\let\endprocedure\relax
%\usepackage[ruled]{algorithm2e} % For algorithms
-\usepackage[noline, noend, nofillcomment]{algorithm2e}
+
+\usepackage[noline, noend, nofillcomment, linesnumbered]{algorithm2e}
+%\usepackage[noline, noend, nofillcomment]{algorithm2e}
%\usepackage[noline, noend, nofillcomment, linesnumbered]{algorithm2e}
% \setlength{\algomargin}{0em}
- \SetArgSty{textnormal}
-% \SetNoFillComment
-% \newcommand{\Xcmfont}[1]{\texttt{\footnotesize{#1}}}\SetCommentSty{Xcmfont}
+
+\usepackage{setspace}
+
+\SetArgSty{textnormal}
+
+% comments in the pseudocode
+\newcommand{\Xcmfont}[1]{{\texttt{#1}}}
+\SetCommentSty{Xcmfont}
+
+\SetNlSty{textnormal}{}{}
+
\renewcommand{\algorithmcfname}{ALGORITHM}
%\SetAlFnt{\small}
%\SetAlCapFnt{\small}
\caption{
An example of constructing IRE for RE $(\epsilon|a^{0,\infty})(a|\epsilon)^{0,3}$ using $mark$ and $enum$\\
and some examples of IPT for the resulting IRE and string ``$a$''.
+S-norm of each subtree is marked with $\#$.
}
\end{figure}
%
Examples of IPTs can be seen on figure \ref{fig_mark_enum}.
+\FloatBarrier
+
\begin{Xdef}
\emph{Indexed parse trees (IPT)} over finite alphabet $\Sigma$, denoted $\XIT_\Sigma$, are:
\begin{enumerate}
IPTs have two definitions of norm, one for $Pos$ and one for $Sub$,
which we call \emph{p-norm} and \emph{s-norm} respectively:
-\FloatBarrier
-
\begin{Xdef}\label{tnorm_of_IPTs}
The \emph{p-norm} and \emph{s-norm} of an IPT $t$ at position $p$ are:
\begin{align*}
\begin{figure}\label{fig_pe2}
\includegraphics[width=\linewidth]{img/pe2.pdf}
\caption{
-An example of all possible IPTs for RE $((a^{1,3})^{1,3})^{1,3}$ and string $aaa$, the corresponding PEs\\
-and the table of their pairwise traces.
-IPTs $t_3$ and $t_5$ are showed in more detail below.
+All possible IPTs for RE $(((a)^{1,3})^{1,3})^{1,3}$ and string $aaa$, the corresponding PEs,\\
+and the table of traces for each pair of PEs.
+Table entry for row $t_i$ and column $t_j$ contains $traces(\Phi_0(t_i), \Phi_0(t_j))$.\\
+IPTs $t_3$ and $t_5$ are showed in more detail
+(dashed lines correspond to frames);
+traces for them are marked in bold.
}
\end{figure}
\end{figure*}
+\clearpage
+\pagebreak
+
+
+\begin{figure*}
+\begin{multicols}{2}
+
+ \newcommand \NOPASS {N\!O\!P\!A\!S\!S}
+ \newcommand \TOPSORT {T\!O\!P\!S\!O\!RT}
+ \newcommand \LINEAR {L\!I\!N\!E\!A\!R}
+
+ \setstretch{0.9}
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+ \Fn {$\underline{closure \Xund goldberg \Xund radzik(X, Q, F, \Delta, B, D)} \smallskip$} {
+
+ \tcc {set up context}
+ $C = context(Q, \Delta)$ \;
+
+ \BlankLine
+ \tcc {add start configurations}
+ \For {$(q, x) \in X$} {
+ $y = result(q)$ \;
+ \If {$y = \bot \vee relax(x, y, B, D)$} {
+ $result(q) = x$ \;
+ $status(q) = \TOPSORT$ \;
+ $push(topsort, q)$ \;
+ }
+ }
+
+ \BlankLine
+ \While {$topsort$ is not empty} {
+
+ \BlankLine
+ \tcc {1st pass: topological sorting}
+ \While {$topsort$ is not empty} {
+ $q = pop(topsort)$ \;
+
+ \If {$status(q) \neq \LINEAR$} {
+ $status(q) = \TOPSORT$ \;
+
+ %\BlankLine
+ \tcc {find next admissible arc}
+ \While {$(p = explore(q, C, B, D)) \neq \bot$ \\
+ \quad and $status(p) \neq \NOPASS$ } {
+ $active(p) = 1$
+ }
+
+ %\BlankLine
+ \If {$p \neq \bot$} {
+ \tcc {follow admissible arc}
+ $push(topsort, q)$ \;
+ $push(topsort, p)$ \;
+ $next(p) = 1$
+ }
+ \Else {
+ \tcc {done: all deps visited}
+ $status(q) = \LINEAR$ \;
+ $push(linear, q)$
+ }
+ }
+ }
+
+ \BlankLine
+ \tcc {2nd pass: linear scan}
+ \While {$linear$ is not empty} {
+ $q = pop(linear)$ \;
+
+ %\BlankLine
+ \If {$active(q)$} {
+
+ \tcc {scan admissible arcs}
+ $next(q) = 1$ \;
+ \While {$(p = explore(q, C, B, D)) \neq \bot$} {
+
+ \If {$status(p) = \NOPASS$} {
+ $push(topsort, p)$ \;
+ $next(p) = 1$ \;
+ }
+ \ElseIf {$status(p) = \LINEAR$} {
+ $active(p) = 1$ \;
+ }
+ }
+ }
+
+ %\BlankLine
+ $status(q) = \NOPASS$ \;
+ $active(q) = 0$ \;
+ }
+ }
+
+%TRIE!!!!!!!
+ \BlankLine
+ \Return $\big\{ (q, x) \mid x = result(q) \wedge core(q, F, \Delta) \}$ \;
+ }
+ \end{algorithm}
+
+ \columnbreak
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+ \Fn {$\underline{context(Q, \Delta)} \smallskip$} {
+ $C : (\; topsort : \text{stack of } q \in Q$ \;
+ \Indp
+ $,\; linear : \text{stack of } q \in Q$ \;
+ $,\; result : Q \rightarrow (\YZ \times \YZ)^* \texttt{ /* tag histories */}$ \;
+ $,\; status : Q \rightarrow \{ \NOPASS, \TOPSORT, \LINEAR \}$ \;
+ $,\; indeg : Q \rightarrow \YZ \texttt{ /* in-degrees */}$ \;
+ $,\; active : Q \rightarrow \YB \texttt{ /* true if needs rescan */}$ \;
+ $,\; arcs : Q \rightarrow 2^\Delta \texttt{ /* linearly ordered */}$ \;
+ $,\; next : Q \rightarrow \YZ) \texttt{ /* next arc to visit */}$ \;
+ \Indm
+
+ \BlankLine
+ \For {$q \in Q$} {
+ $topsort = \emptyset$ \;
+ $linear = \emptyset$ \;
+ $result(q) = \bot$ \;
+ $status(q) = \NOPASS$ \;
+ $indeg(q) = | \{ (q, \Xund, \Xund, \Xund) \in \Delta \} |$ \;
+ $active(q) = 0$ \;
+ $arcs(q) = \{ (q, \epsilon, \Xund, \Xund) \in \Delta \}$ \;
+ $next(q) = 1$ \;
+ }
+
+ \BlankLine
+ \Return $C$ \;
+ }
+ \end{algorithm}
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+ \Fn {$\underline{explore (q, context, B, D)} \smallskip$} {
+ $\{ a_1, \dots, a_n \} = arcs (q)$ \;
+ $i = next (q)$ \;
+
+ %\BlankLine
+ \While {$i < n$} {
+
+ $(q, \epsilon, \chi, p) = a_i$ \;
+ $i = i + 1$ \;
+ $next(q) = i$ \;
+ $x = result(q) \chi$ \;
+ $y = result(p)$ \;
+
+ %\BlankLine
+ \If {$y = \bot$ \texttt{ /* never seen this state */} \\
+ or $indeg(p) < 2$ \texttt{ /* not a join point */} \\
+ or $relax(x, y, B, D)$} {
+ $result(p) = x$ \;
+ \Return $p$ \;
+ }
+ }
+
+ %\BlankLine
+ \Return $\bot$ \;
+ }
+ \end{algorithm}
+
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+ \Fn {$\underline{relax (x, y, B, D)} \smallskip$} {
+ $(\Xund, \Xund, l) = precedence (x, y, B, D)$ \;
+ \Return $l = -1$ \;
+ }
+ \end{algorithm}
+
+ \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+ \Fn {$\underline{core (q, F, \Delta)} \smallskip$} {
+ \tcc{is final or has outgoing transitions on symbols}
+ \Return $q \in F \vee \; \exists (q, \alpha, \Xund, \Xund) \in \Delta^\Sigma$ \;
+ }
+ \end{algorithm}
+
+% \begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{} \SetAlgoInsideSkip{medskip}
+% \Fn {$\underline{find \Xund indegree (q, indegree)} \smallskip$} {
+% $indegree(q) = indegree(q) + 1$ \;
+% \If {$indegree(q) \leq 1$} {
+% \ForEach {outgoing arc $(q, \epsilon, \chi, p) \in \Delta$} {
+% $find \Xund indegree (p, indegree)$ \;
+% }
+% }
+% }
+% \end{algorithm}
+
+\end{multicols}
+\begin{center}
+\caption{GOR1.}
+\end{center}
+\end{figure*}
+
+
\clearpage
\pagebreak
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