\batchmode
\documentclass[9pt]{article}
\RequirePackage{ifthen}




\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage[comma]{natbib}


\usepackage[vlined]{algorithm2e}

%
\providecommand{\etc}{{\it etc.\ }}%
\providecommand{\eg}{{\it e.g.\ }}%
\providecommand{\egcomma}{{\it e.g.,}}%
\providecommand{\ie}{{\it i.e.\ }}%
\providecommand{\false}{\Lambda}%
\providecommand{\leaveout}[1]{} 

%
\providecommand{\proofsystem}[1]{{\sf \small #1}}%
\providecommand{\DPLL}{{\sf \small DPLL}}%
\providecommand{\CL}{{\sf \small CL}}%
\providecommand{\CLminus}{{\sf \small CL{-}-}}%
\providecommand{\RES}{{\sf \small RES}}%
\providecommand{\RESk}[1]{{\sf \small RES($#1$)}} 

%
\providecommand{\mathproofsystem}[1]{{\mathsf{\scriptstyle #1}}}%
\providecommand{\mDPLL}{{\mathsf{\scriptstyle DPLL}}}%
\providecommand{\mCL}{{\mathsf{\scriptstyle CL}}}%
\providecommand{\mCLminus}{{\mathsf{\scriptstyle CL\mbox{{-}-}}}}%
\providecommand{\mRES}{{\mathsf{\scriptstyle RES}}}%
\providecommand{\mRESk}[1]{{\mathsf{\scriptstyle RES($#1$)}}} 



\newtheorem{theorem}{\bf Theorem} 
\newtheorem{corollary}{\bf Corollary} 
\newtheorem{lemma}{\bf Lemma} 
\newtheorem{claim}{\bf Claim} 
\newtheorem{observation}{\bf Observation} 
\newtheorem{proposition}{\bf Proposition} 


\newtheorem{definition}{\bf Definition} 


%
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\begin{algorithm}
% latex2html id marker 512

\par
\par
\caption{{\tt PebSeq1UIP}, generating branching sequence for pebbling formulas}
\par
\dontprintsemicolon
\SetKwFunction{PebSubseqOneUIPWrapper}{PebSubseq1UIPWrapper}
\SetKwFunction{PebSubseqOneUIP}{PebSubseq1UIP}
\SetKwFunction{BottomUpTraversal}{BottomUpTraversal}
\SetKwFunction{Sort}{Sort}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKw{And}{and}
\SetKw{Or}{or}
\SetKw{In}{in}
\SetKw{Downto}{downto}
\par
\begin{small}
\par
\Input{Pebbling graph $G$\  with no repeated labels}
\par
\Output{Branching sequence for $Pbl_G$\  for the 1UIP learning scheme}
\par
\Begin{
  \ForEach {$v$\  \In \BottomUpTraversal{G}}{
    $v.height \leftarrow 1 + max_{u \in v.preds}\{u.height\}$\;
    \Sort{$v.preds$, increasing order by $height$}\;
  }
  \BlankLine
  {\sf // First handle unit clause labeled nodes and generate their sequence}\;
  $U  \leftarrow \{v \in G.nodes : |v.labels| = 1\}$\;
  $G.edges \leftarrow G.edges \setminus \{(u,v) \in G.edges : u \in U\}$\;
  Add to $G.sources$\  any new nodes with now 0 preds\;
  \Sort{$U$, increasing order by $height$}\;
  \ForEach {$u \in U \setminus G.targets $}{
    Output $u.label$\;
    \PebSubseqOneUIPWrapper{$u$}\;
  }
  \BlankLine
  {\sf // Now add branching sequence for targets by increasing height}\;
  \Sort{$G.targets$, increasing order by $height$}\;
  \ForEach {$t \in G.targets$}{
    \PebSubseqOneUIPWrapper{$t$}\;
  }
}
\BlankLine
\PebSubseqOneUIPWrapper{node $v$} \Begin{
  \If {$|v.preds| > 0$}{
    \PebSubseqOneUIP{$v$, $|v.preds|$}\;
  }
}
\BlankLine
\PebSubseqOneUIP{node $v$, int $i$} \Begin{
  $u \leftarrow v.preds[i]$\;
  \BlankLine
  {\sf // If this is the lowest predecessor \ldots}\;
  \If {$i = 1$}{
    \If {$!u.visited$\  \And $u \notin G.sources$}{
      $u.visited \leftarrow$\  {\sc true}\;
      \PebSubseqOneUIPWrapper{$u$}\;
    }
    \Return\;
  }
  \BlankLine
  {\sf // If this is not the lowest one \ldots}\;
  Output $u.labels \setminus \{u.lastLabel\}$\;
  \If{$!u.visitedAsHigh$\  \And $u \notin G.sources$}{
    $u.visitedAsHigh \leftarrow$\  {\sc true}\;
    Output $u.lastLabel$\;
    \If{$!u.visited$}{
      $u.visited \leftarrow$\  {\sc true}\;
      \PebSubseqOneUIPWrapper{$u$}\;
    }
  }
  \PebSubseqOneUIP{$v$, $i-1$}\;
  \BlankLine
  \For {$j \leftarrow (|u.labels|-2)$\  \Downto $1$}{
    Output $u.labels[1], \ldots, u.labels[j]$\;
    \PebSubseqOneUIP{$v$, $i-1$}\;
  }
  \PebSubseqOneUIP{$v$, $i-1$}\;
}
\end{small}
\end{algorithm}%
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\begin{algorithm}
% latex2html id marker 584
[!ht]
\caption{{\tt GridPebSeq1UIP}, generating branching sequence for grid pebbling formulas}
\par
\dontprintsemicolon
\SetKwFunction{GridPebSubseqOneUIP}{GridPebSubseq1UIP}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKw{And}{and}
\par
\begin{small}
\par
\Input{Grid pebbling graph $G$\  with target node $t$}
\par
\Output{Branching sequence for $Pbl_G$\  for the 1UIP learning scheme}
\par
\Begin{
  \GridPebSubseqOneUIP{$t$}\;
}
\BlankLine
\GridPebSubseqOneUIP{node $v$} \Begin{
  \If {$v \in G.sources$}{
    \Return\;
  }
  \BlankLine
    $u \leftarrow v.preds.left$\;
    Output $u.firstLabel$\;
    \If{$!u.visitedAsLeft$\  \And $u \notin G.sources$}{
      $u.visitedAsLeft \leftarrow$\  {\sc true}\;
      Output $u.secondLabel$\;
      \If{$!u.visited$}{
        $u.visited \leftarrow$\  {\sc true}\;
        \GridPebSubseqOneUIP{$u$}\;
      }
    }
    \BlankLine
    $u \leftarrow v.preds.right$\;
    \If {$!u.visited$\  \And $u \notin G.sources$}{
      $u.visited \leftarrow$\  {\sc true}\;
      \GridPebSubseqOneUIP{$u$}\;
  }
}
\end{small}
\end{algorithm}%
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\begin{figure}\begin{displaymath}
\begin{array}{llllll}
-       & x_{2,1} & x_{3,1} & x_{4,1} & \ldots & x_{n-1,1} \\
x_{1,2} & -       & x_{3,2} & x_{4,2} & \ldots & x_{n-1,2} \\
x_{1,3} & x_{2,3} & -       & x_{4,3} & \ldots & x_{n-1,3} \\
x_{1,4} & x_{2,4} & x_{3,4} & -       & \ldots & x_{n-1,4} \\
\vdots \\
x_{1,n} & x_{2,n} & x_{3,n} & x_{4,n} & \ldots & -         \\
x_{1,n} & x_{2,n} & x_{3,n} & x_{4,n} & \ldots & x_{n-1,n} \\
\end{array}
\end{displaymath}

\end{figure}%
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\end{document}
