tl: add some simplifications for first_match
Following a discussion with Victor Khomenko. * doc/tl/tl.tex: Document those rules. * spot/tl/simplify.cc: Implement them. * tests/core/reduccmp.test: Test them.
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Alexandre Duret-Lutz
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b7cd475632
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3 changed files with 73 additions and 20 deletions
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@ -801,7 +801,8 @@ $b_1$, $b_2$ are assumed to be Boolean formulas.
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&&
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f\FSTAR{\mvar{i}..\mvar{j}}\FSTAR{\mvar{k}..\mvar{l}} &\equiv f\FSTAR{\mvar{ik}..\mvar{jl}}\text{~if~}i(k+1)\le jk+1 \\
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f\FSTAR{0}&\equiv \1 & f\FSTAR{1}&\equiv f\text{~if~}\varepsilon\nVDash f\\
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\FIRSTMATCH\code(b\code) &\equiv b & \FIRSTMATCH\code(f\code) &\equiv \eword\text{~if~}\varepsilon\VDash f
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\FIRSTMATCH\code(b\code) &\equiv b & \FIRSTMATCH\code(f\code) &\equiv \eword\text{~if~}\varepsilon\VDash f \\
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&& \FIRSTMATCH\code(\FIRSTMATCH\code(f\code)\code) &\equiv \FIRSTMATCH\code(f\code)
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\end{align*}
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\noindent
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@ -995,22 +996,6 @@ operator, even if the operator has multiple synonyms (like \samp{|},
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\end{align*}
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\section{Operator precedence}
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The following operator precedence describes the current parser of
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@ -1600,7 +1585,14 @@ SERE.
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\sere{b_1\FUSION r_1}\ANDALT\sere{b_2\FUSION r_2} &\equiv \sere{b_1\ANDALT b_2}\FUSION\sere{r_1\ANDALT r_2} &
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\sere{r_1\FUSION b_1}\ANDALT\sere{r_2\FUSION b_2} &\equiv \sere{r_1\ANDALT r_2}\FUSION\sere{b_1\ANDALT b_2} \\
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\sere{b_1\CONCAT r_1}\AND \sere{b_2\CONCAT r_2} &\equiv \sere{b_1\ANDALT b_2}\CONCAT\sere{r_1\AND r_2} \\
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\sere{b_1\FUSION r_1}\AND \sere{b_2\FUSION r_2} &\equiv \sere{b_1\ANDALT b_2}\FUSION\sere{r_1\AND r_2} \mathrlap{\quad\text{if~}\varepsilon\nVDash r_1\land\varepsilon\nVDash r_2}\\
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\sere{b_1\FUSION r_1}\AND \sere{b_2\FUSION r_2} &\equiv \sere{b_1\ANDALT b_2}\FUSION\sere{r_1\AND r_2} \mathrlap{\quad\text{if~}\varepsilon\nVDash r_1\land\varepsilon\nVDash r_2}
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\end{align*}
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\begin{align*}
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\FIRSTMATCH\code(b\STAR{\mvar{i}..\mvar{j}}\code) &\equiv b\STAR{\mvar{i}} \\
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\FIRSTMATCH\code(r\STAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH\code(r\STAR{\mvar{i}}\code) \\
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\FIRSTMATCH\code(r_1\CONCAT{}r_2\STAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH(r_1\CONCAT\mathrlap{r_2\STAR{\mvar{i}})} \\
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\FIRSTMATCH\code(b\CONCAT{}r\code) &\equiv b\CONCAT\FIRSTMATCH\code(r\code) \\
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\FIRSTMATCH\code(r_1\CONCAT{}r_2\code) &\equiv\FIRSTMATCH(r_1)\mathrlap{\quad\text{if~} \varepsilon\VDash r_2}
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\end{align*}
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Starred subformulas are rewritten in Star Normal
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@ -856,7 +856,6 @@ namespace spot
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case op::ap:
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case op::Not:
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case op::FStar:
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case op::first_match:
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return f;
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case op::X:
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{
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@ -1429,6 +1428,58 @@ namespace spot
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}
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return formula::Star(h, min, f.max());
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}
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case op::first_match:
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{
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formula h = f[0];
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if (h.is(op::Star))
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{
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// first_match(b[*i..j]) = b[*i]
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// first_match(f[*i..j]) = first_match(f[*i])
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unsigned m = h.min();
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formula s = formula::Star(h[0], m, m);
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return h[0].is_boolean() ? s : formula::first_match(s);
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}
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else if (h.is(op::Concat))
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{
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// If b is Boolean and e accepts [*0], we have
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// first_match(b;f) = b;first_match(f)
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// first_match(f;e) = first_match(f)
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// first_match(f;g[*i..j]) = first_match(f;g[*i])
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// the first two rules can be repeated, so we will loop
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// to save the recursion and intermediate caching.
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// Extract Boolean formulas at the beginning.
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int i = 0;
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int n = h.size();
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while (i < n && h[i].is_boolean())
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++i;
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vec prefix;
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prefix.reserve(i + 1); // +1 to append first_match()
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for (int ii = 0; ii < i; ++ii)
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prefix.push_back(h[ii]);
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// Extract suffix, minus trailing formulas that accept [*0].
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// "i" is the start of the suffix.
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do
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--n;
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while (i <= n && h[n].accepts_eword());
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vec suffix;
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suffix.reserve(n + 1 - i);
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for (int ii = i; ii <= n; ++ii)
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suffix.push_back(h[ii]);
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if (!suffix.empty() && suffix.back().is(op::Star))
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{
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formula s = suffix.back();
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unsigned smin = s.min();
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suffix.back() = formula::Star(s[0], smin, smin);
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}
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prefix.push_back(formula::first_match
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(formula::Concat(suffix)));
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formula res = formula::Concat(prefix);
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if (res != f)
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return recurse(res);
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}
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return f;
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}
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}
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SPOT_UNREACHABLE();
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}
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@ -1,6 +1,6 @@
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#! /bin/sh
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# -*- coding: utf-8 -*-
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# Copyright (C) 2009-2014, 2016-2018 Laboratoire de Recherche et
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# Copyright (C) 2009-2014, 2016-2019 Laboratoire de Recherche et
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# Developpement de l'Epita (LRDE).
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# Copyright (C) 2004, 2006 Laboratoire d'Informatique de Paris 6 (LIP6),
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# département Systèmes Répartis Coopératifs (SRC), Université Pierre
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@ -436,6 +436,16 @@ GF(a && GF(b) && c), G(F(a & c) & Fb)
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# not reduced
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{a;(b[*2..4];c*;([*0]+{d;e}))*}!, {a;(b[*2..4];c*;([*0]+{d;e}))*}!
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{c[*];e[*]}[]-> a, {c[*];e[*]}[]-> a
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# first_match
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{first_match(a*);b}, b
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{first_match(a)}, a
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{first_match(a;b*)}, a
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{first_match(a;c;b*)}, a & Xc
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{first_match(a;c;b*;e)}, a & X(c & X{first_match(b[*];e)})
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{first_match(a;c;b*;(e | [*0]))}, a & Xc
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{first_match(a;c;b[+];(e | [*0]))}, a & X(c & Xb)
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{first_match(b[+];(e | [*0]))}, b
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{first_match(b[*2..3])}, b & Xb
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EOF
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run 0 ../reduccmp nottau.txt
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