}[\proofof{Theorem~\ref{theorem:rc:2expspace}, upper bound}]
We consider the cases of $\auf\mathbb{N},<\zu$ and
$\auf\mathbb{Z},<\zu$. The case of arbitrary finite flows of time
and that of tt-models with \FSA{} and based on $\auf\mathbb{N},<\zu$
and $\auf\mathbb{Z},<\zu$ will follow from
Theorem~\ref{theorem:rc:2expspace:full}.

One can readily check that as for the propositional temporal logic
$\PTL$, we have the following polynomial reductions for
$\PTL\circ\RC$:
%
\begin{itemize}
\item satisfiability in tt-models based on
$\auf\mathbb{Z},<\zu$ can be polynomially reduced to
satisfiability in tt-models based on $\auf\mathbb{N},<\zu$;
%
\item satisfiability in tt-models
based on $\auf\mathbb{N},<\zu$ can be polynomially reduced to
satisfiability of formulas without past-time temporal operators.
\end{itemize}
%
So, in what follows we consider the simplest case of the
satisfiability problem, that is for $\PTL\circ\RC$-formulas
without past-time temporal operators in tt-models based on
$\auf\mathbb{N},<\zu$.

We present a nondeterministic {\cc 2EXPSPACE} satisfiability
checking algorithm which is similar to that of Sistla and
Clarke~\citeyear{Sistla&Clarke85}. First, one can prove (with the
help of Lemmas~\ref{rc:simula2}~(ii) and
\ref{lemma:rc:quasimodel}) an analogue of \cite[Theorem
24]{Hodkinsonetal00} which states that a $\PTL\circ\RC$-formula
$\varphi$ is satisfiable in tt-model based on
$\auf\mathbb{N},<\zu$ iff there are $l_1,l_2\in\mathbb{N}$ such
that
%
\begin{equation*}
l_1 \leq \sharp(\varphi),\qquad 0\ <\ l_2\ \leq\
|\textit{term}\,\varphi| \cdot \left(2^{\flat(\varphi)}\right)^2
\cdot \sharp(\varphi) + \sharp(\varphi)
\end{equation*}
%
and a `balloon'-like quasimodel
$\mathfrak{Q}=\auf\auf\mathbb{N},<\zu,\boldsymbol{q},\mathfrak{R},\vartriangleleft\zu$
for $\varphi$ with $\boldsymbol{q}(l_1 + n) =
\boldsymbol{q}(l_1+l_2+n)$ for every $n\in\mathbb{N}$. Although
Theorem 24 of \cite{Hodkinsonetal00} was proved for the monodic
fragment of first-order temporal logic, the basic idea of
extracting a `balloon'-like quasimodel from an arbitrary one works
for $\PTL\circ\RC$ as well. The only difference is that
quasistates now are more complex: they can be regarded as sets of
sets of types for $\varphi$ (not just sets of types) and thus,
both $l_1$ and $l_2$ are triple exponential in the length
$\ell(\varphi)$ of $\varphi$. Then a quasimodel $\mathfrak{Q}$ can
be guessed in 2EXPSPACE by an algorithm which is very similar to
that in the proof of \cite[Theorem 4.1]{Hodkinsonetal03}.
\end{proof}


\begin{proof}[\proofof{Theorem~\ref{theorem:rc:2expspace:full}, upper bound}]
The proof is similar to that of
Theorem~\ref{theorem:rc:2expspace}. Again, one can show that all
the cases are polynomially reducible to the case of satisfiability
of $\PTL\times\RC$-formulas without past-time temporal operators
in tt-models with \FSA{} and based on $\auf\mathbb{N},<\zu$. To
take the \FSA{} into account, we can prove (using
Lemmas~\ref{rc:simula2}~(i) and \ref{lemma:rc:quasimodel})
analogues of Theorems 29 and 35 of \cite{Hodkinsonetal00} which
state that a $\PTL\times\RC$-formula $\varphi$ is satisfiable in a
tt-model with \FSA{} and based on $\auf\mathbb{N},<\zu$ iff there
is a \emph{finitary} `balloon'-like quasimodel for $\varphi$ based
on $\auf\mathbb{N},<\zu$. The condition of finiteness for the set
of runs can also be ensured by an algorithm similar to that of
Theorem~\ref{theorem:rc:2expspace}.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection[Upper Complexity Bounds (II): Embedding into
First-Order Temporal Logic]{Upper Complexity Bounds
(II):\\\mbox{}\hfill Embedding into First-Order Temporal
Logic}\label{appendix:embedding}

In this appendix we introduce the \emph{first-order temporal
language} $\QTL$ and  use some known complexity results for
fragments of $\QTL$ to obtain upper complexity bounds for
spatio-temporal logics based on $\RCm$ (and therefore, on
$\BRCCE$).

The alphabet of $\QTL$ consists of individual variables $x_1, x_2,
\dots$, predicate symbols $P_1, P_2, \dots$, each of which is of
some fixed arity, the Booleans, the universal $\forall x$ and
existential $\exists x$ quantifiers for each variable $x$, and the
temporal operators $\until$, $\since$ (with their derivatives
$\nxt$, $\Diamond_F$, $\Box_F$, etc.). Note that our language
contains neither constant symbols nor equality (we simply do not
need them to obtain our complexity results).

$\QTL$ is interpreted in \emph{first-order temporal models} of the
form $\mathfrak{M} = \auf \mathfrak{F}, D, I\zu$, where
$\mathfrak{F} = \auf W, < \zu$ is a flow of time, $D$ a nonempty
set, the \emph{domain} of $\mathfrak{M}$, and $I$ a function
associating with every moment of time $w\in W$ a first-order
structure
%
\begin{equation*}
I(w) = \auf D, P_0^{I(w)}, P_1^{I(w)}, \dots \zu,
\end{equation*}
%
the \emph{state} of $\mathfrak{M}$ at moment $w$, where each
$P_i^{I(w)}$ is a relation on $D$ of the same arity as $P_i$. An
\emph{assignment} in $D$ is a function $\mathfrak{a}$ from the set
of individual variables to $D$. Given such an assignment and a
$\QTL$-formula $\varphi$, we define the \emph{truth-relation}
$(\mathfrak{M}, w)\models^\mathfrak{a}\varphi$ by taking
%
\begin{itemize}
\item $(\mathfrak{M}, w)\models^\mathfrak{a}P_i(x_1, \dots, x_m)$
iff $\auf \mathfrak{a}(x_1), \dots, \mathfrak{a}(x_m)\zu\in
P_i^{I(w)}$,
\item $(\mathfrak{M}, w)\models^\mathfrak{a}\forall x\,\psi$ iff
$(\mathfrak{M}, w)\models^{\mathfrak{a}'}\psi$, for every
assignment $\mathfrak{a}'$ in $D$ which differ from $\mathfrak{a}$
only on $x$,
\end{itemize}
%
plus the standard clauses for the Booleans and temporal operators.
We say that a $\QTL$-formula $\varphi$ is \emph{satisfied} in
$\mathfrak{M}$ if $(\mathfrak{M}, w)\models^\mathfrak{a}\varphi$
for some $w\in W$ and some assignment $\mathfrak{a}$ in $D$. If
all free variables of $\varphi$ are among $x_1, \dots, x_m$, then
instead of $(\mathfrak{M},w)\models^{\mathfrak{a}}\varphi$ we
often write $(\mathfrak{M},w)\models\varphi[d_1,\dots,d_m]$, where
$d_i = \mathfrak{a}(x_i)$ for all $i$, $1 \leq i \leq m$.

Denote by $\QTL^1$ the \emph{one-variable fragment of $\QTL$},
i.e., the set of all $\QTL$-formulas which contain at most one
individual variable, say, $x$. Without loss of generality we may
assume that all predicate symbols of $\QTL^1$ are at most unary.

Now we define an embedding of spatio-temporal languages based on
$\RCm$ into $\QTL^1$. Recall that, by Lemma~\ref{rc:simula3} (i),
if a $\PTL\times\RCm$-formula $\varphi$ of width $n$ is satisfied
in a tt-model with \FSA{} then $\varphi$ is also satisfiable in an
Aleksandrov tt-model based on the same flow of time and a finite
\emph{disjoint union of $n$-brooms}. Similarly, if a
$\PTL\circ\RCm$-formula $\varphi$ of width $n$ is satisfiable then
$\varphi$ is also satisfiable in an Aleksandrov tt-model based on
the same flow of time and possibly infinite \emph{disjoint union
of $n$-brooms}.

To cover both cases, let $\varphi$ be a $\PTL\times\RCm$-formula
of width $n$. We show how to construct a $\QTL^1$-formula
$\varphi^{\dagger_n}$ of length linear in $\ell(\varphi)$ such
that every Aleksandrov tt-model based on a (finite) union of
$n$-brooms satisfying $\varphi$ gives rise to a first-order
temporal model (with finite domain, respectively) satisfying
$\varphi^{\dagger_n}$ and vice versa. Thus, we polynomially reduce
the satisfiability problem for spatio-temporal languages to that
for $\QTL^1$.

%
\begin{figure}
\begin{center}
\setlength{\unitlength}{.08cm}
\begin{picture}(180,40)
%
%\put(0,0){\line(1,0){180}}\put(0,0){\line(0,1){20}}%
%
\put(1,31){\framebox(50,8){}}
\put(5,35){%
\put(0,0){\circle*{2.5}}\put(4,-2){$= \CB\IB p$}%
\put(25,0){\circle{2.5}}\put(29,-2){$= \overline{\CB\IB p}$}%
}%
%
\put(0,4){%
%
\put(108,3){%
%\put(39,2){\Large $\mathfrak{b}$}%
\put(0,10){$P_{j1}[d_\mathfrak{b}]$}%
\put(14,10){$\neg P_{j2}[d_\mathfrak{b}]$}%
\put(32,10){\dots}%
\put(37,10){$\neg P_{jn-1}[d_\mathfrak{b}]$}%
\put(60,10){$P_{jn}[d_\mathfrak{b}]$}%
\put(37,12){\oval(76,25)}%
%\put(36,5){\circle*{2}}
}%
%
\put(92,10){\Large $\Longrightarrow$}%
\put(25,-2){\Large $\mathfrak{b}$}%
\put(138,-6){{\Large $d_\mathfrak{b}$}}%
%
\put(-18,-10){%
\put(40,25){\circle*{2.5}}%
\put(56,25){\circle{2.5}}%
\put(84,25){\circle{2.5}}%
\put(100,25){\circle*{2.5}}%
\put(70,10){\circle*{2.5}}%
\put(71,11.5){\vector(1,1){12}}%
\put(72,11){\vector(2,1){26}}%
\put(69,11.5){\vector(-1,1){12}}%
\put(68,11){\vector(-2,1){26}}%
%
\put(18,8){depth 1}%
\put(18,23){depth 0}%
\put(68,25){\dots}%
\put(70,5){$x^0_{\mathfrak{b}}$}%
\put(37,29){$x^1_{\mathfrak{b}}$}%
\put(53,29){$x^2_{\mathfrak{b}}$}%
\put(80,29){$x^{n-1}_{\mathfrak{b}}$}%
\put(98,29){$x^n_{\mathfrak{b}}$}%
%
}%
}%
\end{picture}
\end{center}
\caption{Representing $n$-broom $\mathfrak{b}$ with region $\CB\IB
p_j$ by a point in a first-order model.}\label{fig:nbroom3}
\end{figure}
%

Suppose that $\varphi$ is satisfied in an Aleksandrov tt-model
$\mathfrak{M}=\auf\mathfrak{F},\mathfrak{G},\mathfrak{V}\zu$,
where $\mathfrak{F}=\auf W,<\zu$ and $\mathfrak{G}$ is a (finite
or infinite) disjoint union of $n$-brooms. With every $n$-broom
$\mathfrak{b}$ of $\mathfrak{G}$ we associate an element
$d_\mathfrak{b}$ of the first-order domain $D$. Then, for every
spatial variable $p_j$ in $\varphi$, we fix $n$ different unary
predicate symbols $P_{j1}(x), \dots, P_{jn}(x)$ with the following
meaning: $P_{ji}(x)$ is true on $d_\mathfrak{b}\in D$ at moment
$w\in W$ iff the $i$-th leaf of $\mathfrak{b}$
($x^i_{\mathfrak{b}}$ in Fig.~\ref{fig:nbroom3}) belongs to region
$\CB\IB{p}$ at $w$. Define $n$ distinct translations
$\cdot^{\dagger^i_n}$, $1 \leq i \leq n$, encoding the truth
values of spatio-temporal \emph{region} terms of $\varphi$ on
leaves of $\mathfrak{G}$ by taking, for a spatial variable $p_j$
and terms $\varrho_1$ and $\varrho_2$,
%
\begin{gather*}
(\CB\IB p_j)^{\dagger^i_n} = P_{ji}(x), \qquad\quad %\\
(\CB\IB\overline{\varrho_1})^{\dagger^i_n}  = \neg(\varrho_1)^{\dagger^i_n}, \qquad\quad %\\
(\CB\IB(\varrho_1\sqcap\varrho_2))^{\dagger^i_n} = (\varrho_1)^{\dagger^i_n}\land (\varrho_2)^{\dagger^i_n}, \\
%(\CB\IB\nxt \varrho_1)^{\dagger i} & = \nxt(\varrho_1)^{\dagger i}, \\
(\CB\IB(\varrho_1\until\varrho_2))^{\dagger^i_n} = (\varrho_1)^{\dagger^i_n}\until (\varrho_2)^{\dagger^i_n}, \qquad\qquad\qquad %\\
(\CB\IB(\varrho_1\since\varrho_2))^{\dagger^i_n} = (\varrho_1)^{\dagger^i_n}\since (\varrho_2)^{\dagger^i_n}. %
\end{gather*}
%
Next we extend these $n$ translations to \emph{arbitrary}
spatio-temporal terms of $\varphi$. First we introduce a
translation $\cdot^{\dagger^0_n}$ to encode the truth value of
arbitrary spatio-temporal terms in the roots of the $n$-brooms of
$\mathfrak{G}$: for a region term $\varrho$, let
%
\begin{equation*}
(\varrho)^{\dagger^0_n} = \bigvee_{k = 1}^n (\varrho)^{\dagger^k_n}. %
\end{equation*}
%
The formula above shows, in particular, that $\cdot^{\dagger^0_n}$
is redundant for \emph{region} terms since their truth values in
the roots can be `computed' as defined by $\cdot^{\dagger^0_n}$.
For a spatio-temporal term of the form $\IB\varrho$, where
$\varrho$ is a region term, we take
%
\begin{equation*}
(\IB\varrho)^{\dagger^0_n} = \bigwedge_{k = 1}^n(\varrho)^{\dagger^k_n}\qquad\text{and}\qquad %
(\IB\varrho)^{\dagger^i_n} = (\varrho)^{\dagger^i_n}\quad\text{for all } i,\ 1 \leq i \leq n, %
\end{equation*}
%
and then, for spatio-temporal terms $\tau_1$ and $\tau_2$,\!\footnote{%
%
For brevity, in this definition we follow the syntax of
$\PTL\times\RC$ rather than $\PTL\times\RCm$.
}%
%
\begin{equation*}
(\overline{\tau_1})^{\dagger^i_n} = \neg(\tau_1)^{\dagger^i_n}\qquad\text{and}\qquad %
(\tau_1\sqcap\tau_2)^{\dagger^i_n} = (\tau_1)^{\dagger^i_n}\land (\tau_2)^{\dagger^i_n} %
\quad\text{for all } i,\  0 \leq i \leq n.  %
\end{equation*}
%
Finally, we define the translation $\cdot^{\dagger_n}$ of
subformulas of $\varphi$: for a spatio-temporal term $\tau$,
%
\begin{equation*}
(\Ubox\tau)^{\dagger_n} =  \forall  x\,(\tau)^{\dagger^0_n} \;\;\land\quad \bigwedge_{k = 1}^n \forall  x\,(\tau)^{\dagger^k_n}%
\end{equation*}
%
and, for spatio-temporal formulas $\psi_1$ and $\psi_2$,
%
\begin{align*}
(\neg\psi_1)^{\dagger_n} &= \neg\psi_1^{\dagger_n}, & %
(\psi_1\land\psi_2)^{\dagger_n} &= \psi_1^{\dagger_n}\land\psi_2^{\dagger_n}, \\
(\psi_1\until\psi_2)^{\dagger_n} & = \psi_1^{\dagger_n}\until\psi_2^{\dagger_n}, & %
(\psi_1\since\psi_2)^{\dagger_n} &= \psi_1^{\dagger_n}\since\psi_2^{\dagger_n}, %
\end{align*}
%
Clearly, the length of $\varphi^{\dagger_n}$ is linear in both $n$
and $\ell(\varphi)$.

\begin{lemma}\label{lemma:rc:embedding}
A $\PTL\times\RCm$-formula $\varphi$ of width $n$ is satisfiable
in an Aleksandrov tt-model based on a \textup{(}finite\textup{)}
disjoint union of $n$-brooms iff $\varphi^{\dagger_n}$ is
satisfiable in a first-order temporal model \textup{(}with a
finite domain\textup{)} based on the same flow of time.
\end{lemma}
%
\begin{proof}
$(\Rightarrow)$ Suppose that $\varphi$ is satisfied in an
Aleksandrov tt-model $\mathfrak{M} = \auf \mathfrak{F},
\mathfrak{G}, \mathfrak{V}\zu$, where $\mathfrak{F} = \auf W,
<\zu$, $\mathfrak{G} = \auf V, R\zu$ is a disjoint union of
$n$-brooms $\mathfrak{b} = \auf W_\mathfrak{b},
R_\mathfrak{b}\zu$, $W_\mathfrak{b} = \{ x^0_\mathfrak{b},
x^1_\mathfrak{b}, \dots, x^n_\mathfrak{b}\}$ and $R_\mathfrak{b}$
is the reflexive closure of $\{ \auf x^0_\mathfrak{b},
x^1_\mathfrak{b}\zu, \dots, \auf x^0_\mathfrak{b},
x^n_\mathfrak{b}\zu\}$ (see Fig.~\ref{fig:nbroom3}).

Construct a first-order temporal model $\mathfrak{N} = \auf
\mathfrak{F}, D, I\zu$ by taking $D$ to be the set of all
$d_\mathfrak{b}$ for $n$-brooms $\mathfrak{b}$ in $\mathfrak{G}$
and, for every $w\in W$,
%
\begin{equation*}
I(w) = \auf D, P_{11}^{I(w)}, \dots, P_{1n}^{I(w)}, P_{21}^{I(w)},
\dots, P_{2n}^{I(w)}, \dots \zu,
\end{equation*}
%
where for each spatial variable $p_j$ in $\varphi$ and each $i$,
$1 \leq i \leq n$,
%
\begin{equation*}
P_{ji}^{I(w)} = \{ d_\mathfrak{b}\in D\mid (\mathfrak{M},\auf w,
x^i_\mathfrak{b}\zu)\models p_j \}.
\end{equation*}
%
Note that $D$ is finite whenever $\mathfrak{G}$ is finite.

Now, by induction on the construction of a spatio-temporal
\emph{region} term $\varrho$ of $\varphi$, it can easily be shown
that for every $w\in W$, every $n$-broom $\mathfrak{b}$ in
$\mathfrak{G}$ and every $i$, $1 \leq i \leq n$,
%
\begin{equation}\label{eq:forkequi}
(\mathfrak{N}, w)\models(\varrho)^{\dagger^i_n}[d_\mathfrak{b}] %
\quad \text{iff} \quad %
(\mathfrak{M},\auf w,x^i_\mathfrak{b}\zu)\models\varrho. %
\end{equation}
%
Next, \eqref{eq:forkequi} can be extended to \emph{arbitrary}
spatio-temporal terms $\tau$ of $\varphi$ and $i$, $0 \leq i \leq
n$:
%
\begin{equation}\label{eq:forkequi:terms}
(\mathfrak{N}, w)\models(\tau)^{\dagger^i_n}[d_\mathfrak{b}] %
\quad \text{iff} \quad %
(\mathfrak{M},\auf w,x^i_\mathfrak{b}\zu)\models\tau. %
\end{equation}
%
The cases for $i$, $1 \leq i \leq n$, trivially follow
from~\eqref{eq:forkequi} and the fact that leaves have no
successors but themselves. Consider now $i = 0$. The case $\tau =
\varrho$ holds simply because region terms are interpreted by
regular closed sets:
%
\begin{equation}\label{eq:forkequi:rc}
(\mathfrak{M},\auf w,x^0_\mathfrak{b}\zu)\models\varrho%
\qquad \text{iff} \qquad %
(\mathfrak{M},\bigl\langle w,x^k_\mathfrak{b}\bigr\rangle)\models\varrho, \quad \text{for some } k,\ 1 \leq k \leq n, %
\end{equation}
%
If $\tau = \IB\varrho$ then, on one hand,
%
\begin{equation*}
(\mathfrak{M},\auf w,x^0_\mathfrak{b}\zu)\models\IB\varrho%
\qquad \text{iff}\qquad %
(\mathfrak{M},\bigl\langle w,x^k_\mathfrak{b}\bigr\rangle)\models\varrho,\quad \text{for all } k,\ 0 \leq k \leq  n, %
\end{equation*}
%
and on the other, by the definition of $\cdot^{\dagger^0_n}$,
%
\begin{equation*}
(\mathfrak{N},w)\models(\IB\varrho)^{\dagger^0_n}[d_\mathfrak{b}]%
\qquad \text{iff}\qquad%
(\mathfrak{N},w)\models(\varrho)^{\dagger^k_n}[d_\mathfrak{b}],\quad \text{for all } k,\ 1 \leq k \leq n, %
\end{equation*}
%
which together with~\eqref{eq:forkequi:rc} and IH
yields~\eqref{eq:forkequi:terms}. The cases of the Booleans are
trivial.

Finally, we show that for every $\psi\in\textit{sub}\,\varphi$, we
have
%
\begin{equation*}
(\mathfrak{N},w)\models\psi^{\dagger_n} \qquad\text{iff}\qquad
(\mathfrak{M},w)\models\psi.
\end{equation*}
%
Case $\psi=\Ubox\tau$:
%
\begin{align*}
  (\mathfrak{N},w)\models(\Ubox\tau)^{\dagger_n}%
& \quad\text{iff}\quad %
\forall d_\mathfrak{b}\in D\ \forall k\in\{0,1,\dots,n\}\ (\mathfrak{N}, w)\models(\tau)^{\dagger^k_n}[d_\mathfrak{b}]\\
&\quad\text{iff}\quad %
\forall \mathfrak{b} \text{ in } \mathfrak{G} \ \forall k\in\{0,1,\dots,n\}\ (\mathfrak{M},\bigl\langle w, x^k_{\mathfrak{b}}\bigr\rangle)\models\tau\\
&\quad\text{iff}\quad %
(\mathfrak{M}, w)\models\Ubox\tau.
\end{align*}
%
The remaining cases are trivial. It follows that
$\varphi^{\dagger_n}$ is satisfied in $\mathfrak{N}$.

\smallskip

$(\Leftarrow)$ Assume that $\varphi^\dagger$ is satisfied in a
first-order temporal model $\mathfrak{N} = \auf \mathfrak{F}, D,
I\zu$, where $\mathfrak{F} = \auf W, <\zu$ and, for every $w\in
W$,
%
\begin{equation*}
I(w) = \Bigl\langle D, P_{11}^{I(w)}, \dots, P_{1n}^{I(w)},
P_{21}^{I(w)}, \dots, P_{2n}^{I(w)},\dots \Bigr\rangle.
\end{equation*}
%
With every point $d\in D$ we associate an $n$-broom
$\mathfrak{b}_d = \auf W_{\mathfrak{b}_d}, R_{\mathfrak{b}_d}\zu$
so that the sets $W_{\mathfrak{b}_d}$, for $d\in D$, are pairwise
disjoint and each contains $n+1$ distinct elements
$x^0_{\mathfrak{b}_d},\dots,x^n_{\mathfrak{b}_d}$. Construct an
Aleksandrov tt-model $\mathfrak{M}=\auf \mathfrak{F},
\mathfrak{G}, \mathfrak{V}\zu$ by taking
%
\begin{itemize}
\item $\mathfrak{G}$ to be the disjoint union of $n$-brooms $\{\mathfrak{b}_d \mid d\in
D\}$,
\item $\mathfrak{V}(p_j,w)= \bigl\{ x^i_{\mathfrak{b}_d}\ \mid\ (\mathfrak{N},
  w)\models (\CB\IB p_j)^{\dagger^i_n}[d],\quad 0 \leq i \leq n\ \text{ and }\ d\in D\bigr\}$.
\end{itemize}
%
Clearly, if $D$ is finite then $\mathfrak{G}$ is finite as well.

By a straightforward induction one can show that for all $w\in W$,
$d\in D$, spatio-temporal region terms $\varrho$, spatio-temporal
terms $\tau$, subformulas $\psi$ of $\varphi$, and all $i$, $0\leq
i \leq n$,
%
\begin{align*}
(\mathfrak{N}, w)\models(\varrho)^{\dagger^i_n}[d] &
\qquad\text{iff}\qquad %
(\mathfrak{M}, \bigl\langle w, x^i_{\mathfrak{b}_d}\bigr\rangle)\models\varrho \qquad(i > 0),\\
(\mathfrak{N}, w)\models(\tau)^{\dagger^i_n}[d] &
\qquad\text{iff}\qquad %
(\mathfrak{M}, \bigl\langle w, x^i_{\mathfrak{b}_d}\bigr\rangle)\models\tau,\\
(\mathfrak{N}, w)\models\psi^{\dagger_n} &
\qquad\text{iff}\qquad%
(\mathfrak{M}, w)\models\psi.
\end{align*}
%
For example,
%
\begin{align*}
(\mathfrak{N},w)\models(\IB\varrho)^{\dagger^0_n}[d] &
\qquad\text{iff}\qquad%
(\mathfrak{N},w)\models(\varrho)^{\dagger^k_n}[d], \text{ for all } k,\ 0 \leq k \leq n\\
&\qquad\text{iff}\qquad %
(\mathfrak{M},\bigl\langle w, x^k_{\mathfrak{b}_d}\bigr\rangle)\models\varrho, \text{ for all } k,\ 0 \leq k \leq n\\
&\qquad\text{iff}\qquad %
(\mathfrak{M},\bigl\langle w, x^0_{\mathfrak{b}_d}\bigr\rangle)\models\IB\varrho. %
\end{align*}
%
It follows that $\varphi$ is satisfied in $\mathfrak{M}$.
\end{proof}

Now we obtain the upper complexity bounds for combinations of
$\PTL$ and $\RCm$:

\begin{proof}[\proofof{Theorem~\ref{theorem:rcm:expspace}, upper bound}]
Follows from Lemmas~\ref{rc:simula3}~(ii) and
\ref{lemma:rc:embedding} together with the results on the
complexity of the one-variable fragment of $\QTL$
\cite{Halpern&Vardi89,Sistla&German87,Hodkinsonetal00,Hodkinsonetal03}.
\end{proof}

\begin{proof}[\proofof{Theorem~\ref{theorem:brcc8:expspace:full}, upper bound}]
Similar to the proof above.
\end{proof}

\begin{proof}[\proofof{Theorem~\ref{theorem:brcc8:expspace:box:dense}, upper bound}]
The proof follows from Lemmas~\ref{rc:simula3}~(i) and
\ref{lemma:rc:embedding} together with the upper complexity bound
of the guarded monodic (and so the one-variable) fragment of
$\QTL$ \cite{Hodkinson03guarded}.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection[Lower Complexity Bounds (II): Embedding First-Order
Temporal Logic]{Lower Complexity Bounds (II): Embedding First-Order
Temporal Logic}\label{section:converse:embedding}

We are now in a position to prove
Theorem~\ref{theorem:brcc8:expspace:box} and establish the lower
complexity bounds for spatio-temporal logics based on $\BRCCE$
(and so for those based on $\RCm$ as well). Denote by
$\QTL_\Box^1$ the one-variable fragment of $\QTL$ with sole
temporal operator $\Box_F$. We define a polynomial embedding of
$\QTL_\Box^1$ into $\PTL_\Box\times\BRCCE$. Note that a similar
embedding of the full one-variable fragment $\QTL^1$ into
$\PTL\times\BRCCE$ can be regarded as an alternative way to prove
the lower complexity bound of
Theorem~\ref{theorem:brcc8:expspace:full}.

A $\QTL_\Box^1$-formula is said to be a \emph{basic Q-formula} if
it is of the form $\forall x\,\vartheta(x)$, where $\vartheta(x)$
is quantifier-free and contains no propositional variables. A
$\QTL_\Box^1$-sentence $\varphi$ is in \emph{Q-normal form} if it
is built from basic Q-formulas using the Booleans and temporal
operator $\Box_F$. In other words, sentences in Q-normal form do
not contain nested quantifiers and use only unary predicate
symbols. The following observation should not come as a surprise
(see, e.g., \citeR{Hughes&Cresswell96}):

\begin{lemma}\label{lemma:normalform}
For every $\QTL_\Box^1$-sentence $\varphi$ one can effectively
construct a $\QTL_\Box^1$-sentence $\widehat\varphi$ in Q-normal
form such that $\varphi$ is satisfiable in a first-order temporal
model with a flow of time $\mathfrak{F}$ \textup{(}and having
finite domain\textup{)} iff $\widehat\varphi$ is satisfiable in a
first-order temporal model based on $\mathfrak{F}$ \textup{(}and
having finite domain\textup{)}. Moreover, the length of
$\widehat\varphi$ is linear in the length of $\varphi$.
\end{lemma}
%
\begin{proof}
Without loss of generality we may assume that $\varphi$ contains
no occurrences of $\exists$.  To transform $\varphi$ into its
Q-normal form, we first introduce a fresh unary predicate symbol
$P_i(x)$ for every propositional variable $p_i$ in $\varphi$ and
replace each occurrence of $p_i$ with $\forall x\,P_i(x)$. Denote
the resulting formula by $\varphi_0$.  For every subformula $\psi$
of $\varphi_0$ define a formula $\psi^{\sharp}$ by taking
inductively
%
\begin{align*}
(P(x))^\sharp &= P(x), & (\forall x\, \psi)^\sharp & = P_{\forall x \psi}(x),\\  %
(\neg \psi)^\sharp &= \neg \psi^\sharp, & %
(\psi_1\land \psi_2)^\sharp &= \psi_1^\sharp \land \psi_2^\sharp, & %
(\Box_F \psi)^\sharp &= \Box_F \psi^\sharp, %
\end{align*}
%
where $P_{\forall x\psi}(x)$ is a fresh unary predicate symbol.
Let
%
\begin{equation*}
\widehat\varphi \ =\ \neg \forall x\, \neg \varphi_0^{\sharp}
\;\land\hspace*{-0.4em} \bigwedge_{\forall x\psi\in\textit{sub}
\varphi_0} \hspace*{-1em}\Box_F^+\bigg[\big(\forall x\, P_{\forall
x \psi}(x) \vee \forall x\, \neg P_{\forall x \psi}(x)\big) \
\wedge\ \big(\forall x\, P_{\forall x \psi}(x) \leftrightarrow
\forall x\,\psi^\sharp\big)\bigg].
\end{equation*}
%
One can readily show by induction that $\widehat\varphi$ is
satisfiable in a first-order temporal model based on
$\mathfrak{F}$ (and having finite domain) iff $\varphi$ is
satisfiable in a first-order temporal model based on
$\mathfrak{F}$ (and having finite domain). Moreover,
$\widehat\varphi$ is in Q-normal form.
\end{proof}

Now, given a $\QTL_\Box^1$-formula $\varphi$ in Q-normal form,
denote by $\varphi^\ast$ the result of replacing all occurrences
of basic Q-formulas $\forall x\,\vartheta(x)$ in $\varphi$ with
$\mathsf{EQ}(\vartheta^\ast, \top)$, where $\top$ is a region term
representing the whole space (for instance, $\CB\IB\textv{u}
\sqcup \CB\IB{\overline{\textv{u}}}$ for a fresh spatial variable
$\textv{u}$), and the translation $\vartheta^\ast$ of
quantifier-free formulas $\vartheta(x)$ is defined by taking:
%
\begin{equation*}
(P(x))^\ast = \CB\IB{p},\quad (\neg \psi)^\ast =
\CB\IB{\;\overline{\psi^\ast}\;},\quad (\psi_1 \land \psi_2)^\ast
= \CB\IB(\psi_1^\ast \sqcap \psi_2^\ast),\quad
(\Box_F \psi)^\ast = \CB\IB{\Box_F \psi^\ast}, %
\end{equation*}
%
where $P(x)$ is a unary predicate symbol and $p$ a spatial
variable standing for $P(x)$. Clearly, $\varphi^\ast$ belongs to
$\PTL_\Box\times\BRCCE$.

\begin{lemma}\label{lemma:rc:backward}
A $\QTL_\Box^1$-sentence $\varphi$ in Q-normal form is satisfiable
in a first-order temporal model based on a flow of time
$\mathfrak{F}$ and having finite domain iff $\varphi^\ast$ is
satisfiable in a tt-model based on $\mathfrak{F}$ and satisfying
\FSA{}.
\end{lemma}
%
\begin{proof}
$(\Rightarrow)$ Suppose that $\varphi$ is in Q-normal form and
$\mathfrak{M} = \auf \mathfrak{F}, D, I\zu$ is a first-order
temporal model, where $\mathfrak{F} = \auf W, <\zu$ and, for all
$w \in W$, $I(w) = \bigl\langle D,P_0^{I(w)}, \dots,
\bigr\rangle$. Let $(\mathfrak{M}, w_0)\models\varphi$ for some
$w_0\in W$. Construct an Aleksandrov tt-model $\mathfrak{M}' =
\auf \mathfrak{F}, \mathfrak{G}, \mathfrak{V}\zu$ by taking
$\mathfrak{G}=\auf D, R\zu$, where $R = \{\auf d,d\zu\mid d\in
D\}$ and $\mathfrak{V}(p_i,w) = \bigl\{\auf w,d\zu\mid d\in
P_i^{I(w)}\bigr\}$. Note that the topological space
$\mathfrak{T}_\mathfrak{G} = \auf D, \mathbb{I}_\mathfrak{G}\zu$
induced by $\mathfrak{G}$ is \emph{discrete}, i.e., for all
$X\subseteq D$,
%
\begin{equation*}
\mathbb{I}_\mathfrak{G}X = \mathbb{C}_\mathfrak{G} X = X.
\end{equation*}
%
It follows by induction that for every quantifier-free
$\QTL_\Box^1$-formula $\vartheta$, every $w\in W$ and every $d\in
D$ we have
%
\begin{equation*}
(\mathfrak{M}, w)\models\vartheta[d] \qquad\text{iff}\qquad
(\mathfrak{M}', \auf w,d\zu)\models \vartheta^\ast.
\end{equation*}
%
Therefore, for every basic Q-formula $\forall x\,\vartheta(x)$ and
every $w\in W$, $(\mathfrak{M}, w)\models\forall x\,\vartheta(x)$
iff $(\mathfrak{M}', w)\models\textsf{EQ}(\vartheta^\ast,\top)$.
It follows by induction that $(\mathfrak{M}',w_0)\models
\varphi^\ast$.

\smallskip
$(\Leftarrow)$ Suppose that $\varphi^\ast$ is satisfied in a
tt-model based on $\mathfrak{F} = \auf W, <\zu$. By
Lemma~\ref{rc:simula2} (i), $\varphi^\ast$  is satisfied in an
Aleksandrov tt-model $\mathfrak{M} = \auf \mathfrak{F},
\mathfrak{G}, \mathfrak{V}\zu$, where $\mathfrak{G} = \auf V,
R\zu$ is a disjoint union of brooms. Denote by $V_0\subseteq V$
the set of leaves of $\mathfrak{G}$ and define a first-order
temporal model $\mathfrak{M'} = \auf \mathfrak{F}, V_0, I\zu$ by
taking, for each $w\in W$,
%
\begin{equation*}
I(w) = \bigl\langle V_0, P_0^{I(w)}, \dots\bigr\rangle\qquad
\text{and}\qquad P_i^{I(w)} = \mathfrak{V}(p_i, w)\cap V_0.
\end{equation*}
%
Clearly, for every $X\subseteq V$, we have
$\mathbb{I}_\mathfrak{G}X \cap V_0= \mathbb{C}_\mathfrak{G} X \cap
V_0 = X \cap V_0$, where $\mathfrak{T}_\mathfrak{G} = \auf V,
\mathbb{I}_\mathfrak{G}\zu$ is the topological space induced by
$\mathfrak{G}$. So we obtain by induction that for every
quantifier-free $\QTL_\Box^1$-formula $\vartheta$, all $w\in W$
and all $d\in V_0$
%
\begin{equation*}
(\mathfrak{M}', w)\models\vartheta[d] \qquad\text{iff}\qquad
(\mathfrak{M}, \auf w,d\zu)\models\vartheta^\ast.
\end{equation*}
%
A regular closed set $X\subseteq V$ in $\mathfrak{T}_\mathfrak{G}$
coincides with $V$ iff it contains $V_0$. So, for all basic
Q-formulas $\forall x\,\vartheta(x)$ and all $w\in W$,
$(\mathfrak{M}', w)\models\forall x\,\vartheta(x)$ iff
$(\mathfrak{M}, w)\models\textsf{EQ}(\vartheta^\ast, \top)$. It
follows by induction that $\varphi$ is satisfied in
$\mathfrak{M}'$.
\end{proof}


\begin{proof}[\proofof{Theorem~\ref{theorem:brcc8:expspace:box}, lower bound}]
By Lemmas~\ref{lemma:normalform} and \ref{lemma:rc:backward} the
satisfiability problem for $\QTL_\Box^1$-formulas in first-order
temporal models with finite domains and based on $\auf
\mathbb{N},<\zu$, $\auf \mathbb{Z},<\zu$ or arbitrary finite flows
of time is polynomially reducible to satisfiability of
$\PTL_\Box\times\BRCCE$ formulas in tt-models with \FSA{}. Since
the former is known to be EXPSPACE-hard~\cite{Hodkinsonetal03} for
$\auf \mathbb{N},<\zu$ and $\auf \mathbb{Z},<\zu$, the latter is
also EXPSPACE-hard in these cases. It should be noted that the
result of Hodkinson and his colleagues \citeyear{Hodkinsonetal03}
can readily be extended to the case of arbitrary finite flows of
time (by reduction of a finite version of the corridor tiling
problem). This gives us the lower complexity bound for
$\PTL_\Box\times\BRCCE$ in the case of finite flows of time.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\subsection{PSPACE-complete Spatio-Temporal Logic}\label{appendix:pspace}

In this appendix we prove Theorem~\ref{st1m:pspace}. In fact, we
show that the satisfiability problem for $\PTL\circ\RC_2$---an
extension of $\PTL\circ\RCCE$---is decidable in PSPACE, where
$\RC_2$ is the sublanguage of $\SFU$ with spatial terms $\tau$
restricted to the following:
%
\begin{align*}
\varrho \quad ::= \quad & \CB\IB p,\\
\sigma \quad ::=\quad & \varrho \quad\mid\quad \overline{\IB\varrho}, \\
\delta \quad ::=\quad & \IB\varrho \quad\mid\quad \overline{\varrho}, \\
\tau   \quad ::=\quad & \sigma_1\sqcup\sigma_2 \quad\mid\quad \delta_1\sqcup\delta_2 \quad\mid\quad \sigma\sqcup\delta. %\\
\end{align*}
%
As before, we denote by $\sigma$ spatial terms representing
regular closed sets (regions) and by $\delta$ those representing
regular open sets (the interiors of regions). Clearly, this
definition is equivalent to the definition on p.~\pageref{def:rc2}
(where we did not make an explicit distinction between $\sigma$
and $\delta$). It is easy to see that $\RC_2$ contains $\RCCE$,
but is less expressive than $\BRCCE$. Spatio-temporal terms $\tau$
of $\PTL\circ\RC_2$ are constructed from region terms of the form
%
\begin{equation*}
\varrho \quad ::= \quad  \CB\IB p \quad\mid\quad \CB\IB\nxt\varrho
\end{equation*}
%
in the same way as spatial terms of $\RC_2$. Finally,
$\PTL\circ\RC_2$-formulas are composed from atomic formulas of the
form $\Ubox\tau$ using the Booleans and the temporal operators.

\smallskip
We will reduce the satisfiability problem for $\PTL\circ\RC_2$ to
that for $\PTL$. This reduction will be done in a number steps.

%a series of transformations.
%In what follows we consider only $\auf\mathbb{N},<\zu$ as a flow of
%time. The cases of $\auf\mathbb{Z},<\zu$ and arbitrary finite flows
%of time are similar and to the reader.

Let $\mathfrak{F} = \auf W,<\zu$ be a flow of time (as in the
formulation of Theorem~\ref{st1m:pspace}) and $\varphi$ a
$\PTL\circ\RC_2$-formula. We begin by removing the next-time
operator from the subterms of~$\varphi$. To this end, let $\psi_0
= \varphi$ and for each variable $p$ from the set
%
\begin{equation*}
\varset_1 \quad = \quad \{p\mid \CB\IB\nxt\CB\IB
p\in\textit{term}\,\psi_0\},
\end{equation*}
%
we introduce a fresh spatial variable $p'$, and then put
%
\begin{equation*}
\varphi_1 \quad=\quad \psi_1\quad\land\quad \bigwedge_{p\in
\varset_1}\Box_P^+\Box_F^+\bigl(\nxt\top\to\Ubox(\CB\IB\nxt\CB\IB
p \equi \CB\IB p')\bigr),
\end{equation*}
%
where $\psi_1$ is the result of replacing each occurrence of
$\CB\IB\nxt\CB\IB p$ in $\psi_0$ with $\CB\IB p'$ and
\mbox{$\Ubox(\varrho_1\equi\varrho_2)$} stands for
$\Ubox(\overline{\varrho_1}\sqcup\varrho_2)\land\Ubox(\varrho_1\sqcup\overline{\varrho_2})$.
Next, for each $p$ from
%
\begin{equation*}
\varset_2 \quad  = \quad  \{p\mid \CB\IB\nxt\CB\IB
p\in\textit{term}\,\psi_1\},
\end{equation*}
%
we introduce a fresh spatial variable $p'$, and set
%
\begin{equation*}
\varphi_2 \quad=\quad \psi_2\quad\land\quad \bigwedge_{p\in
\varset_1\cup
\varset_2}\Box_P^+\Box_F^+\bigl(\nxt\top\to\Ubox(\CB\IB\nxt\CB\IB
p \equi \CB\IB p')\bigr),
\end{equation*}
%
where $\psi_2$ is the result of replacing each occurrence of
$\CB\IB\nxt\CB\IB p$ in $\psi_1$ with $\CB\IB p'$. By repeating
this process sufficiently many times we can obtain a formula
%
\begin{equation}\label{eq:rc2:normal:form}
\widetilde{\varphi}\quad=\quad \psi_\varphi \quad \land\quad
\bigwedge_{p\in
\varset_\varphi}\Box_P^+\Box_F^+\bigl(\nxt\top\to\Ubox(\CB\IB\nxt\CB\IB
p \equi \CB\IB p')\bigr),
\end{equation}
%
where $\varset_\varphi$ is a suitable set of spatial variables,
and $\psi_\varphi$ contains no $\nxt$ in region terms, that is,
$\psi_\varphi$ is a $\PTL[\RC_2]$-formula. (Note that
$\varset_\varphi$ is such that if a spatial variable $p$ occurs in
$\psi_\varphi$ then either $\CB\IB\nxt\CB\IB
p\notin\textit{term}\,\varphi$ or $p\in \varset_\varphi$.)
%
It should be clear that the length of $\widetilde{\varphi}$ is
linear in the length of $\varphi$, and $\varphi$ is satisfiable in
a tt-model based on $\mathfrak{F}$ iff $\widetilde{\varphi}$ is
satisfiable in a tt-model based on $\mathfrak{F}$.

Thus, it suffices to reduce the satisfiability problem for
$\PTL\circ\RC_2$-formulas of the form~\eqref{eq:rc2:normal:form}
to the satisfiability problem for $\PTL$-formulas. Let us now
recall the function $\cdot^\ast$ from
Appendix~\ref{appendix:temporalisation} which maps
$\PTL[\SFU]$-formulas (in particular, $\PTL[\RC_2]$-formulas) to
$\PTL$-formulas. Namely, for every atomic $\RC_2$-formula
$\Ubox\tau$, let $(\Ubox\tau)^\ast = \textp{p}_\tau$, where
$\textp{p}_\tau$ is a fresh propositional variable. Then, given
the $\PTL[\RC_2]$-formula $\psi_\varphi$, define
$\psi_\varphi^\ast$ to be the result of replacing every occurrence
of $\Ubox\tau$ in it with $(\Ubox\tau)^\ast$. As is shown in the
proof of Theorem \ref{PST0:pspace:text}, $\psi_\varphi$ is
satisfiable in a tt-model over $\mathfrak{F} = \auf W,< \zu$ iff
%
\begin{itemize}
\item[(s1)] there exists a temporal model
$\mathfrak{N}=\auf \mathfrak{F},{\mathfrak U}\zu$ satisfying
$\psi_\varphi^{\ast}$ and,
%
\item[(s2)] for every $w\in W$, the set
%
\begin{equation}\label{eq:rc2:phi}
\Phi_w = \{ \Ubox\tau \mid (\mathfrak{N},w)\models
\textp{p}_\tau,\ \tau\in\textit{term}\,\psi_\varphi\}\cup\{
\neg\Ubox\tau \mid (\mathfrak{N},w)\models \neg \textp{p}_\tau,\
\tau\in\textit{term}\,\psi_\varphi\}
\end{equation}
%
of $\RC_2$-formulas is satisfiable.
\end{itemize}
%
To preserve satisfiability of not only $\psi_\varphi$ but the
whole $\widetilde{\varphi}$, we have to ensure somehow that
%
\begin{itemize}
\item[(s3)]
the points satisfying $\Phi_w$ do have predecessors and successors
satisfying $\Phi_{w-1}$ and $\Phi_{w+1}$, respectively.
\end{itemize}

In the remainder of the appendix we first describe an encoding of
the satisfiability problem for sets of $\RC_2$-formulas of the
form~\eqref{eq:rc2:phi} in Boolean logic, which will be used as
part of our final reduction. Then we prove a \emph{completion
property} of $\RC_2$ \cite<cf.>{Balbiani&Condotta02} in the class
of \emph{exhaustive models} that contain `sufficiently many'
points of every type. Roughly, the completion property says that,
given a set $\Phi$ of the form~\eqref{eq:rc2:phi} and an
exhaustive model satisfying some subset of $\Phi$, one can extend
the valuation of the model to satisfy the whole $\Phi$. This
property will make it possible to solve problem (s3) above. It is
worth noting that a similar construction works for stronger
languages such as $\BRCCE$, but then, to enjoy the completion
property, sets~\eqref{eq:rc2:phi} may need exponentially many
formulas (in the number of spatial variables) and, therefore, the
reduction to $\PTL$ will be exponential as well. For $\RC_2$ it
suffices to consider sets~\eqref{eq:rc2:phi} with a quadratic
number of formulas, which results in a quadratic reduction.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection[Properties of RC2-formulas]{Properties of $\RC_2$-formulas}

For any finite set $\varset=\{p_1,\dots,p_n\}$ of spatial
variables, let
%
\begin{equation*}
\aform\quad =\quad \big\{ \Ubox\tau\ \mid\ \mbox{$\tau$ is an
$\RC_2$-term with variables from $\varset$} \big\}.
\end{equation*}
%
Clearly, every $\RC_2$-formula with spatial variables from
$\varset$ is a Boolean combination of spatial formulas from
$\aform$. It should be also clear that $|\aform| \le
16\cdot|\varset|^2$.

As the width of $\RC_2$-formulas is $\leq 2$ (see
p.~\pageref{def:width} for the definition), by
Lemmas~\ref{s4u:simula2} and \ref{rc:simula3}~(ii), an
$\RC_2$-formula is satisfiable iff it is satisfiable in an
Aleksandrov topological model based on a disjoint union of
2-brooms, alias \emph{forks}. In what follows we will regard every
such model $\mathfrak{M}$ as a disjoint union of \emph{fork
models} $\mathfrak{m}=\auf\mathfrak{f},\mathfrak{v}\zu$, where
$\mathfrak{f} = \auf W,R\zu$, $W=\{x^0,x^1,x^2\}$, $R$ is the
reflexive closure of $\{\langle x^0,x^1\rangle,\langle
x^0,x^2\rangle\}$ and $\mathfrak{v}$ a valuation of the spatial
variables. Given $\varset_0\subseteq\varset$, we say that fork
models $\mathfrak{m}_1=\auf\mathfrak{f},\mathfrak{v}_1\zu$ and
$\mathfrak{m}_2=\auf\mathfrak{f},\mathfrak{v}_2\zu$ are
\emph{$\varset_0$-equivalent} and write $\mathfrak{m}_1
\sim_{\varset_0} \mathfrak{m}_2$, if $\mathfrak{v}_1(\CB\IB
p)=\mathfrak{v}_2(\CB\IB p)$ for every $p\in\varset_0$.

Given some $\Phi\subseteq\aform$ and $\psi\in\aform$, we say that
$\psi$ is an \emph{$\mathfrak{f}$-consequence} of $\Phi$ and write
$\Phi\models_\mathfrak{f}\psi$ if $\mathfrak{m}\models\Phi$
implies $\mathfrak{m}\models\psi$ for every fork model
$\mathfrak{m}$ based on $\mathfrak{f}$. $\Phi$ is said to be
\emph{closed} (under $\mathfrak{f}$-consequences) if, for every
$\psi\in\aform$, we have $\psi\in\Phi$ whenever
$\Phi\models_\mathfrak{f}\psi$. Let $\Phi^c = \bigl\{\neg\Ubox\tau
\mid \Ubox\tau\in\aform - \Phi\bigr\}$. Then $\Phi\cup\Phi^c$ is
satisfiable iff $\Phi$ is closed and satisfiable. This means, in
particular, that to check whether the set $\Phi_w$ in
\eqref{eq:rc2:phi} is satisfiable, it is enough to consider only
the closure of $\{ \Ubox\tau \mid (\mathfrak{N},w)\models
\textp{p}_\tau,\ \tau\in\textit{term}\,\psi\}$.

Now we characterise $\models_\mathfrak{f}$ in terms of the Boolean
consequence relation $\models$. As we know from
Appendix~\ref{appendix:embedding}, spatial formulas can be
embedded into the one-variable fragment of first-order logic. More
precisely, it can easily be shown that first-order translations of
formulas from $\aform$ are (equivalent to) formulas of the form
\cite<which are actually \emph{Krom formulas}; see,
e.g.,>{Borgeretal97}:
%
\begin{align}\label{eq:krom1}
(\Ubox(\sigma_1\sqcup\sigma_2))^{\dagger_2} & = \forall x\,\bigl(\sigma_1^{\dagger^1_2} \lor \sigma_2^{\dagger^1_2}\bigr)\ \land\ \forall x\,\bigl(\sigma_1^{\dagger^2_2} \lor \sigma_2^{\dagger^2_2}\bigr),\\%[2pt]  %
%
\label{eq:krom2}%
(\Ubox(\sigma_1\sqcup\delta_2))^{\dagger_2} & = \forall x\,\bigl(\sigma_1^{\dagger^1_2} \lor \delta_2^{\dagger^1_2}\bigr)\ \land\ \forall x\,\bigl(\sigma_1^{\dagger^2_2} \lor \delta_2^{\dagger^2_2}\bigr),%\\%[2pt]  %
%
\end{align}
\begin{align}
\label{eq:krom3}%
(\Ubox(\delta_1\sqcup\delta_2))^{\dagger_2} & = \forall x\,\bigl(\delta_1^{\dagger^1_2} \lor \delta_2^{\dagger^1_2}\bigr)\ \land\ \forall x\,\bigl(\delta_1^{\dagger^1_2} \lor \delta_2^{\dagger^2_2}\bigr)\ \land\ \forall x\,\bigl(\delta_1^{\dagger^2_2} \lor \delta_2^{\dagger^1_2}\bigr)\ \land\ \forall x\,\bigl(\delta_1^{\dagger^2_2} \lor \delta_2^{\dagger^2_2}\bigr),%  \\[2pt]  %
\end{align}
%
where
%
\begin{equation*}
\sigma^{\dagger_2^i} = \begin{cases}P_{ji}(x), &\text{if }
\sigma=\CB\IB p_j,\\\neg P_{ji}(x),&\text{if
}\sigma=\overline{\IB\CB\IB p_j}\end{cases}\quad \text{and}\quad
\delta^{\dagger_2^i} =
\begin{cases}P_{ji}(x), &\text{if } \delta=\IB\CB\IB p_j,\\\neg
P_{ji}(x),&\text{if }\delta=\overline{\CB\IB p_j},\end{cases}\quad
\text{for}\quad i = 1,2.
\end{equation*}
%
It follows from the proof of Lemma~\ref{lemma:rc:embedding} that
an $\RC_2$-formula $\psi$ is satisfied in an Aleksandrov model
$\mathfrak{M}$ based on a disjoint union of forks iff its
first-order translation $\psi^{\dagger_2}$ is satisfied in a
first-order model where every fork $\mathfrak{f} = \auf W,R\zu$ of
$\mathfrak{M}$, $W = \auf x^0,x^1,x^2\zu$, $x^0Rx^1$ and
$x^0Rx^2$, is encoded by a domain element $d_\mathfrak{f}$ with
$P_{ji}(x)$ being true on $d_\mathfrak{f}$ iff
$(\mathfrak{M},x^i)\models\CB\IB p_j$, for $i = 1,2$ (see
Fig.~\ref{fig:nbroom3}). Since in the definition of closed sets we
only consider Aleksandrov models based on a single fork
$\mathfrak{f}$, the domains of respective first-order models
contain a single element $d_\mathfrak{f}$. This means that
\eqref{eq:krom1}--\eqref{eq:krom3} can be encoded by the Boolean
formulas
%
\begin{align*}
(\Ubox(\sigma_1\sqcup\sigma_2))^{\ddagger} & = \bigl(\sigma_1^{\ddagger^1} \lor \sigma_2^{\ddagger^1}\bigr)\ \land\ \bigl(\sigma_1^{\ddagger^2} \lor \sigma_2^{\ddagger^2}\bigr),\\  %
(\Ubox(\sigma_1\sqcup\delta_2))^{\ddagger} & = \bigl(\sigma_1^{\ddagger^1} \lor \delta_2^{\ddagger^1}\bigr)\ \land\ \bigl(\sigma_1^{\ddagger^2} \lor \delta_2^{\ddagger^2}\bigr),\\  %
(\Ubox(\delta_1\sqcup\delta_2))^{\ddagger} & = \bigl(\delta_1^{\ddagger^1} \lor \delta_2^{\ddagger^1}\bigr)\ \land\ \bigl(\delta_1^{\ddagger^1} \lor \delta_2^{\ddagger^2}\bigr)\ \land\ \bigl(\delta_1^{\ddagger^2} \lor \delta_2^{\ddagger^1}\bigr)\ \land\ \bigl(\delta_1^{\ddagger^2} \lor \delta_2^{\ddagger^2}\bigr),%  \\[2pt]  %
\end{align*}
%
where
%
\begin{equation*}
\sigma^{\ddagger^i} = \begin{cases}q_{ji}, &\text{if }
\sigma=\CB\IB p_j,\\\neg q_{ji},&\text{if
}\sigma=\overline{\IB\CB\IB p_j}\end{cases}\qquad\text{and}\qquad
\delta^{\ddagger^i} =
\begin{cases}q_{ji}, &\text{if } \delta=\IB\CB\IB p_j,\\\neg
q_{ji},&\text{if }\delta=\overline{\CB\IB
p_j},\end{cases}\qquad\text{for}\quad i = 1,2.
\end{equation*}
%
Thus, with every $\Phi\subseteq\aform$ we can associate the
conjunction $\Phi^\ddagger$ of the $\cdot^\ddagger$-translations
of formulas in $\Phi$ such that the following holds:
%
\begin{claim}\label{claim:rc2:(i)}
For every $\psi\in\aform$, $\Phi\models_{\mathfrak{f}}\psi$\quad
iff\quad $\Phi^\ddagger\models\psi^\ddagger$.
\end{claim}

To construct the closure of $\Phi \subseteq \aform$ and to check
whether $\Phi$ is satisfiable, we can use  the following
resolution-like inference rules:
%
\begin{gather*}
\text{\small $(\sigma\sigma)$}\quad \begin{array}{c}\Ubox(\sigma_1\sqcup\varrho)\qquad \Ubox(\overline{\IB\varrho}\sqcup\sigma_2)\\\hline\rule{0pt}{12pt}\Ubox(\sigma_1\sqcup\sigma_2) \end{array} %
%
\hspace*{12mm} \text{\small $(\sigma\delta)_1$}\quad \begin{array}{c}\Ubox(\IB\varrho\sqcup\delta_1)\\\hline\rule{0pt}{12pt}\Ubox(\varrho\sqcup\delta_1)\end{array}  %
\hspace*{12mm} \text{\small $(\sigma\delta)_2$}\quad \begin{array}{c}\Ubox(\overline{\varrho}\sqcup\delta_1)\\\hline\rule{0pt}{12pt}\Ubox(\overline{\IB\varrho}\sqcup\delta_1)\end{array} \\[20pt] %
%
%
\text{\small $(\bot)$}\quad \begin{array}{c}\Ubox\varrho\qquad \Ubox\overline{\varrho}\\\hline \bot \end{array} % %
%
\hspace*{16mm}
%
\text{\small $(\delta\delta)$}\quad  \begin{array}{c}\Ubox(\delta_1\sqcup\theta)\qquad \Ubox(\theta'\sqcup\delta_2)\\\hline\rule{0pt}{12pt}\Ubox(\delta_1\sqcup\delta_2) \end{array}\quad \text{\small for\quad %
%
$\begin{cases}
\theta = \overline{\varrho}, & \theta' = \IB\varrho;\\
\theta = \varrho, & \theta' = \overline{\varrho};\\
\theta = \overline{\IB\varrho}, & \theta' = \IB\varrho;%\\
\end{cases}$}%
%
\end{gather*}
%
together with the equivalences:
%
\begin{equation*}
\Ubox\varrho = \Ubox\IB\varrho,\qquad \Ubox\overline{\varrho} = \Ubox\overline{\IB\varrho},\qquad %
\Ubox(\varrho\sqcup\sigma_1) = \Ubox(\IB\varrho\sqcup\sigma_1), \qquad %
\Ubox(\overline{\varrho}\sqcup\sigma_1) = \Ubox(\overline{\IB\varrho}\sqcup\sigma_1),%\\ %
\end{equation*}
%
where $\varrho=\CB\IB p$ for some $p\in\varset$, $\sigma_1$ and
$\sigma_2$ are of the form $\varrho$ or $\overline{\IB\varrho}$,
and $\delta_1$ and $\delta_2$ of the form $\IB\varrho$ or
$\overline{\varrho}$. It is readily checked that the above rules
are sound, and so if $\bot$ is derivable from $\Phi$, then $\Phi$
is not satisfiable. On the other hand, if $\Phi$ is not
satisfiable then $\Phi^\ddagger$ can be regarded as an
unsatisfiable set of binary and unary propositional clauses and,
using the standard resolution procedure, one can construct a
derivation of the empty clause from $\Phi^\ddagger$---which, in
turn, can be mimicked by applications of the above rules (and
equivalences) to derive $\bot$ from $\Phi$. Moreover, since the
propositional resolution is subsumption complete \cite<see,
e.g.,>{Slagleetal69}, we can also derive all consequences of
$\Phi$, thereby obtaining its closure.

Now we encode the above rules and equivalences as Boolean formulas
with variables $\textp{p}_\tau$, for $\Ubox\tau\in\aform$. For
instance, $(\bot)$ and $(\sigma\sigma)$ are encoded by
%
\begin{equation*}
\bigl(\Ubox\varrho\land\Ubox\overline{\varrho}\bigr)^\ast\to\bot\qquad\text{and}\qquad
\bigl(\Ubox(\sigma_1\sqcup\varrho)\land
\Ubox(\overline{\IB\varrho}\sqcup\sigma_2)\to
\Ubox(\sigma_1\sqcup\sigma_2)\bigr)^\ast,
\end{equation*}
%
respectively. Denote by $\Gamma_\varset$ the conjunction of all
such formulas for spatial variables from $\varset$. Then we have
the following:
%
\begin{claim}\label{cl:rc2:final}
For every $\Phi\subseteq\aform$, $\Phi$ is closed and satisfiable
iff the Boolean formula
%
\begin{equation}\label{eq:rc2:saturation}
\Gamma_\varset \quad\land\quad \Bigl[\bigwedge_{\Ubox\tau\in\Phi}\!
\textup{\textp{p}}_\tau \quad\land
\bigwedge_{\Ubox\tau\in\aform-\Phi}\!\!\!\!\!\!\!\!\! \neg
\textup{\textp{p}}_\tau\Bigr]
\end{equation}
%
is satisfiable.
\end{claim}

%To sum up, the above claim solves problem (s2) above:
%identified at the beginning of the appendix:
%to check whether the set
%$\Phi_w$ in~\eqref{eq:rc2:phi} can represent a
%possible state of a tt-model to be constructed, it suffices to
%evaluate the Boolean formula~\eqref{eq:rc2:saturation}
%corresponding to
%
%\[
%\Phi\quad =\quad \{ \Ubox\tau \mid (\mathfrak{N},w)\models
%\textp{p}_\tau,\ \tau\in\textit{term}\,\psi\}.
%\]
%
Finally, to ensure (s3), we need the following \emph{completion
property} of $\RC_2$:
%
\begin{lemma}\label{lemma:rc2:completion}
Let $\Phi$ be a closed subset of $\aform$,
$\varset_0\subseteq\varset$ and $\Phi_0 = \Phi\cap\aformn$. Then
\textup{(i)} $\Phi_0$ is closed and \textup{(ii)} for every fork
model $\mathfrak{m}_0$, if $\mathfrak{m}_0\models\Phi_0$ then
there is a fork model $\mathfrak{m}$ such that
$\mathfrak{m}_0\sim_{\varset_0}\mathfrak{m}$ and
$\mathfrak{m}\models\Phi$.
\end{lemma}
%
\begin{proof}
Claim (i) is clear. To show (ii), we define the characteristic
formula $\chi$ of $\mathfrak{m}_0$ on $\varset_0$ by taking:
%
\begin{equation*}
\chi \quad=\!\!\!\! \bigwedge_{p_j\in\varset_0\!,\ i=1,2}\!\!\!
l^\ast_{ji}\qquad\text{and}\qquad l^\ast_{ji} =
\begin{cases}\neg q_{ji},& \text{if }
(\mathfrak{m}_0,x^i)\not\models\CB\IB p_j,\\q_{ji},& \text{if }
(\mathfrak{m}_0,x^i)\models\CB\IB p_j.\end{cases}
\end{equation*}
%
If $\mathfrak{m}_0\models\Phi_0$ then it follows immediately from
the definitions that $\Phi_0^\ddagger\land\chi$ is satisfiable.
Our aim is to show that $\Phi^\ddagger\land\chi$ is also
satisfiable, which would mean that there is a fork model
$\mathfrak{m}$ as required. Suppose otherwise. Then
$\Phi^\ddagger\models\neg\chi$. We can regard $\Phi^\ddagger$ as a
set of unary and binary clauses and $\neg\chi$ as a clause with
$2\cdot |\varset_0|$ literals $l_{ji}$, the negations of the
$l^\ast_{ji}$. According to the subsumption theorem
\cite{Slagleetal69}, by applying the standard resolution rule to
$\Phi^\ddagger$, we can derive a clause $l_{j_1i_1} \lor
l_{j_2i_2}$ which subsumes $\neg\chi$ (i.e., its both literals
occur in $\neg\chi$). Since $\Phi$ is closed, we have $l_{j_1i_1}
\lor l_{j_2i_2}$ among the clauses of $\Phi^\ddagger$ and as the
$l_{j_ki_k}$ are the $\cdot^{\ddagger^{i_k}}$-translations of
spatial terms for spatial variables from $\varset_0$, we conclude
that $l_{j_1i_1} \lor l_{j_2i_2}$ is indeed among the clauses of
$\Phi_0^\ddagger$, contrary to $\Phi_0^\ddagger\land\chi$ being
satisfiable.
\end{proof}
%

\subsubsection[The Polynomial Translation of PTLoRC2 into PTL]{The Polynomial Translation of $\PTL\circ\RC_2$ into $\PTL$}

Now we are in a position to define a polynomial (at most
quadratic) translation $\cdot^\bullet$ of $\PTL\circ\RC_2$ into
$\PTL$. Starting with a given formula $\varphi$, we construct the
$\PTL\circ\RC_2$-formula $\widetilde{\varphi}$ of the
form~\eqref{eq:rc2:normal:form}:
%
\begin{equation*}
\widetilde{\varphi} \quad=\quad \psi_\varphi \quad \land\quad
\bigwedge_{p\in
\varset_\varphi}\Box_P^+\Box_F^+\bigl(\nxt\top\to\Ubox(\CB\IB\nxt\CB\IB
p \equi \CB\IB p')\bigr),
\end{equation*}
%
where $\psi_\varphi$ is a $\PTL[\RC_2]$-formula. Let
$\varset_\varphi'=\{p'\mid p\in\varset_\varphi\}$ and let
$\varset$ denote the smallest set of spatial variables containing
$\varset_\varphi\cup\varset_\varphi'$ and all spatial variables
occurring in $\psi_\varphi$. Given $\Ubox\tau\in\aformf$, denote
by $\Ubox \tau'$ the formula from $\aformfp$ obtained from
$\Ubox\tau$ by replacing every occurrence of $p\in\varset_\varphi$
with $p'\in\varset_\varphi'$. Consider the $\PTL$-formula
%
\begin{equation*}
\varphi^\bullet \quad=\quad \psi_\varphi^\ast\quad \land\quad
\Box_P^+\Box_F^+\;\Gamma_\varset \quad\land\quad
\Box_P^+\Box_F^+\bigl(\nxt\top\to\!\!
\bigwedge_{\Ubox\tau\in\aformf}\!\!\!\!\!(\nxt \Ubox\tau
\leftrightarrow \Ubox\tau')^\ast\bigr).
\end{equation*}
%

\begin{lemma}
For every $\PTL\circ\RC_2$-formula $\varphi$,
$\widetilde{\varphi}$ is satisfiable in a tt-model based on
$\mathfrak{F} = \auf W,<\zu$ iff $\varphi^\bullet$ is satisfiable
in a temporal model based on $\mathfrak{F}$.
\end{lemma}
%
\begin{proof}
$(\Rightarrow)$ Let
$(\mathfrak{M},w_0)\models\widetilde{\varphi}$. Construct a
temporal model $\mathfrak{N} = \auf\mathfrak{F},\mathfrak{V}\zu$
by taking, for $\Ubox\tau\in\aform$,
%
\begin{equation*}
\mathfrak{V}(\textp{p}_\tau) = \{w \in W\ \mid\ (\mathfrak{M},w)\models \Ubox\tau\}. %
\end{equation*}
%
It is easy to see that $(\mathfrak{N},w_0)\models\varphi^\bullet$.

\medskip

$(\Leftarrow)$ Let $(\mathfrak{N},w_0)\models\varphi^\bullet$ for
some $w_0\in W$. For every $w\in W$, set
%
\begin{equation*}
\Phi_w = \{ \Ubox\tau\in\aform\ \mid\ (\mathfrak{N},w)\models \textp{p}_\tau \}. %
\end{equation*}
%
Let $\Lambda_w$, for $w\in W$, be a set of \emph{all}
non-$\varset$-equivalent fork models $\mathfrak{m}$ with
$\mathfrak{m}\models\Phi_w$. By Claim~\ref{cl:rc2:final}, the
$\Phi_w$ are closed and satisfiable, so the sets $\Lambda_w$ are
nonempty. We use the elements of the $\Lambda_w$ as building
blocks for exhaustive states in the tt-model we are going to
construct in order to satisfy $\varphi$.

First we show that each element of $\Lambda_w$ has a successor in
$\Lambda_{w+1}$ and a predecessor in $\Lambda_{w-1}$ (provided
that $w$ has a successor and predecessor, respectively). More
precisely, we say that a pair of fork models $\mathfrak{m}=
\auf\mathfrak{f},\mathfrak{v}\zu$ and
$\mathfrak{m}'=\auf\mathfrak{f},\mathfrak{v}'\zu$ is
\emph{suitable} and write $\mathfrak{m}\to\mathfrak{m}'$ if
$\mathfrak{v}(\CB\IB p') = \mathfrak{v}'(\CB\IB p)$, for every
$p\in\varset_\varphi$.

\smallskip

(\textit{succ})\quad %
Let $\mathfrak{m}\in\Lambda_w$,
$\mathfrak{m}=\auf\mathfrak{f},\mathfrak{v}\zu$, and let $w\in W$
have a successor $w+1$. By the third conjunct of
$\varphi^\bullet$, we have
%
\begin{align*}
\Phi_w\cap\aformfp\quad & = \quad\{ \Ubox\tau'\in\aformfp\ \mid\
(\mathfrak{N},w)\models \textp{p}_{\tau'} \}\\
 & = \quad
\{ \Ubox\tau'\in\aformfp\ \mid\ (\mathfrak{N},w+1)\models
\textp{p}_{\tau}\}.
\end{align*}
%
Therefore,
%
\begin{equation*}
\Phi_{w+1}\cap\aformf \quad = \quad\{ \Ubox\tau\in\aformf\ \mid\
(\mathfrak{N},w)\models \textp{p}_{\tau'} \}.
\end{equation*}
%
Now, by $\mathfrak{m}\in\Lambda_w$, we have
$\mathfrak{m}\models\Phi_w\cap\aformfp$. So if we define a fork
model $\mathfrak{m}' = \auf\mathfrak{f},\mathfrak{v}'\zu$ by
taking $\mathfrak{v}'(p)=\mathfrak{v}(p')$, for all
$p\in\varset_\varphi$ (and arbitrary otherwise), then
$\mathfrak{m}'\models\Phi_{w+1}\cap\aformf$ follows. Since
$\Phi_{w+1}$ is closed, by Lemma~\ref{lemma:rc2:completion}, we
can find a fork model $\mathfrak{m}'' =
\auf\mathfrak{f},\mathfrak{v}''\zu$ such that
$\mathfrak{m}''\sim_{\varset_\varphi}\mathfrak{m}'$ and
$\mathfrak{m}''\models\Phi_{w+1}$. It follows that
$\mathfrak{m}\to\mathfrak{m}''$ and $\mathfrak{m}''$ is
$\varset$-equivalent to some fork model in $\Lambda_{w+1}$ (i.e.,
we may assume that $\mathfrak{m}''\in\Lambda_{w+1}$).

\smallskip

(\textit{pred})\quad %
Similarly,  for every  $\mathfrak{m}\in\Lambda_m$, $\mathfrak{m} =
\auf\mathfrak{f},\mathfrak{v}\zu$, and every $w\in W$ with a
predecessor $w-1$, there is $\mathfrak{m}'' =
\auf\mathfrak{f},\mathfrak{v}''\zu$ such that
$\mathfrak{m}''\in\Lambda_{w-1}$ and
$\mathfrak{m}''\to\mathfrak{m}$.

\medskip

It should be clear that for every fork model
$\mathfrak{m}\in\Lambda_w$ and every $w\in W$, we can define a
function $r_{\mathfrak{m},w}$ that gives for \emph{each} $u\in W$
a fork model $r_{\mathfrak{m},w}(u)\in\Lambda_u$ such that
$r_{\mathfrak{m},w}(w) = \mathfrak{m}$ and
$r_{\mathfrak{m},w}(u)\to r_{\mathfrak{m},w}(u+1)$, whenever $u+1$
is a successor of $u$. Let $\Delta$ be the set of all such
functions $r_{\mathfrak{m},w}$, for $w\in W$ and
$\mathfrak{m}\in\Lambda_w$.

We are now ready to define an Aleksandrov tt-model
$\mathfrak{M}=\auf\mathfrak{F},\mathfrak{G},\mathfrak{V}\zu$
satisfying $\widetilde{\varphi}$. Let $\mathfrak{G} = \auf W,R\zu$
be a disjoint union of $|\Delta|$-many forks $\mathfrak{f}_r =
\auf W_r,R_r\zu$, $W_r = \{ x_r^0,x_r^1,x_r^2\}$, $x_r^0R_rx_r^1$
and $x_r^0R_rx_r^2$, for each $r\in\Delta$, and let
$\mathfrak{V}(p,w) = \{ x_r^i \in W \mid (r(w),x_r^i)\models
\CB\IB p \}$, for all $p\in\varset$ and $w\in W$. We show by
induction on the construction of
$\chi\in\textit{sub}\,\psi_\varphi$
%(where $\psi_\varphi$ is the
%first conjunct of $\widetilde{\varphi}$)
that, for every $w\in W$,
%
\begin{equation*}
(\mathfrak{M},w)\models\chi \qquad\text{iff}\qquad
(\mathfrak{N},w)\models\chi^\ast.
\end{equation*}
%

\textit{Case} $\psi=\Ubox\tau$. Suppose that
$(\mathfrak{M},w)\models\Ubox\tau$ but
$(\mathfrak{N},w)\not\models \textp{p}_\tau$. Then
$\Ubox\tau\notin\Phi_w$ and, since $\Phi_w$ is closed (by
Claim~\ref{cl:rc2:final} and $\Gamma_\varset$ being true at $w$),
we have $\Phi_w\not\models_{\mathfrak{f}}\Ubox\tau$. It follows
that there is a fork model $\mathfrak{m}\in\Lambda_w$ with
$\mathfrak{m}\models\neg\Ubox\tau$, and so there is $r\in\Delta$
such that $r(w)=\mathfrak{m}$, contrary to
$(\mathfrak{M},w)\models\Ubox\tau$. Conversely, if
$(\mathfrak{N},w)\models \textp{p}_\tau$ then, by construction,
$(\mathfrak{M},w)\models\Ubox\tau$.

The cases of the Booleans and temporal operators are trivial.

As the second conjunct of $\widetilde{\varphi}$ is satisfied by
construction, we obtain
$(\mathfrak{M},w_0)\models\widetilde{\varphi}$.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\bibliographystyle{theapa}
%\bibliography{roman,biba-c,bibd-g,bibh-l,bibm-r,bibs-z}

\begin{thebibliography}{}

\bibitem[\protect\BCAY{Aiello\ \BBA\ van Benthem}{Aiello\ \BBA\ van
  Benthem}{2002a}]{Aiello&Benthem00}
Aiello, M.\BBACOMMA\  \BBA\ van Benthem, J. \BBOP2002a\BBCP.
\newblock \BBOQ Logical patterns in space\BBCQ\
\newblock In Barker-Plummer, D., Beaver, D.~I., van Benthem, J., \BBA\ {Scotto
  di Luzio}, P.\BEDS, {\Bem Words, Proofs and Diagrams}, \BPGS\ 5--25. CSLI
  Publications, Stanford.

\bibitem[\protect\BCAY{Aiello\ \BBA\ van Benthem}{Aiello\ \BBA\ van
  Benthem}{2002b}]{Aiello&Benthem02}
Aiello, M.\BBACOMMA\  \BBA\ van Benthem, J. \BBOP2002b\BBCP.
\newblock \BBOQ A modal walk through space\BBCQ\
\newblock {\Bem Journal of Applied Non-Classical Logics}, {\Bem 12\/}(3--4),
  319--364.

\bibitem[\protect\BCAY{Alexandroff}{Alexandroff}{1937}]{Alexandroff37}
Alexandroff, P. \BBOP1937\BBCP.
\newblock \BBOQ {Diskrete} {R\"aume}\BBCQ\
\newblock {\Bem Matematicheskii Sbornik}, {\Bem 2 (44)}, 501--518.

\bibitem[\protect\BCAY{Allen}{Allen}{1983}]{Allen83}
Allen, J. \BBOP1983\BBCP.
\newblock \BBOQ Maintaining knowledge about temporal intervals\BBCQ\
\newblock {\Bem Communications of the ACM}, {\Bem 26}, 832--843.

\bibitem[\protect\BCAY{Areces, Blackburn,\ \BBA\ Marx}{Areces
  et~al.}{2000}]{Arecesetal00}
Areces, C., Blackburn, P., \BBA\ Marx, M. \BBOP2000\BBCP.
\newblock \BBOQ The computational complexity of hybrid temporal logics\BBCQ\
\newblock {\Bem Logic Journal of the IGPL}, {\Bem 8}, 653--679.

\bibitem[\protect\BCAY{Arhangel'skii\ \BBA\ Collins}{Arhangel'skii\ \BBA\
  Collins}{1995}]{AC95}
Arhangel'skii, A.\BBACOMMA\  \BBA\ Collins, P. \BBOP1995\BBCP.
\newblock \BBOQ On submaximal spaces\BBCQ\
\newblock {\Bem Topology and its Applications}, {\Bem 64}, 219--241.

\bibitem[\protect\BCAY{Asher\ \BBA\ Vieu}{Asher\ \BBA\
  Vieu}{1995}]{Asher&Vieu95}
Asher, N.\BBACOMMA\  \BBA\ Vieu, L. \BBOP1995\BBCP.
\newblock \BBOQ Toward a geometry of common sense: A semantics and a complete
  axiomatization of mereotopology\BBCQ\
\newblock In Mellish, C.\BED, {\Bem Proceedings of the 14th International Joint
  Conference on Artificial Intelligence (IJCAI-95)}, \BPGS\ 846--852. Morgan
  Kaufmann.

\bibitem[\protect\BCAY{Balbiani\ \BBA\ Condotta}{Balbiani\ \BBA\
  Condotta}{2002}]{Balbiani&Condotta02}
Balbiani, P.\BBACOMMA\  \BBA\ Condotta, J.-F. \BBOP2002\BBCP.
\newblock \BBOQ Computational complexity of propositional linear temporal
  logics based on qualitative spatial or temporal reasoning\BBCQ\
\newblock In Armando, A.\BED, {\Bem Proceedings of Frontiers of Combining
  Systems (FroCoS 2002)}, \lowercase{\BVOL}\ 2309 of {\Bem Lecture Notes in
  Computer Science}, \BPGS\ 162--176. Springer.

\bibitem[\protect\BCAY{Balbiani, Tinchev,\ \BBA\ Vakarelov}{Balbiani
  et~al.}{2004}]{Balbianietal04}
Balbiani, P., Tinchev, T., \BBA\ Vakarelov, D. \BBOP2004\BBCP.
\newblock \BBOQ Modal logics for region-based theories of space\BBCQ\
\newblock Manuscript.

\bibitem[\protect\BCAY{Bennett}{Bennett}{1994}]{Bennett94}
Bennett, B. \BBOP1994\BBCP.
\newblock \BBOQ Spatial reasoning with propositional logic\BBCQ\
\newblock In {\Bem Proceedings of the 4th International Conference on Knowledge
  Representation and Reasoning}, \BPGS\ 51--62. Morgan Kaufmann.

\bibitem[\protect\BCAY{Bennett}{Bennett}{1996}]{Bennett96}
Bennett, B. \BBOP1996\BBCP.
\newblock \BBOQ Modal logics for qualitative spatial reasoning\BBCQ\
\newblock {\Bem Logic Journal of the IGPL}, {\Bem 4}, 23--45.

\bibitem[\protect\BCAY{Bennett\ \BBA\ Cohn}{Bennett\ \BBA\
  Cohn}{1999}]{Bennett&Cohn99}
Bennett, B.\BBACOMMA\  \BBA\ Cohn, A. \BBOP1999\BBCP.
\newblock \BBOQ Multi-dimensional multi-modal logics as a framework for
  spatio-temporal reasoning\BBCQ\
\newblock In {\Bem Proceedings of the `Hot topics in Spatio-temporal reasoning'
  workshop, IJCAI-99}, Stockholm.

\bibitem[\protect\BCAY{Bennett, Cohn, Wolter,\ \BBA\ Zakharyschev}{Bennett
  et~al.}{2002}]{Bennettetal02}
Bennett, B., Cohn, A., Wolter, F., \BBA\ Zakharyschev, M.
\BBOP2002\BBCP.
\newblock \BBOQ Multi-dimensional modal logic as a framework for
  spatio-temporal reasoning\BBCQ\
\newblock {\Bem Applied Intelligence}, {\Bem 17}, 239--251.

\bibitem[\protect\BCAY{Blackburn}{Blackburn}{1992}]{Blackburn92}
Blackburn, P. \BBOP1992\BBCP.
\newblock \BBOQ Fine grained theories of time\BBCQ\
\newblock In Aurnague, M., Borillo, A., Borillo, M., \BBA\ Bras, M.\BEDS, {\Bem
  Proceedings of the 4th European Workshop on Semantics of Time, Space, and
  Movement and Spatio-Temporal Reasoning}, \BPGS\ 299--320, Ch\^ateau de Bonas,
  France. Groupe ``Langue, Raisonnement, Calcul'', Toulouse.

\bibitem[\protect\BCAY{B{\"o}rger, Gr{\"a}del,\ \BBA\ Gurevich}{B{\"o}rger
  et~al.}{1997}]{Borgeretal97}
B{\"o}rger, E., Gr{\"a}del, E., \BBA\ Gurevich, Y. \BBOP1997\BBCP.
\newblock {\Bem The Classical Decision Problem}.
\newblock Perspectives in Mathematical Logic. Springer.

\bibitem[\protect\BCAY{Bourbaki}{Bourbaki}{1966}]{Bourbaki66}
Bourbaki, N. \BBOP1966\BBCP.
\newblock {\Bem General topology, Part 1}.
\newblock Hermann, Paris and Addison-Wesley.

\bibitem[\protect\BCAY{Chagrov\ \BBA\ Zakharyaschev}{Chagrov\ \BBA\
  Zakharyaschev}{1997}]{Chagrov&Z97}
Chagrov, A.\BBACOMMA\  \BBA\ Zakharyaschev, M. \BBOP1997\BBCP.
\newblock {\Bem Modal Logic}, \lowercase{\BVOL}~35 of {\Bem Oxford Logic
  Guides}.
\newblock Clarendon Press, Oxford.

\bibitem[\protect\BCAY{Chlebus}{Chlebus}{1986}]{Chlebus86}
Chlebus, B. \BBOP1986\BBCP.
\newblock \BBOQ Domino-tiling games\BBCQ\
\newblock {\Bem Journal of Computer and System Sciences}, {\Bem 32}, 374--392.

\bibitem[\protect\BCAY{Clarke}{Clarke}{1981}]{Clarke81}
Clarke, B. \BBOP1981\BBCP.
\newblock \BBOQ A calculus of individuals based on `connection'\BBCQ\
\newblock {\Bem Notre Dame Journal of Formal Logic}, {\Bem 23}, 204--218.

\bibitem[\protect\BCAY{Clarke}{Clarke}{1985}]{Clarke85}
Clarke, B. \BBOP1985\BBCP.
\newblock \BBOQ Individuals and points\BBCQ\
\newblock {\Bem Notre Dame Journal of Formal Logic}, {\Bem 26}, 61--75.

\bibitem[\protect\BCAY{Clarke\ \BBA\ Emerson}{Clarke\ \BBA\
  Emerson}{1981}]{Clarke&Emerson81}
Clarke, E.\BBACOMMA\  \BBA\ Emerson, E. \BBOP1981\BBCP.
\newblock \BBOQ Design and synthesis of synchronisation skeletons using
  branching time temporal logic\BBCQ\
\newblock In Kozen, D.\BED, {\Bem Logic of Programs}, \lowercase{\BVOL}\ 131 of
  {\Bem Lecture Notes in Computer Science}, \BPGS\ 52--71. Springer.

\bibitem[\protect\BCAY{Clementini, {D}i Felice,\ \BBA\ Hern\'andez}{Clementini
  et~al.}{1997}]{Clementinietal97}
Clementini, E., {D}i Felice, P., \BBA\ Hern\'andez, D.
\BBOP1997\BBCP.
\newblock \BBOQ Qualitative representation of positional information\BBCQ\
\newblock {\Bem Artificial Intelligence}, {\Bem 95}, 317--356.

\bibitem[\protect\BCAY{Cohn}{Cohn}{1997}]{Cohn97}
Cohn, A. \BBOP1997\BBCP.
\newblock \BBOQ Qualitative spatial representation and reasoning
  techniques\BBCQ\
\newblock In Brewka, G., Habel, C., \BBA\ Nebel, B.\BEDS, {\Bem KI-97: Advances
  in Artificial Intelligence}, \lowercase{\BVOL}\ 1303 of {\Bem Lecture Notes
  in Computer Science}, \BPGS\ 1--30. Springer.

\bibitem[\protect\BCAY{Davis}{Davis}{1990}]{Davis90}
Davis, E. \BBOP1990\BBCP.
\newblock {\Bem Representations of Commonsense Knowledge}.
\newblock Morgan Kaufmann.

\bibitem[\protect\BCAY{Degtyarev, Fisher,\ \BBA\ Konev}{Degtyarev
  et~al.}{2003}]{Degtyarevetal03}
Degtyarev, A., Fisher, M., \BBA\ Konev, B. \BBOP2003\BBCP.
\newblock \BBOQ Monodic temporal resolution\BBCQ\
\newblock In Baader, F.\BED, {\Bem Proceedings of the 19th International
  Conference on Automated Deduction (CADE-19)}, \lowercase{\BVOL}\ 2741 of
  {\Bem Lecture Notes in Artificial Intelligence}, \BPGS\ 397--411. Springer.

\bibitem[\protect\BCAY{Demri\ \BBA\ D'Souza}{Demri\ \BBA\
  D'Souza}{2002}]{Demri&Souza02}
Demri, S.\BBACOMMA\  \BBA\ D'Souza, D. \BBOP2002\BBCP.
\newblock \BBOQ An automata-theoretic approach to constraint {LTL}\BBCQ\
\newblock In Agrawal, M.\BBACOMMA\  \BBA\ Seth, A.\BEDS, {\Bem Proceedings of
  the 22nd Conference on Foundations of Software Technology and Theoretical
  Computer Science (FST TCS 2002)}, \lowercase{\BVOL}\ 2556 of {\Bem Lecture
  Notes in Computer Science}, \BPGS\ 121--132. Springer.

\bibitem[\protect\BCAY{Egenhofer\ \BBA\ Franzosa}{Egenhofer\ \BBA\
  Franzosa}{1991}]{Egenhofer&Franzosa91}
Egenhofer, M.\BBACOMMA\  \BBA\ Franzosa, R. \BBOP1991\BBCP.
\newblock \BBOQ Point-set topological spatial relations\BBCQ\
\newblock {\Bem International Journal of Geographical Information Systems},
  {\Bem 5}, 161--174.

\bibitem[\protect\BCAY{Egenhofer\ \BBA\ Herring}{Egenhofer\ \BBA\
  Herring}{1991}]{Egenhofer&Herring91}
Egenhofer, M.\BBACOMMA\  \BBA\ Herring, J. \BBOP1991\BBCP.
\newblock \BBOQ Categorizing topological relationships between regions, lines
  and points in geographic databases\BBCQ\
\newblock \BTR, University of Maine.

\bibitem[\protect\BCAY{Egenhofer\ \BBA\ Sharma}{Egenhofer\ \BBA\
  Sharma}{1993}]{Egenhofer&Sharma93}
Egenhofer, M.\BBACOMMA\  \BBA\ Sharma, J. \BBOP1993\BBCP.
\newblock \BBOQ Assessing the consistency of complete and incomplete
  topological information\BBCQ\
\newblock {\Bem Geographical Systems}, {\Bem 1}, 47--68.

\bibitem[\protect\BCAY{Emerson\ \BBA\ Halpern}{Emerson\ \BBA\
  Halpern}{1985}]{Emerson&Halpern85}
Emerson, E.\BBACOMMA\  \BBA\ Halpern, J. \BBOP1985\BBCP.
\newblock \BBOQ Decision procedures and expressiveness in the temporal logic of
  branching time\BBCQ\
\newblock {\Bem Journal of Computer and System Sciences}, {\Bem 30}, 1--24.

\bibitem[\protect\BCAY{Fine}{Fine}{1974}]{Fine74b}
Fine, K. \BBOP1974\BBCP.
\newblock \BBOQ Logics containing {${\mathcal{K}\mathrm{4}}$}, part {I}\BBCQ\
\newblock {\Bem Journal of Symbolic Logic}, {\Bem 39}, 229--237.

\bibitem[\protect\BCAY{Finger\ \BBA\ Gabbay}{Finger\ \BBA\
  Gabbay}{1992}]{Finger&Gabbay92}
Finger, M.\BBACOMMA\  \BBA\ Gabbay, D. \BBOP1992\BBCP.
\newblock \BBOQ Adding a temporal dimension to a logic system\BBCQ\
\newblock {\Bem Journal of Logic, Language and Information}, {\Bem 2},
  203--233.

\bibitem[\protect\BCAY{Fisher, Dixon,\ \BBA\ Peim}{Fisher
  et~al.}{2001}]{Fisheretal01}
Fisher, M., Dixon, C., \BBA\ Peim, M. \BBOP2001\BBCP.
\newblock \BBOQ Clausal temporal resolution\BBCQ\
\newblock {\Bem ACM Transactions on Computational Logic (TOCL)}, {\Bem 2},
  12--56.

\bibitem[\protect\BCAY{Gabbay, Hodkinson,\ \BBA\ Reynolds}{Gabbay
  et~al.}{1994}]{Gabbayetal94}
Gabbay, D., Hodkinson, I., \BBA\ Reynolds, M. \BBOP1994\BBCP.
\newblock {\Bem Temporal Logic: Mathematical Foundations and Computational
  Aspects, Volume 1}.
\newblock Oxford University Press.

\bibitem[\protect\BCAY{Galton\ \BBA\ Meathrel}{Galton\ \BBA\
  Meathrel}{1999}]{Galton&Meathrel99}
Galton, A.\BBACOMMA\  \BBA\ Meathrel, R. \BBOP1999\BBCP.
\newblock \BBOQ Qualitative outline theory\BBCQ\
\newblock In Dean, T.\BED, {\Bem Proceedings of the 16th International Joint
  Conference on Artificial Intelligence (IJCAI-99)}, \BPGS\ 1061--1066. Morgan
  Kaufmann.

\bibitem[\protect\BCAY{Gerevini\ \BBA\ Nebel}{Gerevini\ \BBA\
  Nebel}{2002}]{Gerevini&Nebel02ecai}
Gerevini, A.\BBACOMMA\  \BBA\ Nebel, B. \BBOP2002\BBCP.
\newblock \BBOQ Qualitative spatio-temporal reasoning with {${\RCCE}$} and
  {A}llen's interval calculus: Computational complexity\BBCQ\
\newblock In {\Bem Proceedings of the 15th European Conference on Artificial
  Intelligence (ECAI'02)}, \BPGS\ 312--316. IOS Press.

\bibitem[\protect\BCAY{Gerevini\ \BBA\ Renz}{Gerevini\ \BBA\
  Renz}{2002}]{Gerevini&Renz02}
Gerevini, A.\BBACOMMA\  \BBA\ Renz, J. \BBOP2002\BBCP.
\newblock \BBOQ Combining topological and size constraints for spatial
  reasoning\BBCQ\
\newblock {\Bem Artificial Intelligence}, {\Bem 137}, 1--42.

\bibitem[\protect\BCAY{G{\"o}del}{G{\"o}del}{1933}]{Godel33a}
G{\"o}del, K. \BBOP1933\BBCP.
\newblock \BBOQ Eine {I}nterpretation des intuitionistischen
  {A}ussagenkalk{\"u}ls\BBCQ\
\newblock {\Bem Ergebnisse eines mathematischen Kolloquiums}, {\Bem 4}, 39--40.

\bibitem[\protect\BCAY{Goldblatt}{Goldblatt}{1976}]{Goldblatt76a}
Goldblatt, R. \BBOP1976\BBCP.
\newblock \BBOQ Metamathematics of modal logic, {P}art {I}\BBCQ\
\newblock {\Bem Reports on Mathematical Logic}, {\Bem 6}, 41--78.

\bibitem[\protect\BCAY{Goranko\ \BBA\ Passy}{Goranko\ \BBA\
  Passy}{1992}]{Goranko&Passy92}
Goranko, V.\BBACOMMA\  \BBA\ Passy, S. \BBOP1992\BBCP.
\newblock \BBOQ Using the universal modality: gains and questions\BBCQ\
\newblock {\Bem Journal of Logic and Computation}, {\Bem 2}, 5--30.

\bibitem[\protect\BCAY{Gotts}{Gotts}{1996}]{Gotts96a}
Gotts, N. \BBOP1996\BBCP.
\newblock \BBOQ An axiomatic approach to topology for spatial information
  systems\BBCQ\
\newblock \BTR\ 96.25, School of Computer Studies, University of Leeds.

\bibitem[\protect\BCAY{Halpern\ \BBA\ Shoham}{Halpern\ \BBA\
  Shoham}{1986}]{Halpern&Shoham86}
Halpern, J.\BBACOMMA\  \BBA\ Shoham, Y. \BBOP1986\BBCP.
\newblock \BBOQ A propositional modal logic of time intervals\BBCQ\
\newblock In {\Bem Proceedings of the 1st Annual {IEEE} Symposium on Logic in
  Computer Science (LICS'86)}, \BPGS\ 279--292. IEEE Computer Society.

\bibitem[\protect\BCAY{Halpern\ \BBA\ Vardi}{Halpern\ \BBA\
  Vardi}{1989}]{Halpern&Vardi89}
Halpern, J.\BBACOMMA\  \BBA\ Vardi, M. \BBOP1989\BBCP.
\newblock \BBOQ The complexity of reasoning about knowledge and time {I}: lower
  bounds\BBCQ\
\newblock {\Bem Journal of Computer and System Sciences}, {\Bem 38}, 195--237.

\bibitem[\protect\BCAY{Hodkinson}{Hodkinson}{2004}]{Hodkinson03guarded}
Hodkinson, I. \BBOP2004\BBCP.
\newblock \BBOQ Complexity of monodic packed fragment over linear and real
  time\BBCQ\
\newblock To appear in {\em Annals of Pure and Applied Logic}, available at
  {\tt http://www.doc.ic.ac.uk/} {\tt /\~{}imh/papers/cxmonlin.pdf}.

\bibitem[\protect\BCAY{Hodkinson, Kontchakov, Kurucz, Wolter,\ \BBA\
  Zakharyaschev}{Hodkinson et~al.}{2003}]{Hodkinsonetal03}
Hodkinson, I., Kontchakov, R., Kurucz, A., Wolter, F., \BBA\
Zakharyaschev, M.
  \BBOP2003\BBCP.
\newblock \BBOQ On the computational complexity of decidable fragments of
  first-order linear temporal logics\BBCQ\
\newblock In Reynolds, M.\BBACOMMA\  \BBA\ Sattar, A.\BEDS, {\Bem Proceedings
  of {TIME-ICTL} 2003}, \BPGS\ 91--98. IEEE Computer Society.

\bibitem[\protect\BCAY{Hodkinson, Wolter,\ \BBA\ Zakharyaschev}{Hodkinson
  et~al.}{2000}]{Hodkinsonetal00}
Hodkinson, I., Wolter, F., \BBA\ Zakharyaschev, M. \BBOP2000\BBCP.
\newblock \BBOQ Decidable fragments of first-order temporal logics\BBCQ\
\newblock {\Bem Annals of Pure and Applied Logic}, {\Bem 106}, 85--134.

\bibitem[\protect\BCAY{Hodkinson, Wolter,\ \BBA\ Zakharyaschev}{Hodkinson
  et~al.}{2001}]{Hodkinsonetal01lpar}
Hodkinson, I., Wolter, F., \BBA\ Zakharyaschev, M. \BBOP2001\BBCP.
\newblock \BBOQ Monodic fragments of first-order temporal logics: 2000--2001
  {A.D}\BBCQ\
\newblock In Nieuwenhuis, R.\BBACOMMA\  \BBA\ Voronkov, A.\BEDS, {\Bem Logic
  for Programming, Artificial Intelligence and Reasoning}, \lowercase{\BVOL}\
  2250 of {\Bem Lecture Notes in Artificial Intelligence}, \BPGS\ 1--23.
  Springer.

\bibitem[\protect\BCAY{Hodkinson, Wolter,\ \BBA\ Zakharyaschev}{Hodkinson
  et~al.}{2002}]{Hodkinsonetal02lics}
Hodkinson, I., Wolter, F., \BBA\ Zakharyaschev, M. \BBOP2002\BBCP.
\newblock \BBOQ Decidable and undecidable fragments of first-order branching
  temporal logics\BBCQ\
\newblock In {\Bem Proceedings of the 17th Annual IEEE Symposium on Logic in
  Computer Science (LICS 2002)}, \BPGS\ 393--402. IEEE Computer Society.

\bibitem[\protect\BCAY{Hughes\ \BBA\ Cresswell}{Hughes\ \BBA\
  Cresswell}{1996}]{Hughes&Cresswell96}
Hughes, G.\BBACOMMA\  \BBA\ Cresswell, M. \BBOP1996\BBCP.
\newblock {\Bem A New Introduction to Modal Logic}.
\newblock Methuen, London.

\bibitem[\protect\BCAY{Hustadt\ \BBA\ Konev}{Hustadt\ \BBA\
  Konev}{2003}]{Hustadt&Konev03}
Hustadt, U.\BBACOMMA\  \BBA\ Konev, B. \BBOP2003\BBCP.
\newblock \BBOQ {TRP{\small $++$}} 2.0: A temporal resolution prover\BBCQ\
\newblock In Baader, F.\BED, {\Bem Proceedings of the 19th International
  Conference on Automated Deduction (CADE-19)}, \lowercase{\BVOL}\ 2741 of
  {\Bem Lecture Notes in Computer Science}, \BPGS\ 274--278. Springer.

\bibitem[\protect\BCAY{Kontchakov, Lutz, Wolter,\ \BBA\
  Zakharyaschev}{Kontchakov et~al.}{2004}]{Kontchakovetal02}
Kontchakov, R., Lutz, C., Wolter, F., \BBA\ Zakharyaschev, M.
\BBOP2004\BBCP.
\newblock \BBOQ Temporalising tableaux\BBCQ\
\newblock {\Bem Studia Logica}, {\Bem 76}, 91--134.

\bibitem[\protect\BCAY{Kutz, Sturm, Suzuki, Wolter,\ \BBA\ Zakharyaschev}{Kutz
  et~al.}{2003}]{Kutzetal03}
Kutz, O., Sturm, H., Suzuki, N.-Y., Wolter, F., \BBA\ Zakharyaschev,
M.
  \BBOP2003\BBCP.
\newblock \BBOQ Logics of metric spaces\BBCQ\
\newblock {\Bem ACM Transactions on Computational Logic}, {\Bem 4}, 260--294.

\bibitem[\protect\BCAY{Ladner}{Ladner}{1977}]{Ladner77}
Ladner, R. \BBOP1977\BBCP.
\newblock \BBOQ The computational complexity of provability in systems of modal
  logic\BBCQ\
\newblock {\Bem SIAM Journal on Computing}, {\Bem 6}, 467--480.

\bibitem[\protect\BCAY{Lemon\ \BBA\ Pratt}{Lemon\ \BBA\
  Pratt}{1998}]{Lemon&Pratt98}
Lemon, O.\BBACOMMA\  \BBA\ Pratt, I. \BBOP1998\BBCP.
\newblock \BBOQ On the incompleteness of modal logics of space: advancing
  complete modal logics of place\BBCQ\
\newblock In Kracht, M., de~Rijke, M., Wansing, H., \BBA\ Zakharyaschev,
  M.\BEDS, {\Bem Advances in Modal Logic, Volume 1}, \BPGS\ 115--132. CSLI
  Publications, Stanford.

\bibitem[\protect\BCAY{Lewis\ \BBA\ Langford}{Lewis\ \BBA\
  Langford}{1932}]{Lewis&Langford32}
Lewis, C.\BBACOMMA\  \BBA\ Langford, C. \BBOP1932\BBCP.
\newblock {\Bem Symbolic Logic}.
\newblock Appleton-Century-Crofts, New York.

\bibitem[\protect\BCAY{Ligozat}{Ligozat}{1998}]{Ligozat98}
Ligozat, G. \BBOP1998\BBCP.
\newblock \BBOQ Reasoning about cardinal directions\BBCQ\
\newblock {\Bem Journal of Visual Languages and Computing}, {\Bem 9}, 23--44.

\bibitem[\protect\BCAY{Manna\ \BBA\ Pnueli}{Manna\ \BBA\
  Pnueli}{1992}]{Manna&Pnueli92}
Manna, Z.\BBACOMMA\  \BBA\ Pnueli, A. \BBOP1992\BBCP.
\newblock {\Bem The Temporal Logic of Reactive and Concurrent Systems}.
\newblock Springer.

\bibitem[\protect\BCAY{McKinsey}{McKinsey}{1941}]{McKinsey41}
McKinsey, J. \BBOP1941\BBCP.
\newblock \BBOQ A solution of the decision problem for the {L}ewis systems
  {${\mathcal{S}\mathrm{2}}$} and {${\mathcal{S}\mathrm{4}}$}, with an
  application to topology\BBCQ\
\newblock {\Bem Journal of Symbolic Logic}, {\Bem 6}, 117--134.

\bibitem[\protect\BCAY{McKinsey\ \BBA\ Tarski}{McKinsey\ \BBA\
  Tarski}{1944}]{McKinsey&Tarski44}
McKinsey, J.\BBACOMMA\  \BBA\ Tarski, A. \BBOP1944\BBCP.
\newblock \BBOQ The algebra of topology\BBCQ\
\newblock {\Bem Annals of Mathematics}, {\Bem 45}, 141--191.

\bibitem[\protect\BCAY{Muller}{Muller}{1998a}]{Muller98}
Muller, P. \BBOP1998a\BBCP.
\newblock \BBOQ A qualitative theory of motion based on spatio-temporal
  primitives\BBCQ\
\newblock In Cohn, A., Schubert, L., \BBA\ Shapiro, S.\BEDS, {\Bem Proceedings
  of the 6th International Conference on Principles of Knowledge Representation
  and Reasoning (KR'98)}, \BPGS\ 131--142. Morgan Kaufmann.

\bibitem[\protect\BCAY{Muller}{Muller}{1998b}]{Muller98a}
Muller, P. \BBOP1998b\BBCP.
\newblock \BBOQ Space-time as a primitive for space and motion\BBCQ\
\newblock In Guarino, N.\BED, {\Bem Proceedings of the International Conference
  on Formal Ontology in Information Systems (FOIS'98)}, \lowercase{\BVOL}~46 of
  {\Bem Frontiers in Artificial Intelligence and Applications}, \BPGS\ 63--76.
  IOS Press.

\bibitem[\protect\BCAY{Nebel\ \BBA\ B\"{u}rckert}{Nebel\ \BBA\
  B\"{u}rckert}{1995}]{Nebel&Buerckert95}
Nebel, B.\BBACOMMA\  \BBA\ B\"{u}rckert, H. \BBOP1995\BBCP.
\newblock \BBOQ Reasoning about relations: A maximal tractable subclass of
  {A}llen's interval algebra\BBCQ\
\newblock {\Bem Journal of the ACM}, {\Bem 42}, 43--66.

\bibitem[\protect\BCAY{Nebel}{Nebel}{1996}]{Nebel96}
Nebel, B. \BBOP1996\BBCP.
\newblock \BBOQ Artificial intelligence: A computational perspective\BBCQ\
\newblock In Brewka, G.\BED, {\Bem Principles of Knowledge Representation},
  \BPGS\ 237--266. CSLI Publications.

\bibitem[\protect\BCAY{Nutt}{Nutt}{1999}]{Nutt99}
Nutt, W. \BBOP1999\BBCP.
\newblock \BBOQ On the translation of qualitative spatial reasoning problems
  into modal logics\BBCQ\
\newblock In Burgard, W., Christaller, T., \BBA\ Cremers, A.\BEDS, {\Bem
  Advances in Artificial Intelligence. Proceedings of the 23rd Annual German
  Conference on Artificial Intelligence (KI'99)}, \lowercase{\BVOL}\ 1701 of
  {\Bem Lecture Notes in Computer Science}, \BPGS\ 113--124. Springer.

\bibitem[\protect\BCAY{Ono\ \BBA\ Nakamura}{Ono\ \BBA\
  Nakamura}{1980}]{Ono&Nakamura80}
Ono, H.\BBACOMMA\  \BBA\ Nakamura, A. \BBOP1980\BBCP.
\newblock \BBOQ On the size of refutation {K}ripke models for some linear modal
  and tense logics\BBCQ\
\newblock {\Bem Studia Logica}, {\Bem 39}, 325--333.

\bibitem[\protect\BCAY{Orlov}{Orlov}{1928}]{Orlov28}
Orlov, I. \BBOP1928\BBCP.
\newblock \BBOQ The calculus of compatibility of propositions\BBCQ\
\newblock {\Bem Mathematics of the USSR, Sbornik}, {\Bem 35}, 263--286.
\newblock (In Russian).

\bibitem[\protect\BCAY{Post}{Post}{1946}]{Post46}
Post, E. \BBOP1946\BBCP.
\newblock \BBOQ A variant of a recursively unsolvable problem\BBCQ\
\newblock {\Bem Bulletin of the AMS}, {\Bem 52}, 264--268.

\bibitem[\protect\BCAY{Pratt-Hartmann}{Pratt-Hartmann}{2002}]{PrattHartmann02}
Pratt-Hartmann, I. \BBOP2002\BBCP.
\newblock \BBOQ A topological constraint language with component counting\BBCQ\
\newblock {\Bem Journal of Applied Non-Classical Logics}, {\Bem 12}, 441--467.

\bibitem[\protect\BCAY{Randell, Cui,\ \BBA\ Cohn}{Randell
  et~al.}{1992}]{Randelletal92}
Randell, D., Cui, Z., \BBA\ Cohn, A. \BBOP1992\BBCP.
\newblock \BBOQ A spatial logic based on regions and connection\BBCQ\
\newblock In Nebel, B., Rich, C., \BBA\ Swartout, W.\BEDS, {\Bem Proceedings of
  the 3rd International Conference on Principles of Knowledge Representation
  and Reasoning (KR'92)}, \BPGS\ 165--176. Morgan Kaufmann.

\bibitem[\protect\BCAY{Renz}{Renz}{1998}]{Renz98}
Renz, J. \BBOP1998\BBCP.
\newblock \BBOQ A canonical model of the region connection calculus\BBCQ\
\newblock In Cohn, A., Schubert, L., \BBA\ Shapiro, S.\BEDS, {\Bem Proceedings
  of the 6th International Conference on Principles of Knowledge Representation
  and Reasoning (KR'98)}, \BPGS\ 330--341. Morgan Kaufmann.

\bibitem[\protect\BCAY{Renz\ \BBA\ Nebel}{Renz\ \BBA\
  Nebel}{1998}]{Renz&Nebel98}
Renz, J.\BBACOMMA\  \BBA\ Nebel, B. \BBOP1998\BBCP.
\newblock \BBOQ Spatial reasoning with topological information\BBCQ\
\newblock In Freksa, C., Habel, C., \BBA\ Wender, K.\BEDS, {\Bem Spatial
  Cognition---An Interdisciplinary Approach to Representation and Processing of
  Spatial Knowledge}, \lowercase{\BVOL}\ 1404 of {\Bem Lecture Notes in
  Computer Science}, \BPGS\ 351--372. Springer.

\bibitem[\protect\BCAY{Renz\ \BBA\ Nebel}{Renz\ \BBA\
  Nebel}{1999}]{Renz&Nebel99}
Renz, J.\BBACOMMA\  \BBA\ Nebel, B. \BBOP1999\BBCP.
\newblock \BBOQ On the complexity of qualitative spatial reasoning\BBCQ\
\newblock {\Bem Artificial Intelligence}, {\Bem 108}, 69--123.

\bibitem[\protect\BCAY{Renz}{Renz}{2002}]{Renz02}
Renz, J. \BBOP2002\BBCP.
\newblock \BBOQ A canonical model of the region connection calculus\BBCQ\
\newblock {\Bem Journal of Applied Non-Classical Logics}, {\Bem 12}, 469--494.

\bibitem[\protect\BCAY{Renz\ \BBA\ Nebel}{Renz\ \BBA\
  Nebel}{2001}]{Renz&Nebel01}
Renz, J.\BBACOMMA\  \BBA\ Nebel, B. \BBOP2001\BBCP.
\newblock \BBOQ Efficient methods for qualitative spatial reasoning\BBCQ\
\newblock {\Bem Journal of Artificial Intelligence Research}, {\Bem 15},
  289--318.

\bibitem[\protect\BCAY{Reynolds}{Reynolds}{2003}]{Reynolds03}
Reynolds, M. \BBOP2003\BBCP.
\newblock \BBOQ The complexity of the temporal logic with `until' over general
  linear time\BBCQ\
\newblock {\Bem Journal of Computer and System Science}, {\Bem 66}, 393--426.

\bibitem[\protect\BCAY{Reynolds}{Reynolds}{2004}]{Reynolds99}
Reynolds, M. \BBOP2004\BBCP.
\newblock \BBOQ The complexity of the temporal logic over the reals\BBCQ\
\newblock Submitted, available at {\tt
  http://www.csse.uwa.edu.au/\~{}mark/research/Online/CORT.htm}.

\bibitem[\protect\BCAY{Schwendimann}{Schwendimann}{1998}]{Schwendimann98b}
Schwendimann, S. \BBOP1998\BBCP.
\newblock \BBOQ A new one-pass tableau calculus for {PLTL}\BBCQ\
\newblock In de~Swart, H.\BED, {\Bem Proceedings of the International
  Conference on Automated Reasoning with Analytic Tableaux and Related Methods
  ({TABLEAUX}-98)}, \lowercase{\BVOL}\ 1397 of {\Bem Lecture Notes in
  Artificial Intelligence}, \BPGS\ 277--291. Springer.

\bibitem[\protect\BCAY{Sistla\ \BBA\ Clarke}{Sistla\ \BBA\
  Clarke}{1985}]{Sistla&Clarke85}
Sistla, A.\BBACOMMA\  \BBA\ Clarke, E. \BBOP1985\BBCP.
\newblock \BBOQ The complexity of propositional linear temporal logics\BBCQ\
\newblock {\Bem Journal of the Association for Computing Machinery}, {\Bem 32},
  733--749.

\bibitem[\protect\BCAY{Sistla\ \BBA\ German}{Sistla\ \BBA\
  German}{1987}]{Sistla&German87}
Sistla, A.\BBACOMMA\  \BBA\ German, S. \BBOP1987\BBCP.
\newblock \BBOQ Reasoning with many processes\BBCQ\
\newblock In {\Bem Proceedings of the Second IEEE Symposium on Logic in
  Computer Science (LICS'87)}, \BPGS\ 138--153. IEEE Computer Society.

\bibitem[\protect\BCAY{Slagle, Chang,\ \BBA\ Lee}{Slagle
  et~al.}{1969}]{Slagleetal69}
Slagle, J., Chang, C.-L., \BBA\ Lee, R. \BBOP1969\BBCP.
\newblock \BBOQ Completeness theorems for semantic resolution in
  consequence-finding\BBCQ\
\newblock In {\Bem Proceedings of the 1st International Joint Conference on
  Artificial Intelligence (IJCAI'69)}, \BPGS\ 281--286. William Kaufmann.

\bibitem[\protect\BCAY{Smith\ \BBA\ Park}{Smith\ \BBA\
  Park}{1992}]{Smith&Park92}
Smith, T.\BBACOMMA\  \BBA\ Park, K. \BBOP1992\BBCP.
\newblock \BBOQ An algebraic approach to spatial reasoning\BBCQ\
\newblock {\Bem International Journal of Geographical Information Systems},
  {\Bem 6}, 177--192.

\bibitem[\protect\BCAY{Stockmeyer}{Stockmeyer}{1974}]{Stockmeyer74}
Stockmeyer, L. \BBOP1974\BBCP.
\newblock {\Bem The Complexity of Decision Problems in Automata Theory and
  Logic}.
\newblock Ph.D.\ thesis, MIT.

\bibitem[\protect\BCAY{Stockmeyer}{Stockmeyer}{1987}]{Stockmeyer87}
Stockmeyer, L. \BBOP1987\BBCP.
\newblock \BBOQ Classifying the computational complexity of problems\BBCQ\
\newblock {\Bem Journal of Symbolic Logic}, {\Bem 52}, 1--43.

\bibitem[\protect\BCAY{Stone}{Stone}{1937}]{Stone37}
Stone, M. \BBOP1937\BBCP.
\newblock \BBOQ Application of the theory of {B}oolean rings to general
  topology\BBCQ\
\newblock {\Bem Transactions of the AMS}, {\Bem 41}, 321--364.

\bibitem[\protect\BCAY{Tarski}{Tarski}{1938}]{Tarski38}
Tarski, A. \BBOP1938\BBCP.
\newblock \BBOQ Der {A}ussagenkalk{\"u}l und die {T}opologie\BBCQ\
\newblock {\Bem Fundamenta Mathematicae}, {\Bem 31}, 103--134.

\bibitem[\protect\BCAY{{Tsao {C}hen}}{{Tsao {C}hen}}{1938}]{Tsao-Chen38}
{Tsao {C}hen}, T. \BBOP1938\BBCP.
\newblock \BBOQ Algebraic postulates and a geometric interpretation of the
  {L}ewis calculus of strict implication\BBCQ\
\newblock {\Bem Bulletin of the AMS}, {\Bem 44}, 737--744.

\bibitem[\protect\BCAY{van Benthem}{van Benthem}{1995}]{Benthem95}
van Benthem, J. \BBOP1995\BBCP.
\newblock \BBOQ Temporal logic\BBCQ\
\newblock In Gabbay, D., Hogger, C., \BBA\ Robinson, J.\BEDS, {\Bem Handbook of
  Logic in Artificial Intelligence and Logic Programming}, \lowercase{\BVOL}~4,
  \BPGS\ 241--350. Oxford Scientific Publishers.

\bibitem[\protect\BCAY{van Emde~Boas}{van Emde~Boas}{1997}]{EmdeBoas97}
van Emde~Boas, P. \BBOP1997\BBCP.
\newblock \BBOQ The convenience of tilings\BBCQ\
\newblock In Sorbi, A.\BED, {\Bem Complexity, Logic and Recursion Theory},
  \lowercase{\BVOL}\ 187 of {\Bem Lecture Notes in Pure and Applied
  Mathematics}, \BPGS\ 331--363. Marcel Dekker Inc.

\bibitem[\protect\BCAY{Wolper}{Wolper}{1985}]{Wolper85}
Wolper, P. \BBOP1985\BBCP.
\newblock \BBOQ The tableau method for temporal logic: An overview\BBCQ\
\newblock {\Bem Logique et Analyse}, {\Bem 28}, 119--152.

\bibitem[\protect\BCAY{Wolter}{Wolter}{1996}]{Wolter96e}
Wolter, F. \BBOP1996\BBCP.
\newblock \BBOQ Properties of tense logics\BBCQ\
\newblock {\Bem Mathematical Logic Quarterly}, {\Bem 42}, 481--500.

\bibitem[\protect\BCAY{Wolter\ \BBA\ Zakharyaschev}{Wolter\ \BBA\
  Zakharyaschev}{2000a}]{Wolter&Z00ecai}
Wolter, F.\BBACOMMA\  \BBA\ Zakharyaschev, M. \BBOP2000a\BBCP.
\newblock \BBOQ Spatial reasoning in {${\RCCE}$} with {B}oolean region
  terms\BBCQ\
\newblock In Horn, W.\BED, {\Bem Proceedings of the 14th European Conference on
  Artificial Intelligence (ECAI 2000)}, \BPGS\ 244--248. IOS Press.

\bibitem[\protect\BCAY{Wolter\ \BBA\ Zakharyaschev}{Wolter\ \BBA\
  Zakharyaschev}{2000b}]{Wolter&Z00kr}
Wolter, F.\BBACOMMA\  \BBA\ Zakharyaschev, M. \BBOP2000b\BBCP.
\newblock \BBOQ Spatio-temporal representation and reasoning based on
  {${\RCCE}$}\BBCQ\
\newblock In Cohn, A., Giunchiglia, F., \BBA\ Seltman, B.\BEDS, {\Bem
  Proceedings of the 7th Conference on Principles of Knowledge Representation
  and Reasoning (KR 2000)}, \BPGS\ 3--14. Morgan Kaufmann.

\bibitem[\protect\BCAY{Wolter\ \BBA\ Zakharyaschev}{Wolter\ \BBA\
  Zakharyaschev}{2002}]{Wolter&Z02millenium}
Wolter, F.\BBACOMMA\  \BBA\ Zakharyaschev, M. \BBOP2002\BBCP.
\newblock \BBOQ Qualitative spatio-temporal representation and reasoning: a
  computational perspective\BBCQ\
\newblock In Lakemeyer, G.\BBACOMMA\  \BBA\ Nebel, B.\BEDS, {\Bem Exploring
  Artificial Intelligence in the New Millenium}, \BPGS\ 175--216. Morgan
  Kaufmann.

\bibitem[\protect\BCAY{Zimmermann}{Zimmermann}{1995}]{Zimmermann95}
Zimmermann, K. \BBOP1995\BBCP.
\newblock \BBOQ Measuring without measures: the delta-calculus\BBCQ\
\newblock In Frank, A.\BBACOMMA\  \BBA\ Kuhn, W.\BEDS, {\Bem Proceedings of the
  2nd International Conference on Spatial Information Theory (COSIT)},
  \lowercase{\BVOL}\ 988 of {\Bem Lecture Notes in Computer Science}, \BPGS\
  59--67. Springer.

\end{thebibliography}

\end{document}
