\begin{tikzpicture}[>=stealth, ->, auto, node distance=0.2in]
-\tikzstyle{every node}=[draw=none, shape=rectangle, , outer sep = 0in];
-
-% $(\epsilon|a^{0,\infty})(a|\epsilon)^{0,\infty}$
+\tikzstyle{every node}=[draw=none, shape=rectangle, outer sep = 0in];
\small{
-\begin{scope}[xshift=0.7in, yshift=0in]
- \node (a) {{
- $\begin{aligned}
- & mark ((\epsilon|a^{0,\infty})(a|\epsilon)^{0,3} )) = \big[ \\
- %
- & \quad mark ( (\epsilon|a^{0,\infty}) ) = \\
- & \quad mark ( (\epsilon|a^{0,\infty})^{1,1} ) = \big[ \\
- & \quad\quad mark ( \epsilon|a^{0,\infty} ) = \big[ \\
- & \quad\quad\quad mark ( \epsilon ) = (0,0,\epsilon), \\
- & \quad\quad\quad mark ( a^{0,\infty} ) = \big[ \\
- & \quad\quad\quad\quad mark ( a ) = (0,0,a) \\
- & \quad\quad\quad \big] = (0,0,(0,0,a)^{0,\infty}) \\
- & \quad\quad \big] = (0,0,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty})) \\
- & \quad \big] = (1,0,(1,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1}) \\
- %
- & \quad mark ( (a|\epsilon)^{0,3} ) = \big[ \\
- & \quad\quad mark ( a|\epsilon ) = \big [ \\
- & \quad\quad\quad mark ( a ) = (0,0,a) \\
- & \quad\quad\quad mark ( \epsilon ) = (0,0,\epsilon) \\
- & \quad\quad \big] = (0,0,(0,0,a) \mid (0,0,\epsilon)) \\
- & \quad \big] = (1,0,(1,1,(0,0,a) \mid (0,0,\epsilon))^{0,3}) \\
- %
- & \big] = (1,0,
- (1,0,(1,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1}) \\
- & \quad\quad\quad \cdot (1,0,(1,1,(0,0,a) \mid (0,0,\epsilon))^{0,3})
- )
- \end{aligned}$
- }};
-\end{scope}
-
-\begin{scope}[xshift=4in, yshift=0in]
- \node (a) {{
- $\begin{aligned}
- & enum (1,1, \; (1,0,
- (1,0,(1,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1}) \\
- & \quad\quad\quad\quad\quad\quad \cdot (1,0,(1,1,(0,0,a) \mid (0,0,\epsilon))^{0,3})
- ) \;) = \big[ \\
- %
- & \quad enum (2,1, \; (1,0,(1,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1}) \;) = \big[ \\
- & \quad\quad enum (3,1, \; (1,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty})) \;) = \big[ \\
- & \quad\quad\quad enum (4,2, \; (0,0,\epsilon)) = (4,2, \; (0,0,\epsilon) \;) \\
- & \quad\quad\quad enum (4,2, \; (0,0,(0,0,a)^{0,\infty}) \;) = \big[ \\
- & \quad\quad\quad\quad enum (4,2, \; (0,0,a)) = (4,2, \; (0,0,a) \;) \\
- & \quad\quad\quad \big] = (4,2, \; (0,0,(0,0,a)^{0,\infty}) \;) \\
- & \quad\quad \big] = (4,2, \; (3,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty})) \;) \\
- & \quad \big] = (4,2, \; (2,0,(3,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1}) \;) \\
- %
- & \quad enum (4,2, \; (1,0,(1,1,(0,0,a) \mid (0,0,\epsilon))^{0,3}) \;) = \big[ \\
- & \quad\quad enum (5,2, \; (1,1,(0,0,a) \mid (0,0,\epsilon)) \;) = \big[ \\
- & \quad\quad\quad enum (6,3, \; (0,0,a) ) = (6, 3, \; (0,0,a) \;) \\
- & \quad\quad\quad enum (6,3, \; (0,0,\epsilon) ) = (6, 3, \; (0,0,\epsilon) \;) \\
- & \quad\quad \big] = (6,3, \; (5,2,(0,0,a) \mid (0,0,\epsilon)) \;) \\
- & \quad \big] = (6,3, \; (4,0,(5,2,(0,0,a) \mid (0,0,\epsilon))^{0,3}) \;) \\
- %
- & \big] = (6,3, \; (1,0,
- (2,0,(3,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1}) \\
- & \quad\quad\quad\quad\quad \cdot (4,0,(5,2,(0,0,a) \mid (0,0,\epsilon))^{0,3})
- ) \;) \\
- \end{aligned}$
- }};
-\end{scope}
-
-%\begin{scope}[xshift=2.5in, yshift=-2.4in]
-% \node (c) {{
-% $\begin{aligned}
-% & \IRE ((\epsilon|a^{0,\infty})(a|\epsilon)^{0,3} ))
-% = (1,0, (2,0,(3,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1})
-% \cdot (4,0,(5,2,(0,0,a) \mid (0,0,\epsilon))^{0,3}))
-% \end{aligned}$
-% }};
-%\end{scope}
-
-\begin{scope}[xshift=2.4in, yshift=-1.95in]
+\begin{scope}[xshift=2.4in, yshift=-1.7in]
\graph [tree layout, grow=down, fresh nodes, level distance=0.35in] {
"${(1, 0, \cdot)}_{\Lambda}$" -- {
"${(2, 0, \{1,1\})}_{1}$" -- {
};
\end{scope}
-\newcommand\xsp[1]{\hphantom{\hspace{#1em}\hspace{-0.2em}}}
+\begin{scope}[xshift=2.4in, yshift=-3.3in]
+ \node (c) {{
+ $(1,0, (2,0,(3,1,(0,0,\epsilon) \mid (0,0,(0,0,a)^{0,\infty}))^{1,1})
+ \cdot (4,0,(5,2,(0,0,a) \mid (0,0,\epsilon))^{0,3}))$
+ }};
+\end{scope}
+
+\newcommand\xsp[1]{\hphantom{\hspace{#1em}\hspace{-0.0em}}}
\begin{scope}[xshift=-0.3in, yshift=-3.6in, sibling distance=0.6in, level distance=0.35in]
\node[xshift=-0.2in, draw=none] {$s:$};
\node at (a1211) {\xsp{3.5}\footnotesize{$\#\infty$}};
\node at (a1212) {\xsp{3.5}\footnotesize{$\#\infty$}};
- \node[xshift=0.4in, yshift=-1.65in, draw=none]
+ \node[xshift=0.2in, yshift=-1.65in, draw=none]
{${T}^{1}\big(
{T}^{2}(
{T}^{3}(
\node at (a1221) {\xsp{3.5}\footnotesize{$\#\infty$}};
\node at (a1222) {\xsp{3.5}\footnotesize{$\#\infty$}};
- \node[xshift=0.3in, yshift=-1.35in, draw=none]
+ \node[xshift=0.2in, yshift=-1.65in, draw=none]
{${T}^{1}\big(
{T}^{2}(
{T}^{3}(
\newcommand{\YT}{\mathbb{T}}
\newcommand{\YQ}{\mathbb{Q}}
\newcommand{\YZ}{\mathbb{Z}}
+\newcommand{\PT}{PT}
+\newcommand{\PE}{P\!E}
\newcommand{\IPT}{I\!PT}
\newcommand{\IRE}{I\!RE}
performs pairwise comparison of all configurations in the new set,
recording results in $B$ and $D$ matrices.
On the next step $q$-components become $o$-components.
-If a pair of paths originating from current configurations meet in the next $closure ()$,
-it will use origin states to lookup comparison results in $B$ and $D$ matrices.
+If paths originating from current configurations meet on some future step,
+$closure ()$ will use origin states to lookup comparison results in $B$ and $D$ matrices.
On the other hand, if the paths do not meet, then comparison performed by $update \Xund ptables ()$ is redundant.
Unfortunately we cannot get rid of redundant comparisons:
we do not know in advance which configurations will spawn ambiguous paths.
-An alternative approach is to disambiguate lazily, only if paths meet ---
-but that requires tracking paths which grow with the length of input.
-Eager pre-comparison is a tradeoff that
-allows the algorithm work in bounded memory at the expense of some time overhead.
+It is possible to delay disambiguation until the paths meet,
+but that would require keeping arbitrary long paths in memory.
+Eager comparison is a tradeoff that allows the algorithm work in bounded memory at the expense of some time overhead.
\\
%\vfill\null
-\section{Representation of match results}
-
-\begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
-\begin{multicols}{2}
- \setstretch{0.8}
-
- \Fn {$\underline{initial \Xund result (T)} \smallskip$} {
- \For {$i = \overline {0, |T| / 2}$} {
- $pmatch[i].rm \Xund so = -1$ \;
- $pmatch[i].rm \Xund eo = -1$ \;
- }
- \Return $pmatch$ \;
- }
- \BlankLine
-
- \Fn {$\underline{update \Xund result (X, k, \Xund)} \smallskip$} {
- \Return $\big\{ (q, o, u, apply \Xund tags (u, r, k)) \mid (q, o, u, r) \in X \big\}$ \;
- %\For {$(q, o, t_1 \hdots t_n, pmatch) \in X$} {
- % \For {$i = \overline {1, n}$} {
- % \lIf {$t_i < 0$} {$l = -1$}
- % \lElse {$l = k$}
- % \lIf {$t_i mod \, 2 \equiv 0$} {$pmatch[|t_i|/2].rm \Xund so = l$}
- % \lElse {$pmatch[(|t_i| - 1)/2].rm \Xund eo = l$}
- % }
- %}
- %\Return $X$ \;
- }
- \BlankLine
-
- \Fn {$\underline{f\!inal \Xund result (u, r, k)} \smallskip$} {
- $pmatch = apply \Xund tags (u, r, k)$ \;
- $pmatch[0].rm \Xund so = 0$ \;
- $pmatch[0].rm \Xund eo = k$ \;
- \Return $pmatch$ \;
- }
- \BlankLine
-
- \Fn {$\underline{apply \Xund tags (t_1 \hdots t_n, pmatch, k)} \smallskip$} {
- \For {$i = \overline {1, n}$} {
- \lIf {$t_i < 0$} {$l = -1$}
- \lElse {$l = k$}
- \lIf {$t_i mod \, 2 \equiv 0$} {$pmatch[|t_i|/2].rm \Xund eo = l$}
- \lElse {$pmatch[(|t_i| \!+\! 1)/2].rm \Xund so = l$}
- }
- \Return $pmatch$ \;
- }
- \BlankLine
-
- \vfill
-
-\columnbreak
-
- \Fn {$\underline{initial \Xund result (\Xund)} \smallskip$} {
- \Return $\epsilon$ \;
- }
- \BlankLine
-
- \Fn {$\underline{update \Xund result (X, \Xund, \alpha)} \smallskip$} {
- \Return $\big\{ (q, o, u, r \cdot u \cdot \alpha) \mid (q, o, u, r) \in X \big\}$ \;
- }
- \BlankLine
-
- \Fn {$\underline{f\!inal \Xund result (u, r, \Xund)} \smallskip$} {
- \Return $parse \Xund tree (r \cdot u, 1)$ \;
- }
- \BlankLine
-
- \Fn {$\underline{parse \Xund tree (u, i)} \smallskip$} {
- \If {$u = (2i \!-\! 1) \cdot (2i)$} {
- \Return $T(\epsilon)$
- }
- \If {$u = (1 \!-\! 2i) \cdot \hdots $} {
- \Return $T(\varnothing)$
- }
- \If {$u = (2i \!-\! 1) \cdot \alpha_1 \hdots \alpha_n \cdot (2i) \wedge \alpha_1, \hdots, \alpha_n \in \Sigma $} {
- \Return $T(a_1, \hdots, a_n)$
- }
- \If {$u = (2i \!-\! 1) \cdot \beta_1 \hdots \beta_m \cdot (2i) \wedge \beta_1 = 2j \!-\! 1 \in T$} {
- $n = 0, k = 1$ \;
- \While {$k \leq m$} {
- $l = k$ \;
- \lWhile {$|\beta_{k+1}| > 2j$} {
- $k = k + 1$
- }
- $n = n + 1$ \;
- $t_n = parse \Xund tree (\beta_l \dots \beta_k, j)$
- }
- \Return $T(t_1, \dots, t_n)$
- }
- \Return $\varnothing$ \tcp{ill-formed PE}
- }
- \BlankLine
-
-\end{multicols}
-\vspace{1.5em}
-\caption{Construction of match results: POSIX offsets (on the left) and parse tree (on the right).}
-\end{algorithm}
-\medskip
-
-\section{Regular Expressions and Ordered Parse Trees}
-
-In this section we revise the basic definitions of
-regular expressions,
-parse trees,
-ambiguity
-and the order on parse trees.
-
- \begin{Xdef}
- \emph{Regular expressions (REs)} over finite alphabet $\Sigma$, denoted $\XR_\Sigma$, are:
- \begin{enumerate}
- \item Atomic REs:
-% \emph{empty set} $\emptyset \in \XE_\Sigma$,
- \emph{empty word} $\epsilon \in \XR_\Sigma$ and
- \emph{unit word} $\alpha \in \XR_\Sigma$, where $\alpha \in \Sigma$.
- \item Compound REs: if $e_1, e_2 \in \XR_\Sigma$, then
- \emph{union} $e_1 | e_2 \in \XR_\Sigma$,
- \emph{product} $e_1 e_2 \in \XR_\Sigma$,
- \emph{repetition} $e_1^{n, m} \in \XR_\Sigma$ (where $0 \leq n \leq m \leq \infty$), and
- \emph{submatch group} $(e_1) \in \XR_\Sigma$.
- \end{enumerate}
- \end{Xdef}
-
-REs originate in the work of Kleene\cite{Kle51}.
-A mathematically rigorous definition of REs is given in terms of the Kleene algebra\cite{Koz94}.
-Our definition is close to the one widely used in literature\cite{HU90}\cite{SS88},
-with a couple of minor adaptations to the POSIX standard.
-First, we consider parentheses as a distinct construct: in POSIX $(e)$ is semantically different from $e$.
-Second, we use generalized repetition $e^{n, m}$ instead of iteration $e^*$:
-expressing repetition in terms of iteration and product requires duplication of the subexpression,
-which may change semantics: e.g. in POSIX $(e)(e)$ is not the same as $(e)^{2,2}$.
-As usual, we assume that repetition has precedence over product and product over union,
-and parentheses may be used to override it (all parentheses are capturing).
-%Additionally, we use the following shortcut notation:
-%$e^*$ for $e^{0,\infty}$,
-%$e^+$ for $e^{1,\infty}$,
-%$e^?$ for $e^{0,1}$,
-%and $e^n$ for $e^{n,n}$.
-% %\\
-% %
-% %One possible interpretation of RE is the \emph{tree} interpretation,
-% %in which every RE denotes a set of \emph{parse trees}.
-
- \begin{Xdef}
- \emph{Parse trees (PT)} over finite alphabet $\Sigma$, denoted $\XT_\Sigma$, are:
- \begin{enumerate}
- \item Atomic PT:
- \emph{nil tree} ${\varnothing} \in \XT_\Sigma$,
- \emph{empty tree} ${\epsilon} \in \XT_\Sigma$ and
- \emph{unit tree} ${\alpha} \in \XT_\Sigma$, where $\alpha \in \Sigma$.
- \item Compound PT: if $t_1, \dots, t_n \in \XT_\Sigma$, where $n \geq 1$, then
- ${T}(t_1, \dots, t_n) \in \XT_\Sigma$.
- \end{enumerate}
- \end{Xdef}
-
-Note that our PTs are different from \ref{OS13}:
-we have a \emph{nil} tree (a placeholder for absent alternative and zero repetitions)
-and do not differentiate between various kinds of compound trees.
-Each RE denotes a set of PTs given by the operator $PT: \XR_\Sigma \rightarrow 2^{\XT_\Sigma}$:
- \begin{align*}
- PT(\epsilon) &= \{ {\epsilon} \}
- \\
- PT(\alpha) &= \{ {\alpha} \}
- \\
- PT(r_1 \mid r_2) &=
- \big\{ {T}(t, \varnothing) \mid t \in PT(r_1) \big\} \cup
- \big\{ {T}(\varnothing, t) \mid t \in PT(r_2) \big\}
- \\
- PT(r_1 \cdot r_2) &=
- \big\{ {T}(t_1, t_2) \mid
- t_1 \in PT(r_1),
- t_2 \in PT(r_2)
- \big\} \\
- PT(r^{n, m}) &=
- \begin{cases}
- \big\{ {T}(t_1, \dots, t_m) \mid t_i \in PT(r) \;
- \forall i = \overline{1, m} \big\} \cup \{ {T}(\varnothing) \} &\text{if } n = 0 \\
- \big\{ {T}(t_n, \dots, t_m) \mid t_i \in PT(r) \;
- \forall i = \overline{n, m} \big\} &\text{if } n > 0
- \end{cases}
- \\
- PT( (r) ) &= PT(r)
- \end{align*}
-
-The \emph{string} induced by a PT $t$, denoted $str(t)$, is the concatenation of all alphabet symbols in the left-to-right traversal of $t$.
-For a RE $r$ and a string $w$, we write $PT(r, w)$ to denote the set $\{ t \in PT(r) \mid str(t) = w \}$
-(note that this set is potentially infinite).
-
- \begin{Xdef}\label{ambiguity_of_parse_trees}
- PTs $s$ and $t$ are \emph{ambiguous} iff $s \neq t$ and $s, t \in PT(r, w)$ for some RE $r$ and string $w$.
- $\square$
- \end{Xdef}
-
-Following \ref{OS13}, we assign \emph{positions} to the nodes of RE and PT.
-The root position is $\Lambda$, and position of the $i$-th subtree of a tree with position $p$ is $p.i$.
-The \emph{length} of position $p$, denoted $|p|$, is defined as $0$ for $\Lambda$ and $|p| + 1$ for $p.i$.
-%The set of all positions is denoted $\XP$.
-The subtree of a tree $t$ at position $p$ is denoted $t|_p$.
-Position $p$ is a \emph{prefix} of position $q$ iff $q = p.p'$ for some $p'$,
-and a \emph{proper prefix} if additionaly $p \neq q$.
-Position $p$ is a \emph{sibling} of position $q$ iff $q = q'.i, p = q'.j$ for some $q'$ and $i,j \in \YN$.
-Positions are ordered lexicographically, as in \ref{OS13}.
-The set of all positions of a tree $t$ is denoted $Pos(t)$.
-
- \begin{Xdef}\label{norm_of_parse_tree}
- The \emph{norm} of PT $t$ at position $p$ is:
- \begin{align*}
- \|t\|_p &=
- \begin{cases}
- -1 &\text{if } p \in Pos(t) \text{ and } t|_p = \varnothing \\
- |str(t|_p)| &\text{if } p \in Pos(t) \text{ and } t|_p \neq \varnothing \\
- \infty &\text{if } p \not\in Pos(t)
- \end{cases}
- \end{align*}
- \end{Xdef}
-
-We shorten $\|t\|_\Lambda$ as $\|t\|$.
-Generally the norm of a subtree means the number of alphabet symbols in the leaves of this subtree, with two exceptions.
-First, for nil subtrees the norm is $-1$: intuitively, they have the lowest ``ranking'' among all possible subtrees.
-Second, for inexistent subtrees (those with positions out of $Pos(t)$) the norm is infinite.
-This may seem counterintuitive at first, but it makes sense in the presense of REs with empty repetition.
-According to the POSIX standard, optional empty repetitions are not allowed,
-and our definition of the norm reflects this:
-if a tree $s$ has a subtree $s|_p$ corresponding to an empty repetition,
-and another tree $t$ has no subtree at position $p$ ($p \not\in Pos(t)$),
-then the infinite norm $\|t\|_p$ ``outranks'' $\|s\|_p$.
+\section{Formalization}
- \begin{Xdef}\label{order_on_PTs}
- \emph{Order on PTs.}
- Given parse trees $t, s \in PT(e, w)$, we say that $t <_p s$ w.r.t. \emph{desision position} $p$ % $p \in Sub(t) \cup Sub(s)$
- iff $\|t\|_p > \|s\|_p$ and $\|t\|_q = \|s\|_q \; \forall q < p$.
- We say that $t < s$ iff $t <_p s$ for some $p$.
- \end{Xdef}
-
-Note that in the definition \ref{order_on_PTs}
-we do not explicitly demand that $p, q \in Pos(t) \cup Pos(s)$:
-if this is not the case, the norms of both subtrees are $\infty$ and thus equal.
-Also note that our definition of order differs from the one given by Okui and Suzuki \ref{OS13}:
-we define order on the whole set of PTs,
-while they defined it on a finite subset of PTs called \emph{canonical parse trees (CPTs)} ---
-a subset of PTs that do not contain subtrees corresponding to optional empty repetitions.
-We extend their definition to non-canonical PTs.
-
- \begin{XThe}
- Relation $<$ is a strict total order on PTs.
- \end{XThe}
-
-
-
-\section{Partially Ordered Parse Trees}
-
-The POSIX standard uses the terms \emph{subexpression} and \emph{subpattern}
-when speaking about parenthesized and non-parenthesized sub-REs respectively.
-The difference between them is that
-subexpressions (also called \emph{submatch groups}) are subject to submatch extraction:
-if a RE matches some string, we need to know what part of the string matches each subexpression.
-For subpatterns we don't need this information.
-According to the POSIX standard, both subexpressions and subpatterns
-obey the same hierarchical rules,
-where the outer sub-REs are prior to the inner sub-REs and the left sub-REs are prior to the right sub-REs.
-Disambiguation starts with the topmost sub-RE and proceeds down to each most nested subexpression,
-considering all prior sub-RE on its way (both subexpressions and subpatterns),
-and stopping at the first difference.
-%
-In order to reflect these disambiguation rules, we introduce the notion of explicit and implicit submatch groups:
-\emph{explicit submatch group} is a subexpression
-and \emph{implicit submatch group} is either a subexpression, or a subpattern which contains nested or sibling subexpressions.
-In this section we rewrite REs in a form
-where every sub-RE is equipped with a pair of numbers called
-\emph{implicit submatch index} and \emph{explicit submatch index}.
-As the reader might guess, these numbers indicate if the given sub-RE is an implicit or explicit submatch group.
-%
-For a given sub-RE,
-if both indices are zero, it means that this sub-RE is ignored from submatch extraction perspective.
-If only the second index is zero, it means that the sub-RE itself is not subject to submatch extraction,
-but it may be involved in disambiguation.
-If both indices are non-zero, then the sub-RE is a submatch group.
-%
-Indices are convenient because they allow us to consider individual sub-REs
-without loosing the submatch context of the whole RE.
-We call this representation \emph{indexed regular expressions (IREs)}.
-%An example of IRE can be seen on figure \ref{fig_parse_trees}.
+According to the POSIX standard,
+submatch extraction applies only to parenthesized subexpressions,
+but the longest-match rule applies to all subexpressions regardless of parentheses.
+Therefore first of all we rewrite REs in a form that makes submatch information explicit.
+Namely, we equip every subexpression with an \emph{implicit} and \emph{explicit} submatch index.
+Explicit indices enumerate submatch groups; for all other subexpressions they are zero.
+Implicit indices enumerate submatch groups and subexpressions that are not submatch groups,
+but contain nested or sibling groups and need to be considered by disambiguation.
+%If both indices are zero, then the subexpression is completely ignored from disambiguation perspective.
+%Indices are convenient because they allow us to consider individual sub-REs
+%without loosing the submatch context of the whole RE.
\begin{Xdef}
- \emph{Indexed regular expressions (IREs)} over finite alphabet $\Sigma$, denoted $\XIR_\Sigma$, are:
+ \emph{Indexed regular expressions (IRE)} over finite alphabet $\Sigma$, denoted $\XIR_\Sigma$:
\begin{enumerate}
- \item Atomic IREs:
- \emph{empty tree} $(i, j, \epsilon) \in \XIR_\Sigma$ and
- \emph{unit tree} $(i, j, \alpha) \in \XIR_\Sigma$, where $\alpha \in \Sigma$ and $i, j \in \YZ$.
-
- \item Compound IREs: if $r_1, r_2 \in \XIR_\Sigma$ and $i, j \in \YZ$, then
- \emph{union} $(i, j, r_1 \mid r_2) \in \XIR_\Sigma$,
- \emph{product} $(i, j, r_1 \cdot r_2) \in \XIR_\Sigma$ and
- \emph{repetition} $(i, j, r_1^{n, m}) \in \XIR_\Sigma$ (where $0 \leq n \leq m \leq \infty$).
+ \item
+ Empty IRE $(i, j, \epsilon)$ and
+ unit IRE $(i, j, \alpha)$, where $\alpha \in \Sigma$ and $i, j \in \YZ$,
+ are in $\XIR_\Sigma$.
+
+ \item If $r_1, r_2 \in \XIR_\Sigma$ and $i, j \in \YZ$, then
+ union $(i, j, r_1 \mid r_2)$,
+ product $(i, j, r_1 \cdot r_2)$ and
+ repetition $(i, j, r_1^{n, m})$, where $0 \leq n \leq m \leq \infty$,
+ are in $\XIR_\Sigma$.
\end{enumerate}
\end{Xdef}
-The function $\IRE : \XR_\Sigma \rightarrow \XIR_\Sigma$ transforms REs into IREs.
+Function $\IRE$ transforms RE into IRE.
It is defined via a composition of two functions,
-$mark$ that transforms REs into IREs with submatch indices in the boolean range $\{0, 1\}$,
-and $enum$ that substitutes boolean indices with consecutive numbers:
-$\IRE(e) = r$ where $(\Xund, \Xund, r) = enum(1, 1, mark(e))$.
-Note that we consider $(e)$ as a special case of repetition $(e)^{1,1}$:
-this allows us to handle all parenthesized sub-RE uniformly.
+$mark()$ that transforms RE into IRE with submatch indices in the boolean range $\{0, 1\}$,
+and $enum()$ that substitutes boolean indices with consecutive numbers.
+%$\IRE(e) = r$ where $(\Xund, \Xund, r) = enum(1, 1, mark(e))$.
+%Note that we consider $(e)$ as a special case of repetition $(e)^{1,1}$:
+%this allows us to handle all parenthesized sub-RE uniformly.
An example of constructing an IRE from a RE is given on figure \ref{fig_mark_enum}.
-The reverse transformation is also possible by erasing all indices
-and adding parentheses around subexpressions with nonzero explicit submatch index.
-Therefore RE and IRE are equivalent representations.
+%The reverse transformation is also possible by erasing all indices
+%and adding parentheses around subexpressions with nonzero explicit submatch index.
+%Therefore RE and IRE are equivalent representations.
+
\begin{align*}
&\begin{aligned}
mark &: \XR_\Sigma \longrightarrow \XIR_\Sigma \\
- mark &(x) \mid_{x \in \{\epsilon, \alpha\}} = (0, 0, x) \\
+ mark &(x) \mid_{x \in \{\epsilon, \alpha\}} = (0, 0, x) \\[-0.2em]
%
mark &(e_1 \circ e_2) \mid_{\circ \in \{|,\cdot\}} = (i, 0,
(i, j_1, r_1) \circ
(i, j_2, r_2)
- ) \\
- &\text{where } (i_1, j_1, r_1) = mark(e_1) \\
- &\space{\hphantom{where }}(i_2, j_2, r_2) = mark(e_2) \\
- &\space{\hphantom{where }}i = i_1 \vee i_2 \\
+ ) \\[-0.2em]
+ &\text{where } (i_1, j_1, r_1) = mark(e_1) \\[-0.2em]
+ &\space{\hphantom{where }}(i_2, j_2, r_2) = mark(e_2) \\[-0.2em]
+ &\space{\hphantom{where }}i = i_1 \vee i_2 \\[-0.2em]
%
- mark &(e^{n, m}) \mid_{e = (e_1)} = (1, 0, (1, 1, r)) \\
- &\text{where } (\Xund, \Xund, r) = mark(e_1) \\
+ mark &(e^{n, m}) \mid_{e = (e_1)} = (1, 0, (1, 1, r)) \\[-0.2em]
+ &\text{where } (\Xund, \Xund, r) = mark(e_1) \\[-0.2em]
%
- mark &(e^{n, m}) \mid_{e \neq (e_1)} = (i, 0, (i, j, r)) \\
- &\text{where } (i, j, r) = mark(e) \\
+ mark &(e^{n, m}) \mid_{e \neq (e_1)} = (i, 0, (i, j, r)) \\[-0.2em]
+ &\text{where } (i, j, r) = mark(e) \\[-0.2em]
%
mark &((e)) = mark((e)^{1, 1})
\end{aligned}
enum &: \YZ \times \YZ \times \XIR_\Sigma \longrightarrow \YZ \times \YZ \times \XIR_\Sigma \\
enum &(\bar{i}, \bar{j}, (i, j, x)) \mid_{x \in \{\epsilon, \alpha\}}
= (\bar{i} + i, \bar{j} + j, (\bar{i} \times i, \bar{j} \times j, x))
- \\
+ \\[-0.2em]
enum &(\bar{i}, \bar{j}, (i, j, r_1 \circ r_2)) \mid_{\circ \in \{|,\cdot\}}
- = (i_2, j_2, (\bar{i} \times i, \bar{j} \times j, \bar{r}_1 \circ \bar{r}_2)) \\
- &\text{where } (i_1, j_1, \bar{r}_1) = enum(\bar{i} + i, \bar{j} + j, r_1) \\
+ = (i_2, j_2, (\bar{i} \times i, \bar{j} \times j, \bar{r}_1 \circ \bar{r}_2)) \\[-0.2em]
+ &\text{where } (i_1, j_1, \bar{r}_1) = enum(\bar{i} + i, \bar{j} + j, r_1) \\[-0.2em]
&\space{\hphantom{where }}(i_2, j_2, \bar{r}_2) = enum(i_1, j_1, r_2)
- \\
- enum &(\bar{i}, \bar{j}, (i, j, r^{n,m})) = (i_1, j_1, (\bar{i} \times i, \bar{j} \times j, \bar{r}^{n,m})) \\
+ \\[-0.2em]
+ enum &(\bar{i}, \bar{j}, (i, j, r^{n,m})) = (i_1, j_1, (\bar{i} \times i, \bar{j} \times j, \bar{r}^{n,m})) \\[-0.2em]
&\text{where }
(i_1, j_1, \bar{r}) = enum(\bar{i} + i, \bar{j} + j, r)
+ \\[-0.2em]
+ \\[-0.2em]
+ \IRE &: \XR_\Sigma \rightarrow \XIR_\Sigma \\[-0.2em]
+ \IRE&(e) = r \\[-0.2em]
+ &\text{where }(\Xund, \Xund, r) = enum(1, 1, mark(e))
+ \\[-0.2em]
\end{aligned}
\end{align*}
+ \medskip
-Just like REs denote sets of PTs, IREs denote sets of \emph{IPTs} --- \emph{indexed parse trees}.
-The only difference between PTs and IPTs is that each node of an IPT has an associated number ---
-the implicit submatch index inherited from the corresponding IRE node (denoted with a superscript).
-Explicit submatch index is not used in IPTs.
-%
-As for PTs, $str(t)$ denotes the string formed by concatenation of all alphabet symbols in the left-to-right traversal of $t$,
-and $\IPT(r, w) = \{ t \in \IPT(r) \mid str(t) = w \}$ is the set of all IPTs for an IRE $r$ and a string $w$.
-%
-Examples of IPTs can be seen on figure \ref{fig_mark_enum}.
+The relation between regular expressions and parse trees is given by the operator $\PT$.
+Each IRE denotes a set of PTs:
- \begin{Xdef}
- \emph{Indexed parse trees (IPT)} over finite alphabet $\Sigma$, denoted $\XIT_\Sigma$, are:
- \begin{enumerate}
- \item Atomic IPTs:
- \emph{nil tree} ${\varnothing}^i \in \XIT_\Sigma$,
- \emph{empty tree} ${\epsilon}^i \in \XIT_\Sigma$ and
- \emph{unit tree} ${\alpha}^i \in \XIT_\Sigma$, where $\alpha \in \Sigma$ and $i \in \YZ$.
- \item Compound IPTs: if $t_1, \dots, t_n \in \XIT_\Sigma$, where $n \geq 1$, and $i \in \YZ$, then
- ${T}^i(t_1, \dots, t_n) \in \XIT_\Sigma$.
- \end{enumerate}
- \end{Xdef}
-
-The operator $\IPT: \XIR_\Sigma \rightarrow 2^{\XIT_\Sigma}$ gives the set of all IPTs denoted by the given IRE:
-%(its definition is very similar to the one of operator $PT$):
\begin{align*}
- \IPT\big((i, \Xund, \epsilon)\big) &= \{ {\epsilon}^{i} \}
- \\
- \IPT\big((i, \Xund, \alpha)\big) &= \{ {\alpha}^{i} \}
- \\
- \IPT\big((i, \Xund, (i_1, j_1, r_1) \mid (i_2, j_2, r_2))\big) &=
- \big\{ {T}^{i}(t, \varnothing^{i_2}) \mid t \in \IPT\big((i_1, j_1, r_1)\big) \big\} \cup
- \big\{ {T}^{i}(\varnothing^{i_1}, t) \mid t \in \IPT\big((i_2, j_2, r_2)\big) \big\}
+ \PT &: \XIR_\Sigma \rightarrow 2^{\XT_\Sigma}
\\
- \IPT\big((i, \Xund, (i_1, j_1, r_1) \cdot (i_2, j_2, r_2))\big) &=
+ \PT\big((i, \Xund, \epsilon)\big) &= \{ {\epsilon}^{i} \}
+ \\[-0.2em]
+ \PT\big((i, \Xund, \alpha)\big) &= \{ {\alpha}^{i} \}
+ \\[-0.2em]
+ \PT\big((i, \Xund, (i_1, j_1, r_1) \mid (i_2, j_2, r_2))\big) &=
+ \big\{ {T}^{i}(t, \varnothing^{i_2}) \mid t \in \PT\big((i_1, j_1, r_1)\big) \big\} \cup
+ \big\{ {T}^{i}(\varnothing^{i_1}, t) \mid t \in \PT\big((i_2, j_2, r_2)\big) \big\}
+ \\[-0.2em]
+ \PT\big((i, \Xund, (i_1, j_1, r_1) \cdot (i_2, j_2, r_2))\big) &=
\big\{ {T}^{i}(t_1, t_2) \mid
- t_1 \in \IPT\big((i_1, j_1, r_1)\big),
- t_2 \in \IPT\big((i_2, j_2, r_2)\big)
- \big\} \\
- \IPT\big((i, \Xund, (i_1, j_1, r_1)^{n, m})\big) &=
+ t_1 \in \PT\big((i_1, j_1, r_1)\big),
+ t_2 \in \PT\big((i_2, j_2, r_2)\big)
+ \big\} \\[-0.2em]
+ \PT\big((i, \Xund, (i_1, j_1, r_1)^{n, m})\big) &=
\begin{cases}
- \big\{ {T}^{i}(t_1, \dots, t_m) \mid t_k \in \IPT\big((i_1, j_1, r_1)\big) \;
- \forall k = \overline{1, m} \big\} \cup \{ {T}^{i}(\varnothing^{i_1}) \} &\text{if } n = 0 \\
- \big\{ {T}^{i}(t_n, \dots, t_m) \mid t_k \in \IPT\big((i_1, j_1, r_1)\big) \;
+ \big\{ {T}^{i}(t_1, \dots, t_m) \mid t_k \in \PT\big((i_1, j_1, r_1)\big) \;
+ \forall k = \overline{1, m} \big\} \cup \{ {T}^{i}(\varnothing^{i_1}) \} &\text{if } n = 0 \\[-0.2em]
+ \big\{ {T}^{i}(t_n, \dots, t_m) \mid t_k \in \PT\big((i_1, j_1, r_1)\big) \;
\forall k = \overline{n, m} \big\} &\text{if } n > 0
\end{cases}
\end{align*}
+ \medskip
+
+We write $str(t)$ to denote the string formed by concatenation of all alphabet symbols in the left-to-right traversal of $t$,
+and $\PT(r, w)$ denotes the set $\big\{ t \in \PT(\IRE(r)) \mid str(t) = w \big\}$ of all PTs for a RE $r$ and a string $w$.
+
+ \begin{Xdef}\label{ambiguity_of_parse_trees}
+ PTs $s$ and $t$ are \emph{ambiguous} iff $s \neq t$ and $s, t \in PT(r, w)$ for some RE $r$ and string $w$.
+ $\square$
+ \end{Xdef}
-\emph{Positions} for IPTs are defined in the same way as for PTs, and
-$Pos(t)$ denotes the set of all positions of an IPT $t$.
+Following \ref{OS13}, we assign \emph{positions} to the nodes of RE and PT.
+The root position is $\Lambda$, and position of the $i$-th subtree of a tree with position $p$ is $p.i$
+(we shorten $\|t\|_\Lambda$ as $\|t\|$).
+The \emph{length} of position $p$, denoted $|p|$, is defined as $0$ for $\Lambda$ and $|p| + 1$ for $p.i$.
+%The set of all positions is denoted $\XP$.
+The subtree of a tree $t$ at position $p$ is denoted $t|_p$.
+Position $p$ is a \emph{prefix} of position $q$ iff $q = p.p'$ for some $p'$,
+and a \emph{proper prefix} if additionaly $p \neq q$.
+Position $p$ is a \emph{sibling} of position $q$ iff $q = q'.i, p = q'.j$ for some $q'$ and $i,j \in \YN$.
+Positions are ordered lexicographically.
+The set of all positions of a tree $t$ is denoted $Pos(t)$.
Additionally, we define a set of \emph{submatch positions} as
-$Sub(t) = \{ p \mid \exists t|_p = s^i : i \neq 0 \}$.
-In other words, $Sub(t)$ is a subset of $Pos(t)$ that contains positions of subtrees with nonzero implicit submatch index.
-Intuitively, this is the set of positions that are important for submatch extraction:
+$Sub(t) = \big\{ p \mid \exists t|_p = s^i : i \neq 0 \big\}$ ---
+a subset of $Pos(t)$ that contains positions of subtrees with nonzero implicit submatch index.
+Intuitively, this is the set of positions important from disambiguation perspective:
in the case of ambiguity we do not need to consider the full trees,
just the relevant parts of them.
%
-IPTs have two definitions of norm, one for $Pos$ and one for $Sub$,
+PTs 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{figure}\label{fig_mark_enum}
+\includegraphics[width=\linewidth]{img/mark_enum.pdf}
+\vspace{-2em}
+\caption{
+IRE for RE $(\epsilon|a^{0,\infty})(a|\epsilon)^{0,3}$
+and examples of PTs for string $a$.
+S-norm is marked with $\#$.
+}
+\end{figure}
+
+%\FloatBarrier
- \begin{Xdef}\label{tnorm_of_IPTs}
- The \emph{p-norm} and \emph{s-norm} of an IPT $t$ at position $p$ are:
+ \begin{Xdef}\label{tnorm_of_PTs}
+ The \emph{p-norm} and \emph{s-norm} of a PT $t$ at position $p$ are:
\begin{align*}
\pnorm{t}{p} =
\begin{cases}
- -1 &\text{if } p \in Pos(t) \text{ and } t|_p = \varnothing^i \\
- |str(t|_p)| &\text{if } p \in Pos(t) \text{ and } t|_p \neq \varnothing^i \\
+ -1 &\text{if } p \in Pos(t) \text{ and } t|_p = \varnothing^i \\[-0.2em]
+ |str(t|_p)| &\text{if } p \in Pos(t) \text{ and } t|_p \neq \varnothing^i \\[-0.2em]
\infty &\text{if } p \not\in Pos(t)
\end{cases}
\quad\quad\quad
\snorm{t}{p} =
\begin{cases}
- -1 &\text{if } p \in Sub(t) \text{ and } t|_p = \varnothing^i \\
- |str(t|_p)| &\text{if } p \in Sub(t) \text{ and } t|_p \neq \varnothing^i \\
+ -1 &\text{if } p \in Sub(t) \text{ and } t|_p = \varnothing^i \\[-0.2em]
+ |str(t|_p)| &\text{if } p \in Sub(t) \text{ and } t|_p \neq \varnothing^i \\[-0.2em]
\infty &\text{if } p \not\in Sub(t)
\end{cases}
\end{align*}
\end{Xdef}
-%In other words, the norm is infinite for all positions not in $Sub(t)$,
-%and for positions in $Sub(t)$ it depends on the form of subtree:
-%for nil subtrees it equals $-1$,
-%and for other subtrees it equals the number of symbols.
-%We shorten $\|t\|_\Lambda$ as $\|t\|$.
-
-Accordingly, IPTs have two definitions of order:
-\emph{p-order} based on p-norm
-and \emph{s-order} based on s-norm.
+Generally the norm of a subtree means the number of alphabet symbols in its leaves, with two exceptions.
+First, for nil subtrees the norm is $-1$: intuitively, they have the lowest ``ranking'' among all possible subtrees.
+Second, for inexistent subtrees (those with positions not in $Pos(t)$) the norm is infinite.
+This may seem counterintuitive at first, but it makes sense in the presense of REs with empty repetition.
+According to the POSIX, optional empty repetitions are not allowed, and our definition reflects this:
+if a tree $s$ has a subtree $s|_p$ corresponding to an empty repetition,
+and another tree $t$ has no subtree at position $p$,
+then the infinite norm $\|t\|_p$ ``outranks'' $\|s\|_p$.
+We define two orders on PTs:
- \begin{Xdef}\label{total_order_on_IPTs}
- \emph{P-order on IPTs.}
+ \begin{Xdef}\label{total_order_on_PTs}
+ \emph{P-order on PTs.}
Given parse trees $t, s \in PT(e, w)$, we say that $t <_p s$ w.r.t. \emph{desision position} $p$
iff $\pnorm{t}{p} > \pnorm{s}{p}$ and $\pnorm{t}{q} = \pnorm{s}{q} \; \forall q < p$.
We say that $t < s$ iff $t <_p s$ for some $p$.
\end{Xdef}
- \begin{Xdef}\label{partial_order_on_IPTs}
- \emph{S-order on IPTs.}
+ \begin{Xdef}\label{partial_order_on_PTs}
+ \emph{S-order on PTs.}
Given parse trees $t, s \in PT(e, w)$, we say that $t \prec_p s$ w.r.t. \emph{desision position} $p$ % $p \in Sub(t) \cup Sub(s)$
iff $\snorm{t}{p} > \snorm{s}{p}$ and $\snorm{t}{q} = \snorm{s}{q} \; \forall q < p$.
We say that $t \prec s$ iff $t \prec_p s$ for some $p$.
\end{Xdef}
-P-order is defined in exactly the same way as the order on PTs.
-S-order, however, has an important difference: is not total, as
-there might be two distinct IPTs that coincide in all submatch positions, yet differ in some non-submatch positions.
-For such trees s-order is not defined: none of them is less than the other one,
-and therefore they are called \emph{incomparable}.
-From submatch extraction perspective, incomparable IPTs are indistinguishable.
-
\begin{Xdef}\label{incomparable_IPTs}
IPTs $t$ and $s$ are \emph{incomparable}, denoted $t \sim s$,
iff neither $t \prec s$, nor $s \prec t$.
\end{Xdef}
-Incomparability relation is an equivalence relation: it is
-reflexive (obviously $s \sim s$),
-symmetric (obviously $s \sim t$ implies $t \sim s$) and
-transitive (lemma \ref{lemma_ptorder_transitivity_of_incomparability} in appendix shows that $s \sim t \wedge t \sim u$ implies $s \sim u$).
-Therefore all IPTs of a given RE can be partitioned into \emph{incomparability classes}.
-%
-Unlike the case of PTs, where the minimal PT is unique,
-in case of IPTs there is a whole class of minimal trees:
-$\IPT_{min}(r,w) = \{ t \in \IPT(r,w) \mid \forall u \in \IPT(r,w) : t \sim u \vee t < u \}$.
-
-%as theorem \ref{theorem_porder_on_IPTs} states, p-order is a strict total order.
-%As the theorem \ref{theorem_sorder_on_IPTs} states, s-order is a strict weak order.
-
- \begin{XThe}\label{theorem_porder_on_IPTs}
+ \begin{XThe}\label{theorem_porder_on_PTs}
P-order $<$ is a strict total order on IPTs.
+ (Proof given in appendix [??].)
\end{XThe}
- \begin{XThe}\label{theorem_sorder_on_IPTs}
+ \begin{XThe}\label{theorem_sorder_on_PTs}
S-order $\prec$ is a strict weak order on IPTs.
+ (Proof given in appendix [??].)
\end{XThe}
-\begin{figure}\label{fig_mark_enum}
-\includegraphics[width=\linewidth]{img/mark_enum.pdf}
-\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}
-
-What is the relation between the total p-order $<$ and the partial s-order $\prec$ on IPTs?
-Can we say that p-order is an extension of s-order on non-comparable IPTs?
-We show by the means of a counterexample that the answer to the above question is negative.
-Consider the IPTs $t$ and $u$ on figure \ref{fig_mark_enum}.
-On one hand, we have $t \prec_{2.2} u$, because $\snorm{t}{2.2} = \infty > 0 = \snorm{u}{2.2}$ and s-norms at all preceding submatch positions agree.
-On the other hand, we have $u <_{1.1} t$, because $\pnorm{t}{1.1} = -1 < 0 = \pnorm{u}{1.1}$
-and p-norms at all preceding positions agree.
-So, p-order is not necessarily an extension of s-order.
+P-order is total --- the $<$-minimal tree is unique.
+S-order is partial:
+there might be two distinct PTs that coincide in all submatch positions, yet differ in some non-submatch positions.
+Such trees are called \emph{incomparable},
+and from disambiguation perspective they are indistinguishable.
%
-However, the two orders are ``compatible'' in the following sense:
-they agree on the notion of the minimal tree (the exact meaning of this is formalized by theorem \ref{theorem_order_compat}).
+Incomparability is an equivalence relation: it is
+reflexive (obviously $s \sim s$),
+symmetric (obviously $s \sim t$ implies $t \sim s$) and
+transitive (see lemma \ref{lemma_ptorder_transitivity_of_incomparability} in appendix).
+Consequently there is a whole class of $\prec$-minimal trees.
+%$\PT_{min}(r,w) = \{ t \in \PT(r,w) \mid \forall u \in \PT(r,w) : t \sim u \vee t < u \}$.
+%
+One might ask, what is the relation between p-order $<$ and s-order $\prec$?
+We show by the means of a counterexample that p-order is not an extension of s-order.
+Consider trees $t$ and $u$ on figure \ref{fig_mark_enum}:
+on one hand $t \prec_{2.2} u$, because $\snorm{t}{2.2} = \infty > 0 = \snorm{u}{2.2}$ and s-norms at all preceding submatch positions agree;
+on the other hand $u <_{1.1} t$, because $\pnorm{t}{1.1} = -1 < 0 = \pnorm{u}{1.1}$
+and p-norms at all preceding positions agree.
+So the two orders may disagree.
+However, as theorem \ref{theorem_order_compat} shows,
+the two orders agree on the notion of minimal tree.
%
-This is an important result becuse it means that
-if we keep refining submatch granularity by tunrning more and more sub-RE into submatch groups,
-we will continuously narrow down the class of $\prec$-minimal trees,
+This is important, as it means that adding a few parentheses in RE will not drastically change submatch results:
+we will continuously narrow down the class of $\prec$-minimal trees
until we are left with a unique $<$-minimal tree.
-In practice this means that adding a few parentheses in RE will not drastically change the previous results of submatch extraction;
-it will only add more details.
+In the rest of the paper the words ``order'' and ``norm`` refer to s-order and s-norm.
\begin{XThe}\label{theorem_order_compat}
For an IRE $r$ and string $w$,
- let $t_{min}$ be the $<$-minimal tree in $\IPT(r,w)$
- and let $T_{min} = \IPT_{min}(r,w)$ be the class of the $\prec$-minimal trees in $\IPT(r,w)$.
+ let $t_{min}$ be the $<$-minimal tree in $\PT(r,w)$
+ and let $T_{min} = \PT_{min}(r,w)$ be the class of the $\prec$-minimal trees in $\PT(r,w)$.
Then $t_{min} \in T_{min}$.
+ (Proof given in appendix [??].)
\end{XThe}
-In the rest of the paper the words ``order'' and ``norm`` refer to s-order and s-norm.
-
-\FloatBarrier
-
-\section{Parenthesized expressions}
-
-In this section we introduce linearised representation of parse trees: \emph{parenthesized expressions}.
-%The structure of the section follows \cite{OS13},
-%however there are subtle, but important differences in some of the definitions and proofs.
-
- \begin{Xdef}
- \emph{Parenthesized expressions (PEs)} over finite alphabet $\Sigma$, denoted $\XP_\Sigma$, are:
- \begin{enumerate}
- \item Atomic PE:
- \emph{nil expression} $\Xm \in \XP_\Sigma$,
- \emph{empty expression} $\epsilon \in \XP_\Sigma$ and
- \emph{unit expression} $a \in \XP_\Sigma$, where $\alpha \in \Sigma$.
- \item Compound PE: if $e_1, e_2 \in \XP_\Sigma$, then
- $e_1 e_2 \in \XP_\Sigma$ and
- $\Xl e_1 \Xr \in \XP_\Sigma$.
- \end{enumerate}
- \end{Xdef}
-
-Parenthesis $\Xl$ is called \emph{opening} and
-parenthesis $\Xr$ is called \emph{closing};
+Following the idea of Okui and Suzuki,
+we go from comparison of parse trees to comparison of their linearized representation --- parenthesized expressions.
+Parenthesis $\Xl$ is opening, and
+parenthesis $\Xr$ is closing;
the \emph{nil}-parenthesis $\Xm$ is both opening and closing.
-The \emph{height} of parenthesis in a PE is the number of preceding opening parentheses (including this one)
+For convenience we sometimes annotate parentheses with \emph{height},
+which we define as the number of preceding opening parentheses (including this one)
minus the number of preceding closing parentheses (including this one).
-For convenience we sometimes explicitly denote height with subscript:
-this allows us to consider (not necessarily correctly nested) fragments of the given PE in isolation,
-without losing the context of the whole PE.
-However, height is not a part of parenthesis itself
-and is not taken into account when comparing the elements of PEs.
-Function $\Phi : \YZ \times \XT_\Sigma \rightarrow \XP_\Sigma$ transforms PTs into PEs:
- $$
- \Phi_{h}(t^{i}) = \begin{cases}
- str(t^{i}) &\text{if } i = 0 \\
- \Xm_h &\text{if } i \neq 0 \wedge t = \varnothing \\
- \Xl_{h+1} \Xr_h &\text{if } i \neq 0 \wedge t = \epsilon \\
- \Xl_{h+1} a \Xr_h &\text{if } i \neq 0 \wedge t = a \in \Sigma \\
+Explicit height annotations allow us to consider PE fragments in isolation
+without losing the context of the whole expression.
+However, height is not a part of parenthesis itself,
+and it is not taken into account when comparing the elements of PEs.
+Function $\Phi$ transforms PT at the given height into PE:
+
+ \begin{align*}
+ \Phi &: \YZ \times \XT_\Sigma \rightarrow \XP_\Sigma
+ \\
+ \Phi_{h}(t^{i}) &= \begin{cases}
+ str(t^{i}) &\text{if } i = 0 \\[-0.2em]
+ \Xm_h &\text{if } i \neq 0 \wedge t = \varnothing \\[-0.2em]
+ \Xl_{h+1} \Xr_h &\text{if } i \neq 0 \wedge t = \epsilon \\[-0.2em]
+ \Xl_{h+1} a \Xr_h &\text{if } i \neq 0 \wedge t = a \in \Sigma \\[-0.2em]
\Xl_{h+1} \Phi_{h+1}(t_1) \dots \Phi_{h+1}(t_n) \Xr_h &\text{if } i \neq 0 \wedge t = T(t_1, \dots, t_n)
\end{cases}
- $$
+ \end{align*}
+ \medskip
-For a given RT $r$ and string $w$, the set $\big\{ \Phi_{0}(t) \mid t \in PT(r, w) \big\}$ is denoted $PE(r, w)$.
-If $\alpha, \beta \in PE(r, w)$ for some $r$ and $w$, we say that $\alpha$ and $\beta$ are \emph{comparable}.
+For a given RE $r$ and string $w$, the set $\big\{ \Phi_{0}(t) \mid t \in PT(r, w) \big\}$ is denoted $\PE(r, w)$.
+If $\alpha, \beta \in \PE(r, w)$ for some $r$ and $w$, we say that $\alpha$ and $\beta$ are \emph{comparable}.
%
-For PE $\alpha$, $\beta$,
+For PE $\alpha$ and $\beta$,
$\alpha \sqcap \beta$ denotes the longest common prefix of $\alpha$ and $\beta$,
$\alpha \backslash \beta$ denotes the suffix of $\alpha$ after removing $\alpha \sqcap \beta$,
-$lasth(\alpha)$ denotes the height of the last parenthesis in $\alpha$ or $\infty$ if $\alpha$ is empty or begins with a letter,
-$minh(\alpha)$ denotes the minimal height of parenthesis in $\alpha$ or $\infty$ if $\alpha$ is empty or begins with a letter,
-$first(\alpha)$ denotes the first parenthesis in $\alpha$ or $\bot$ if $\alpha$ is empty or begins with a letter.
+$lasth(\alpha)$ denotes the height of the last parenthesis in $\alpha$ (or $\infty$ if $\alpha$ is empty or begins with a letter),
+$minh(\alpha)$ denotes the minimal height of parenthesis in $\alpha$ (or $\infty$ if $\alpha$ is empty or begins with a letter),
+$first(\alpha)$ denotes the first parenthesis in $\alpha$ (or $\bot$ if $\alpha$ is empty or begins with a letter).
Each PE $\alpha$ can be represented as $\alpha_0 a_1 \alpha_1 \dots a_n \alpha_n$,
-where $\alpha_i$ is the $i$-th \emph{frame} --- the possibly empty sequence of parentheses between
-subsequent alphabet symbols $a_i$ and $a_{i+1}$ or the beginning or end of $\alpha$.
-Note that comparable PE have the same number of frames.
-For distinct comparable PE $\alpha$, $\beta$
-the index of the first distinct pair of frames is called \emph{fork}.
+where $\alpha_i$ is the $i$-th \emph{frame} --- a possibly empty sequence of parentheses between
+subsequent alphabet symbols $a_i$ and $a_{i+1}$ (or the beginning or end of $\alpha$).
+For comparable PE $\alpha$ and $\beta$ the index of the first distinct pair of frames is called \emph{fork}
+(note that comparable PE have the same number of frames).
+
+\begin{figure}\label{fig_pe}
+\includegraphics[width=\linewidth]{img/pe.pdf}
+\vspace{-2em}
+\caption{
+Examples: (a) -- (d): four main rules of POSIX comparison,
+(e) -- pairwise comparison of PEs.
+}
+\end{figure}
+
+%\FloatBarrier
\begin{Xdef}
Let $\alpha$, $\beta$ be distinct comparable PE, such that
We define $trace (\alpha, \beta)$ as the sequence $(\rho_0, \dots, \rho_n)$, where:
$$
\rho_i = \begin{cases}
- -1 &\text{if } i < k \\
- min (lasth (\alpha_i \sqcap \beta_i), minh(\alpha_i \backslash \beta_i)) &\text{if } i = k \\
+ -1 &\text{if } i < k \\[-0.2em]
+ min (lasth (\alpha_i \sqcap \beta_i), minh(\alpha_i \backslash \beta_i)) &\text{if } i = k \\[-0.2em]
min (\rho_{i-1}, minh(\alpha_i)) &\text{if } i > k
\end{cases}
$$
We write $traces(\alpha, \beta)$ to denote $\big( trace (\alpha, \beta), trace (\beta, \alpha) \big)$.
\end{Xdef}
-\begin{figure}\label{fig_pe}
-\includegraphics[width=\linewidth]{img/pe.pdf}
-\caption{
-Examples: (a) -- (d): four main rules of POSIX comparison,
-(e) -- pairwise comparison of PEs.
-}
-\end{figure}
-
\begin{Xdef}\label{prec1}
Let $\alpha$, $\beta$ be comparable PE and
$traces(\alpha, \beta) = \big( (\rho_0, \dots, \rho_n), (\rho'_0, \dots, \rho'_n) \big)$.
$\alpha \sqsubset \beta \Leftrightarrow \exists i \leq n:
\big( \rho_i > \rho'_i \big) \wedge
\big( \rho_j = \rho'_j \; \forall j > i \big)$.
-% $$
-% \alpha \sqsubset \beta \quad \Leftrightarrow \quad \exists i \leq n:
-% \prescript{}{\wedge}{\left[
-% \begin{aligned}
-% &\rho_i > \rho'_i \\
-% &\forall j > i: \rho_j = \rho'_j
-% \end{aligned}
-% \right.}
-% $$
If neither $\alpha \sqsubset \beta$, nor $\beta \sqsubset \alpha$,
then $\alpha$, $\beta$ are \emph{longest-equivalent}: $\alpha \sim \beta$
(note that in this case $\rho_i = \rho'_i \; \forall i = \overline {1, n}$).
$\alpha \subset \beta \Leftrightarrow x < y$, where
the set of possible values of $x$ and $y$ is ordered as follows:
$\bot < \Xr < \Xl < \Xm$.
-% Let $\alpha$, $\beta$ be comparable PE, and let
-% $x = first (\alpha \backslash \beta)$,
-% $y = first (\beta \backslash \alpha)$.
-% The \emph{leftmost-precedence} relation $\subset$ is defined as follows:
-% $\alpha \subset \beta \Leftrightarrow
-% \big(x = \bot \wedge y \not= \bot \big) \vee
-% \big(x = \Xr \wedge y \not= \bot \big) \vee
-% \big(x = \Xl \wedge y = \bar{\Xl} \big)$.
-%
-% $$
-% \alpha \subset \beta \quad \Leftrightarrow \quad
-% \prescript{}{\vee \;}{\left[
-% \begin{aligned}
-% &x = \bot &&\wedge && y \not= \bot \\
-% &x = \Xr &&\wedge && y \not= \bot \\
-% &x = \Xl &&\wedge && y = \bar{\Xl}
-% \end{aligned}
-% \right.}
-% $$
\end{Xdef}
\begin{Xdef}\label{pe_order}
$\alpha < \beta \Leftrightarrow
\big( \alpha \sqsubset \beta \big) \vee
\big( \alpha \sim \beta \wedge \alpha \subset \beta \big)$.
-% \begin{align*}
-% \alpha < \beta \quad \Leftrightarrow \quad
-% \prescript{}{\vee}{\left[
-% \begin{aligned}
-% &\alpha \sqsubset \beta \\
-% &\alpha \sim \beta \wedge \alpha \subset \beta
-% \end{aligned}
-% \right.}
-% \end{align*}
\end{Xdef}
\begin{XThe}\label{theorem_order_on_pe_same_as_on_pt}
$s <_p t \Leftrightarrow \Phi_{h}(s) < \Phi_{h}(t) \; \forall h$.
\end{XThe}
-\FloatBarrier
+
+\section{Representation of match results}
+
+\begin{algorithm}[H] \DontPrintSemicolon \SetKwProg{Fn}{}{}{}
+\begin{multicols}{2}
+ \setstretch{0.8}
+
+ \Fn {$\underline{initial \Xund result (T)} \smallskip$} {
+ \For {$i = \overline {0, |T| / 2}$} {
+ $pmatch[i].rm \Xund so = -1$ \;
+ $pmatch[i].rm \Xund eo = -1$ \;
+ }
+ \Return $pmatch$ \;
+ }
+ \BlankLine
+
+ \Fn {$\underline{update \Xund result (X, k, \Xund)} \smallskip$} {
+ \Return $\big\{ (q, o, u, apply \Xund tags (u, r, k)) \mid (q, o, u, r) \in X \big\}$ \;
+ %\For {$(q, o, t_1 \hdots t_n, pmatch) \in X$} {
+ % \For {$i = \overline {1, n}$} {
+ % \lIf {$t_i < 0$} {$l = -1$}
+ % \lElse {$l = k$}
+ % \lIf {$t_i mod \, 2 \equiv 0$} {$pmatch[|t_i|/2].rm \Xund so = l$}
+ % \lElse {$pmatch[(|t_i| - 1)/2].rm \Xund eo = l$}
+ % }
+ %}
+ %\Return $X$ \;
+ }
+ \BlankLine
+
+ \Fn {$\underline{f\!inal \Xund result (u, r, k)} \smallskip$} {
+ $pmatch = apply \Xund tags (u, r, k)$ \;
+ $pmatch[0].rm \Xund so = 0$ \;
+ $pmatch[0].rm \Xund eo = k$ \;
+ \Return $pmatch$ \;
+ }
+ \BlankLine
+
+ \Fn {$\underline{apply \Xund tags (t_1 \hdots t_n, pmatch, k)} \smallskip$} {
+ \For {$i = \overline {1, n}$} {
+ \lIf {$t_i < 0$} {$l = -1$}
+ \lElse {$l = k$}
+ \lIf {$t_i mod \, 2 \equiv 0$} {$pmatch[|t_i|/2].rm \Xund eo = l$}
+ \lElse {$pmatch[(|t_i| \!+\! 1)/2].rm \Xund so = l$}
+ }
+ \Return $pmatch$ \;
+ }
+ \BlankLine
+
+ \vfill
+
+\columnbreak
+
+ \Fn {$\underline{initial \Xund result (\Xund)} \smallskip$} {
+ \Return $\epsilon$ \;
+ }
+ \BlankLine
+
+ \Fn {$\underline{update \Xund result (X, \Xund, \alpha)} \smallskip$} {
+ \Return $\big\{ (q, o, u, r \cdot u \cdot \alpha) \mid (q, o, u, r) \in X \big\}$ \;
+ }
+ \BlankLine
+
+ \Fn {$\underline{f\!inal \Xund result (u, r, \Xund)} \smallskip$} {
+ \Return $parse \Xund tree (r \cdot u, 1)$ \;
+ }
+ \BlankLine
+
+ \Fn {$\underline{parse \Xund tree (u, i)} \smallskip$} {
+ \If {$u = (2i \!-\! 1) \cdot (2i)$} {
+ \Return $T^i(\epsilon)$
+ }
+ \If {$u = (1 \!-\! 2i) \cdot \hdots $} {
+ \Return $T^i(\varnothing)$
+ }
+ \If {$u = (2i \!-\! 1) \cdot \alpha_1 \hdots \alpha_n \cdot (2i) \wedge \alpha_1, \hdots, \alpha_n \in \Sigma $} {
+ \Return $T^i(a_1, \hdots, a_n)$
+ }
+ \If {$u = (2i \!-\! 1) \cdot \beta_1 \hdots \beta_m \cdot (2i) \wedge \beta_1 = 2j \!-\! 1 \in T$} {
+ $n = 0, k = 1$ \;
+ \While {$k \leq m$} {
+ $l = k$ \;
+ \lWhile {$|\beta_{k+1}| > 2j$} {
+ $k = k + 1$
+ }
+ $n = n + 1$ \;
+ $t_n = parse \Xund tree (\beta_l \dots \beta_k, j)$
+ }
+ \Return $T^i(t_1, \dots, t_n)$
+ }
+ \Return $\varnothing$ \tcp{ill-formed PE}
+ }
+ \BlankLine
+
+\end{multicols}
+\vspace{1.5em}
+\caption{Construction of match results: POSIX offsets (on the left) and parse tree (on the right).}
+\end{algorithm}
+\medskip
\section{Tagged NFA}
projects / re2c / commitdiff