simplify: more rules for first_match
* spot/tl/simplify.cc: Implement the rules. * tests/core/reduccmp.test: Test them. * doc/tl/tl.tex: Document them.
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Alexandre Duret-Lutz
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3 changed files with 118 additions and 12 deletions
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@ -1590,9 +1590,17 @@ SERE.
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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(r\FSTAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH\code(r\FSTAR{\mvar{i}}\code) \\
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\FIRSTMATCH\code(r_1\CONCAT{}r_2\STAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH\code(r_1\CONCAT{}r_2\STAR{\mvar{i}}\code) \\
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\FIRSTMATCH\code(r_1\CONCAT{}r_2\FSTAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH\code(r_1\CONCAT{}r_2\FSTAR{\mvar{i}}\code) \\
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\FIRSTMATCH\code(r_1\FUSION{}r_2\STAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH\code(r_1\FUSION{}r_2\STAR{\mvar{\max(i,1)}}\code) \\
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\FIRSTMATCH\code(r_1\FUSION{}r_2\FSTAR{\mvar{i}..\mvar{j}}\code) &\equiv \FIRSTMATCH\code(r_1\FUSION{}r_2\FSTAR{\mvar{i}}\code) \\
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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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\FIRSTMATCH\code(b\STAR{\mvar{i}..\mvar{j}}\CONCAT{}r\code) &\equiv b\STAR{\mvar{i}}\CONCAT\FIRSTMATCH\code(b\STAR{0..\mvar{j-i}}\CONCAT r\code) \\
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\FIRSTMATCH\code(b\STAR{\mvar{i}..\mvar{j}}\FUSION{}r\code) &\equiv b\STAR{\mvar{i-1}}\CONCAT\FIRSTMATCH\code(b\STAR{1..\mvar{j-i+1}}\FUSION r\code)\quad\text{if~}i>1\\
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\FIRSTMATCH\code(r_1\CONCAT{}r_2\code) &\equiv\FIRSTMATCH\code(r_1\code)\quad\text{if~} \varepsilon\VDash r_2\\
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\FIRSTMATCH\code(\FIRSTMATCH\code(r_1\code)\CONCAT{}r_2\code) &\equiv\FIRSTMATCH\code(r_1\code)\CONCAT\FIRSTMATCH\code(r_2\code)\\
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\FIRSTMATCH\code(\FIRSTMATCH\code(r_1\code)\FUSION{}r_2\code) &\equiv\FIRSTMATCH\code(r_1\code)\FUSION\FIRSTMATCH\code(\1\FUSION{}r_2\code)
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\end{align*}
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Starred subformulas are rewritten in Star Normal
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@ -1436,25 +1436,45 @@ namespace spot
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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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if (m < h.max())
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{
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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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s = h[0].is_boolean() ? s : formula::first_match(s);
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return recurse(s);
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}
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}
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else if (h.is(op::FStar))
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{
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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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if (m < h.max())
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{
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formula s = formula::FStar(h[0], m, m);
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return recurse(formula::first_match(s));
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}
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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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// 1. first_match(b;f) ≡ b;first_match(f)
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// 2. first_match(f;e) ≡ first_match(f)
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// 3. first_match(first_match(f);g) =
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// first_match(f);first_match(g)
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// 4. first_match(f;g[*i:j]) ≡ first_match(f;g[*i])
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// 5. first_match(f;g[:*i..j]) = first_match(f;g[:*i])
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// 6. first_match(b[*i:j];f) ≡ b[*i];first_match(b[*0:j-i];f)
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// Rules 1-3 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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while (i < n
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&& (h[i].is_boolean() || h[i].is(op::first_match)))
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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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// +1 to append first_match(suffix), +1 to for rule 6.
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prefix.reserve(i + 2);
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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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@ -1466,11 +1486,25 @@ namespace spot
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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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// Rules 4-5
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if (!suffix.empty() && suffix.back().is(op::Star, op::FStar))
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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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suffix.back() = formula::bunop(s.kind(),
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s[0], smin, smin);
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}
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// Rule 6
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if (!suffix.empty() && suffix.front().is(op::Star))
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{
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formula s = suffix.front();
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unsigned smin = s.min();
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if (smin > 0 && s[0].is_boolean())
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{
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prefix.push_back(formula::Star(s[0], smin, smin));
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suffix.front() =
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formula::Star(s[0], 0, s.max() - smin);
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}
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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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@ -1478,6 +1512,58 @@ namespace spot
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if (res != f)
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return recurse(res);
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}
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else if (h.is(op::Fusion))
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{
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// 1. first_match(b:f) = b:first_match(f) if f rejects [*0]
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// (not implemented, because first_match will build a DFA
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// and limiting its choices is good)
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// 2. first_match(b[*i..j]:f) =
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// b[*i-1];first_match(b[*1..j-i+1]:f) if i>1
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if (h[0].is(op::Star))
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{
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formula s = h[0];
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unsigned smin = s.min();
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if (smin > 1 && s[0].is_boolean())
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{
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--smin;
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unsigned smax = s.max();
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if (smax != formula::unbounded())
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smax -= smin;
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formula s2 = formula::Star(s[0], 1, smax);
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formula in = formula::Fusion({s2, h.all_but(0)});
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in = formula::first_match(in);
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formula s3 = formula::Star(s[0], smin, smin);
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return recurse(formula::Concat({s3, in}));
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}
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}
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// 3. first_match(first_match(f):g) =
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// first_match(f):first_match(1:g)
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if (h[0].is(op::first_match))
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{
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formula rest = h.all_but(0);
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if (rest.accepts_eword())
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rest = formula::Fusion({formula::tt(), rest});
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rest = formula::first_match(rest);
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return recurse(formula::Fusion({h[0], rest}));
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}
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// 4. first_match(f:g[*i..j]) = first_match(f:g[*max(1,i)])
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// 5. first_match(f:g[:*i..j]) = first_match(f:g[:*i])
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unsigned last = h.size() - 1;
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formula tail = h[last];
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if (tail.is(op::Star, op::FStar))
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{
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unsigned smin = tail.min();
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if (smin < tail.max())
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{
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if (smin == 0 && tail.is(op::Star))
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++smin;
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formula s2 =
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formula::bunop(tail.kind(), tail[0], smin, smin);
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formula in = formula::Fusion({h.all_but(last), s2});
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return recurse(formula::first_match(in));
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}
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}
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}
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return f;
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}
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}
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@ -446,6 +446,18 @@ GF(a && GF(b) && c), G(F(a & c) & Fb)
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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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{first_match(a;b[*2..3];a)}, a & X(b & X(b & X{first_match(b[*0..1];a)}))
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{first_match(b:a[*2][+])}, {first_match(b:a[*2])}
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{first_match(b[*3]:a[*2][+])}, b & X(b & X{first_match(b:a[*2])})
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{first_match((a | (b;b))[*2..3])}, {first_match((a | b[*2])[*2])}
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{first_match((a | (b;b))[:*2..3])}, {first_match((a | b[*2])[:*2])}
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{first_match(a;(a | (b;b))[*2..3])}, a & X{first_match((a | b[*2])[*2])}
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{first_match(a:(a | (b;b))[*2..3])}, {first_match(a:(a | b[*2])[*2])}
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{first_match(a;(a | (b;b))[:*2..3])}, a & X{first_match((a | b[*2])[:*2])}
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{first_match(a:(a | (b;b))[:*2..3])}, {first_match(a:(a | b[*2])[:*2])}
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{first_match(1:e[*0..3])[*]}[]-> c, c W !e
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{first_match(first_match(a*;e);b)}[]->a, {first_match(a[*];e)}[]-> X(a | !b)
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{first_match(first_match(a*;e):b*)}[]->a, {first_match(a[*];e)}[]-> (a | !b)
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EOF
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run 0 ../reduccmp nottau.txt
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