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\begin{document}
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$ 1/3$%
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$ 1.16$%
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$ 79.87$%
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$ 2.19$%
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$ 77.65$%
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$ 2.04$%
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$ 88.88$%
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$ (E,N)$%
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$ 89.86$%
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$ 4.58$%
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$ 79.56$%
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$ 99.87$%
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$ q_{w_2}$%
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\fbox{\parbox{18cm}{
\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Partial Rule Learning Algorithm}\\
\>{\em (Initialize)}\\
\>\>$FD \leftarrow $\  Set of features detectors\\
\>\>$EA \leftarrow $\  Set of elementary actions\\
\>\>$C \leftarrow \{w_{\emptyset}\}\cup\{(v(\text{\em fd}),c(ea)), (v(\neg \text{\em fd}),c(ea))| \text{\em fd}\in FD, ea \in EA\}$\\
\>\>{\bf For each} $w$\  {\bf in} $C$\  \\
\>\>\>$q_w \leftarrow 0$\\
\>\>\>$e_w \leftarrow 0$\\
\>\>\>$i_w \leftarrow 0$\\
\>\>{\bf endfor}\\
\>\>$\overline{e} \leftarrow 0$\\
\>{\bf Repeat} (for each episode):\\
\>\>\>$C' \leftarrow \{w \in C| w$\  is active$\}$\\
\>\>\>{\bf Repeat} (for each step in the episode):\\
\>\>\>\>{\em (Action Selection)}\\
\>\>\>\>\>\>Action Evaluation \hspace{1cm} {\em(Computes $guess(a') \: \forall a'$)}\\
\>\>\>\>\>\>$a \leftarrow$\  arg $\underset{\forall a'}{\max}\{guess(a')\}$\\
\>\>\>\>\>\>Execute $a$\\
\>\>\>\>{\em (System Update)}\\
\>\>\>\>\>\>$r_{a} \leftarrow Reward$\  generated by $a$\\
\>\>\>\>\>\>$C'_{ant} \leftarrow C'$\\
\>\>\>\>\>\>$C' \leftarrow \{w \in C| w$\  is active$\}$\\
\>\>\>\>\>\>Statistics Update\\
\>\>\>\>\>\>Partial-Rule Management\\
\>\>\>{\bf until} terminal situation
\end{tabbing}
}}
\end{center}
}
\end{figure}%
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\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Action Evaluation}\\
\>\>{\bf For each action} $a'$\\
\>\>\>\>$w \leftarrow winner(C',a')$\\
\>\>\>\>$guess(a') \leftarrow q_w + 2 \: random(-\epsilon_w,\epsilon_w) $\\
\>\>{\bf endfor}
\end{tabbing}
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$ w_\emptyset$%
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{\newpage\clearpage
\lthtmlfigureA{figure414}%
\begin{figure}{\small
\begin{center}
\fbox{\parbox{18cm}{
\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Statistics Update}\\
\>\>{\bf if} terminal situation {\bf then}\\
\>\>\>\>$v \leftarrow 0$\\
\>\>{\bf else}\\
\>\>\>\>$v \leftarrow \underset{\forall a'}{\max}\{q_w| w=winner(C',a')\}$\\
\>\>{\bf endif}\\
\>\>$q \leftarrow r_{a} + \gamma \: v$\\
\>\>{\bf For each} $w=(v,c)$\  {\bf in} $C'_{ant}$\\
\>\>\>\>{\bf if} $c$\  is in accordance with $a$\  {\bf then}\\
\>\>\>\>\>{\bf if} $q \: \in \: I_w$\  {\bf then}\\
\>\>\>\>\>\>\>$i_w \leftarrow i_w+1$\\
\>\>\>\>\>{\bf else} \\
\>\>\>\>\>\>\>$i_w \leftarrow \min (\eta-1,i_w-1)$\\
\>\>\>\>\>{\bf endif}\\
\>\>\>\>\>$q_w \leftarrow  q_w \: (1-m_w) + q \: m_w $\\
\>\>\>\>\>$e_w \leftarrow  e_w \: (1-m_w) + |q_w-q| \: m_w $\\
\>\>\>\>{\bf endif}\\
\>\>{\bf endfor}\\
\>\>$\overline{e} \leftarrow e_{w_{\emptyset}}$
\end{tabbing}
}}
\end{center}
}
\end{figure}%
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$ m_w$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4292}%
$ \tau$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4294}%
$ C'_{ant}(a)$%
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$ w_{1} \oplus w_{2}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4300}%
$ w_{1}$%
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{\newpage\clearpage
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$ w_{2}$%
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{\newpage\clearpage
\lthtmlfigureA{figure465}%
\begin{figure}{\small
\begin{center}
\fbox{\parbox{18cm}{
\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Partial Rule Management}\\
\par
\>$w \leftarrow$\  {\em winner}$(C'_{ant},a)$\\
\>{\bf if} $|q_w-q|>\overline{e}$\  \hspace{1cm} {\em (If it is time to create new rules)}\\
\par
\>\>{\em (Partial Rule Elimination)}\\
\>\>\>{\em (Test if there is no room for new rules)}\\
\>\>\>{\bf if} $\|C\| > \mu-\tau$\  {\bf then} \hspace{1cm}\\
\>\>\>\>\>{\em (Rule elimination based on redundancy)}\\
\>\>\>\>\>$C \leftarrow C - \{w \: \in \: C \: | \: redundancy(w)>\lambda \}$\\
\\
\>\>\>\>\>{\em (Rule elimination based on creation error)}\\
\>\>\>\>\>{\bf if} $\|C\| > \mu-\tau$\  {\bf then} {\em (If there is still no room)}\\
\par
\>\>\>\>\>\>\>$SC \leftarrow$\  The $\tau$\  partial rules from $C$\  with:\\
\>\>\>\>\>\>\>\>\>- Lowest {\em creation\_error}$(w)$, and\\
\>\>\>\>\>\>\>\>\>- {\em creation\_error}$(w)<|q_w-q|$\\
\>\>\>\>\>\>\>$C \leftarrow C - SC$\\
\>\>\>\>\>{\bf endif} \\
\>\>\>{\bf endif} \\
\\
\>\>{\em (Partial Rule Generation)}\\
\par
\>\>\>$t \leftarrow 0$\\
\>\>\>{\bf while} $\|C\| < \mu$\  {\bf and} $t<\tau$\  \\
\>\>\>\>\>{\em (Create a new rule $w'$)}\\
\>\>\>\>\>Select two different rules $w_{1}$, $w_{2}$\  from $C'_{ant}(a)$\\
\>\>\>\>\>\>preferring those that minimize\\
\>\>\>\>\>\>\>$|q_{w_i}-q| \: c_{w_i}+ \overline{e} \: (1-c_{w_i})$\\
\>\>\>\>\>$w'=(w_{1} \oplus w_{2})$\\
\>\>\>\>\>{\em creation\_error}$(w') \leftarrow |q_w-q|$\\
\\
\>\>\>\>\>{\em (Insert the new rule in the controller)}\\
\>\>\>\>\>$C \leftarrow C \cup \{w'\}$\\
\>\>\>\>\>$t \leftarrow t+1$\\
\>\>\>{\bf endwhile}\\
\par
\>{\bf endif}
\par
\end{tabbing}
}}
\end{center}
}
\end{figure}%
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$ w_{i}$%
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$ q_{w_{i}}$%
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{\newpage\clearpage
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$ \mu$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay4345}%
$\displaystyle similarity(w,w')=\frac{\|I_w \cap I_{w'}\|}{\max\{\|I_w\|,\|I_{w'}\|\}}                        \frac{U(w \oplus w')}{\min\{U(w),U(w')\}},$%
\lthtmlindisplaymathZ
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$ U(w)$%
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$ w=(w_{1} \oplus w_{2})$%
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$\displaystyle redundancy(w)=\max\{similarity(w,w_1),similarity(w,w_2)\}.$%
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$ w \oplus w_{1}=w$%
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{\newpage\clearpage
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$ U(w) \leq
U(w_1)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4361}%
$\displaystyle \frac{U(w \oplus w_1)}{\min\{U(w),U(w_1)\}}=\frac{U(w)}{\min\{U(w),U(w_1)\}}=\frac{U(w)}{U(w)}=1.$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay4365}%
$\displaystyle redundancy(w)=\max\{\frac{\|I_w \cap I_{w_1}\|}{\max\{\|I_w\|,\|I_{w_1}\|\}},\frac{\|I_w \cap              I_{w_2}\|}{\max\{\|I_w\|,\|I_{w_2}\|\}}\}.$%
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$ \lambda$%
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\lthtmlinlinemathA{tex2html_wrap_inline4375}%
$ w'$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4379}%
$ w \in C$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4381}%
$ w=w' \oplus w''$%
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{\newpage\clearpage
\lthtmlfigureA{figure548}%
\begin{figure}{\small
\begin{center}
\fbox{\parbox{18cm}{
\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Action Evaluation}\\
\>{\em (Initialization)}\\
\>\>\>$L \leftarrow$\  List of active rules sorted by relevance.\\
\>\>\>$EA \leftarrow$\  Set of elementary actions\\
\>\>\>$A \leftarrow$\  Set of combinations of $EA$\\
\>\>\>$v \leftarrow -\infty$\  \>\>\>\>\>\>\> {\em (Situation assessment)}\\
\>\>\>$a \leftarrow \emptyset$\   \>\>\>\>\>\>\> {\em (Optimal action)}\\
\>\>\>$g \leftarrow -\infty$\  \>\>\>\>\>\>\> {\em (Optimal action reward prediction)}\\
\>{\em (Process)}\\
\>\>\>$w \leftarrow ${\em first\_element}$(L)$\\
\>\>\>{\bf do}\\
\>\>\>\>\>$co_w \leftarrow$\  partial command of $w$\\
\>\>\>\>\>$g_w \leftarrow q_w + 2 \:random(-\epsilon_w,\epsilon_w)$\\
\>\>\>\>\>$A_w \leftarrow \{ a \in A | co_w $\  in accordance with $ a \}$\\
\>\>\>\>\>{\bf if} $q_w>v$\  {\bf then}\\
\>\>\>\>\>\>\>$v \leftarrow q_w$\\
\>\>\>\>\>{\bf endif }\\
\>\>\>\>\>{\bf if } $g_w>g$\  {\bf then}\\
\>\>\>\>\>\>\>$g \leftarrow g_w$\\
\>\>\>\>\>\>\>$a \leftarrow co_w$\\
\>\>\>\>\>{\bf endif}\\
\>\>\>\>\>$A \leftarrow A - A_w$\  \\
\>\>\>\>\>$w \leftarrow ${\em next\_element}$(L)$\\
\>\>\>{\bf until} $A \neq \emptyset$
\end{tabbing}
}}
\end{center}
}

\end{figure}%
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$ co_w$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4397}%
$ g_w$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4399}%
$ (m \leftarrow k)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4401}%
$ m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4405}%
$ A$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlfigureA{figure577}%
\begin{figure}{\small
\begin{center}
\fbox{\parbox{18cm}{
\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Action Evaluation}\\
\>{\em (Initialization)}\\
\>\>\>$L \leftarrow$\  List of active rules sorted by relevance.\\
\>\>\>$v \leftarrow -\infty$\\
\>\>\>$a \leftarrow \emptyset$\\
\>\>\>$g \leftarrow -\infty$\\
\>\>\>$tree \leftarrow$\  {\em new\_node}$(\emptyset_c)$\\
\>\>\>$open \leftarrow 1$\\
\>\>\>$closed \leftarrow 0$\\
\>{\em (Process)}\\
\>\>\>$w \leftarrow ${\em first\_element}$(L)$\\
\>\>\>{\bf do}\\
\>\>\>\>\>$g_w \leftarrow q_w + 2 \:random(-\epsilon_w,\epsilon_w)$\\
\>\>\>\>\>Include Rule$(tree,w,g_w)$\\
\>\>\>\>\>$w \leftarrow ${\em next\_element}$(L)$\\
\>\>\>{\bf until} $closed=open$
\end{tabbing}
}}
\end{center}
}

\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlfigureA{figure595}%
\begin{figure}{\small
\begin{center}
\fbox{\parbox{18cm}{
\begin{tabbing} i :\= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii \= ii
\kill
{\bf Include rule(}$n, w, g_w${\bf )}\\
\par
\>{\bf if not(}{\em is\_leaf}$(n)${\bf ) then}\\
\>\>\>$co_w \leftarrow command(w)$\\
\>\>\>$co_n \leftarrow command(n)$\\
\par
\>\>\>{\bf if} $motor(n) \neq \perp$\  {\bf then} \hspace{0.5cm} {\em (Closed Node: Search for compatible sub-nodes)}\\
\par
\>\>\>\>\>{\bf if} $\exists ea \in co_w$\  {\bf with} $motor(ea)=motor(n)$\  {\bf then}\\
\>\>\>\>\>\>\>Include Rule$(get\_subtree(value(ea),n),w,g_w)$\\
\>\>\>\>\>{\bf else}\\
\par
\>\>\>\>\>\>\>{\bf for all} $s$\  {\bf in} $subtrees(n)$\  {\bf do}\\
\>\>\>\>\>\>\>\>\>Include Rule$(s,w,g_w)$\\
\>\>\>\>\>\>\>{\bf endfor}\\
\par
\>\>\>\>\>{\bf endif}\\
\par
\>\>\>{\bf else} \hspace{0.5cm} {\em (Open Node: Specialize the node)}\\
\par
\>\>\>\>\>{\bf if} $co_w - co_n \neq \varnothing$\  {\bf then}  \hspace{0.5cm} {\em (Extend a node)}\\
\par
\>\>\>\>\>\>\>$ea \leftarrow$\  {\em action\_in}$(co_w - co_n)$\\
\>\>\>\>\>\>\>{\em set\_motor}$(n,motor(ea))$\\
\>\>\>\>\>\>\>$closed \leftarrow closed + 1$\\
\>\>\>\>\>\>\>{\bf for all} $k$\  {\bf in} {\em values(motor(ea))} {\bf do}\\
\>\>\>\>\>\>\>\>{\em new\_subtree}$(n,\{k,new\_node(co_n \oplus (motor(ea)\leftarrow k))\})$\\
\>\>\>\>\>\>\>\>$open \leftarrow open + 1$\\
\>\>\>\>\>\>\>{\bf endfor}\\
\>\>\>\>\>\>\>Include Rule$(n,w,g_w)$\\
\par
\>\>\>\>\>{\bf else} \hspace{0.5cm} {\em (Transform a node into a leaf)}\\
\par
\>\>\>\>\>\>\>{\em transform\_to\_leaf}$(n,q_w,g_w,\rho_w,co_w)$\\
\par
\>\>\>\>\>\>\>$closed \leftarrow closed + 1$\\
\par
\>\>\>\>\>\>\>{\bf if} $q_w>v$\  {\bf then}\\
\>\>\>\>\>\>\>\>\>$v \leftarrow q_w$\\
\>\>\>\>\>\>\>{\bf endif}\\
\>\>\>\>\>\>\>{\bf if} $g_w>guess$\  {\bf then}\\
\>\>\>\>\>\>\>\>\>$g \leftarrow g_w$\\
\>\>\>\>\>\>\>\>\>$a \leftarrow co_w$\\
\>\>\>\>\>\>\>{\bf endif}\\
\par
\>\>\>\>\>{\bf endif}\\
\par
\>\>\>{\bf endif}\\
\par
\>{\bf endif}
\par
\end{tabbing}
}}
\end{center}
}
\end{figure}%
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{\newpage\clearpage
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$ \perp$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4449}%
$ co_n$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4463}%
$ (m_1\leftarrow v_1)\wedge (m_2 \leftarrow v_1)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4467}%
$ (m_1\leftarrow v_1)                           $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4471}%
$ (m_2\leftarrow v_1)\wedge (m_3 \leftarrow v_1)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4475}%
$ (m_2\leftarrow v_1)                           $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4479}%
$ (m_1\leftarrow v_0)                           $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4483}%
$ (m_2\leftarrow v_0)                           $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4487}%
$ (m_3\leftarrow v_1)                           $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4491}%
$ (m_3\leftarrow v_0)                           $%
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{\newpage\clearpage
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\includegraphics[width=1.0\linewidth]{images/figure_jair_A.eps}%
\lthtmlpictureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4575}%
$ v_0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4577}%
$ v_1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4581}%
$ (m_1\leftarrow v_1,m_2\leftarrow v_0,m_3\leftarrow \sharp)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4583}%
$ \sharp$%
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{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap4585}%
\includegraphics[width=0.9\linewidth]{images/originals/figure_jair_11}%
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_indisplay4592}%
$\displaystyle O(n_r \: l^{n_m}),$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4594}%
$ n_r$%
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{\newpage\clearpage
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$ l$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4602}%
$\displaystyle \sum_{i=0}^{n_m}l^i=\frac{l^{n_m+1}-1}{l-1}=O(l^{n_m}).$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4604}%
$ l^{n_m}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4606}%
$ n_c$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4608}%
$\displaystyle O(n_r \: n_c).$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4610}%
$\displaystyle \Theta(n_r\:n_c).$%
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\lthtmlinlinemathA{tex2html_wrap_inline4617}%
$ s,s'$%
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{\newpage\clearpage
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$ ea_i=(m_i \leftarrow k)$%
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{\newpage\clearpage
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$ a=(ea_{1},\ldots,ea_{n_m})$%
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{\newpage\clearpage
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$ C=\{w_i \: | \: i=1..n_r\}$%
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$ C',C'_{ant}$%
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{\newpage\clearpage
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$ \epsilon_w = e_w \: c_w +  \overline{e} \: (1-c_w)$%
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{\newpage\clearpage
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$ \rho_w=1/(1+\epsilon_w)$%
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{\newpage\clearpage
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$ I_w=[q_w \pm 2 \epsilon_w]$%
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{\newpage\clearpage
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$ \alpha$%
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{\newpage\clearpage
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$ winner(C',a)$%
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$ guess(a)$%
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{\newpage\clearpage
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$ r_a$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4734}%
$ q=r_a+\gamma v$%
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\end{document}
