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https://github.com/Z3Prover/z3
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683cb4ec03
Light-weight Antimirov cofactors for seq_monadic
================================================
Overview
--------
The seq_monadic solver explores symbolic regex derivatives when
computing live
states and product transitions. Its original transition representation
used
Brzozowski cofactors.
This change adds a light-weight Antimirov representation. It first
computes the
existing Brzozowski cofactors and then decomposes targets whose outer
shape is
s1 | ... | sn
or
(s1 | ... | sn) . tail
into separate transitions. Concatenations are maintained in
right-associative
form, so a distributable union occurs as the head of the concatenation.
Targets are reconstructed with mk_regex_concat to preserve
normalization.
After decomposition, transitions with the same target are merged by
disjoining
their guards. The existing path-aware cofactor traversal and range
predicates
are therefore retained.
Example
-------
For
r = .* a .{k}
the Brzozowski cofactors have the shape
[a, r | .{k}]
[^a, r]
The light-weight Antimirov transformation produces
[., r]
[a, .{k}]
This preserves the language while avoiding the deterministic subset
states
that grow exponentially on this family.
Modes
-----
seq_monadic exposes two transition modes:
light_antimirov default
brzozowski retained as an explicit option
The implementation is seq_rewriter::light_ant_derivative_cofactors.
Correctness
-----------
The seq_monadic unit tests run in both modes. They cover character and
generic
element sequences, multiple and repeated variables, variable
constraints,
bounded loops, conjunctions of memberships, and witness construction. A
focused test checks the cofactor transformation above. The complete
94-test
unit suite used during evaluation passed. The benchmark harness is not
registered as a normal unit test; after detaching it, all 93 registered
tests
pass.
Benchmark evaluation
--------------------
The final optimized comparison used all 1,545 SMT2 files under
C:\git\bench\inputs\regexes. Each file was run in a separate process
with a
15-second timeout. There were 1,513 cases where both modes completed
without a
process failure or timeout.
Brzozowski Light-Ant
paired solver time 94.98 s 63.26 s
median solver time 1.734 ms 0.710 ms
derivative calls 5.70 M 3.39 M
cofactors 13.33 M 6.99 M
live states 2.74 M 0.44 M
product states 652.8 K 642.8 K
Light-Ant reduced paired solver time by 33.4%, derivative calls by
40.5%,
cofactors by 47.5%, and live states by 84.0%. It was faster on 1,091
cases;
Brzozowski was faster on 421 cases.
Light-Ant changed 131 Brzozowski undef results to sat. There were no
reverse
verdict changes and no mismatches against known sat/unsat statuses. Both
modes
had two timeouts. Process failures decreased from 30 to 27.
By corpus, paired solver time improved by 39.1% on ClemensRegex and by
11.9% on
MargusRegex.
Alternatives considered
-----------------------
Direct use of full Antimirov derivatives was also evaluated. It
introduced
large performance outliers, particularly around intersections, and
produced
additional undef results and timeouts. Disabling intersection-over-union
distribution improved some of these cases but remained slower and less
robust
than the light-weight transformation. The direct full-Ant mode was
therefore
removed from this change.
Copilot-Session: a2ce3573-4e15-4a4a-afb5-21e3cb04e4a2
369 lines
17 KiB
C++
369 lines
17 KiB
C++
/*++
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Copyright (c) 2026 Microsoft Corporation
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Module Name:
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seq_monadic.cpp
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Abstract:
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Unit tests for the whole-language monadic-decomposition membership solver in
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ast/rewriter/seq_monadic.cpp. Mirrors the validated Python prototype
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(files/solve_proto.py): single-variable repeated-membership shapes x.a.x in R.
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Author:
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Nikolaj Bjorner / Margus Veanes 2026
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--*/
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#include "ast/ast.h"
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#include "ast/reg_decl_plugins.h"
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#include "ast/seq_decl_plugin.h"
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#include "ast/arith_decl_plugin.h"
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#include "ast/rewriter/seq_rewriter.h"
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#include "ast/rewriter/seq_monadic.h"
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#include "ast/rewriter/expr_safe_replace.h"
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#include <iostream>
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namespace {
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struct plugin_registrar {
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plugin_registrar(ast_manager& m) { reg_decl_plugins(m); }
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};
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class seq_monadic_test {
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ast_manager m;
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plugin_registrar m_reg;
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seq_rewriter m_rw;
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seq_monadic m_mon;
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seq_util u;
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sort_ref m_str; // String sort
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sort_ref m_re; // RegEx sort over m_str
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seq_monadic::transition_mode m_mode;
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unsigned m_fail = 0;
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seq_util::rex& re() { return u.re; }
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// regex builders
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expr_ref word(char const* s) { return expr_ref(re().mk_to_re(u.str.mk_string(zstring(s))), m); }
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expr_ref cat(expr* a, expr* b) { return expr_ref(re().mk_concat(a, b), m); }
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expr_ref alt(expr* a, expr* b) { return expr_ref(re().mk_union(a, b), m); }
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expr_ref star(expr* a) { return expr_ref(re().mk_star(a), m); }
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expr_ref inter(expr* a, expr* b) { return expr_ref(re().mk_inter(a, b), m); }
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expr_ref comp(expr* a) { return expr_ref(re().mk_complement(a), m); }
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expr_ref dot() { return expr_ref(re().mk_full_char(m_re), m); }
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expr_ref dotstar() { return expr_ref(re().mk_full_seq(m_re), m); }
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expr_ref rng(char lo, char hi) {
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char sl[2] = { lo, 0 }, sh[2] = { hi, 0 };
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return expr_ref(re().mk_range(u.str.mk_string(zstring(sl)), u.str.mk_string(zstring(sh))), m);
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}
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expr_ref loop(expr* r, unsigned lo, unsigned hi) { return expr_ref(re().mk_loop(r, lo, hi), m); }
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// string-term builders
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expr_ref var(char const* nm) { return expr_ref(m.mk_const(nm, m_str), m); }
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expr_ref sword(char const* s) { return expr_ref(u.str.mk_string(zstring(s)), m); }
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expr_ref sconcat(expr* a, expr* b) { return expr_ref(u.str.mk_concat(a, b), m); }
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// term x . w . x (w a constant word)
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expr_ref xwx(expr* x, char const* w) { return sconcat(x, sconcat(sword(w), x)); }
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// term x . a . y (two distinct variables)
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expr_ref xay(expr* x, expr* y) { return sconcat(x, sconcat(sword("a"), y)); }
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// term x . y . x
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expr_ref xyx(expr* x, expr* y) { return sconcat(x, sconcat(y, x)); }
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static char const* s(lbool l) { return l == l_true ? "sat" : l == l_false ? "unsat" : "undef"; }
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char const* mode_name() const {
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switch (m_mode) {
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case seq_monadic::transition_mode::brzozowski: return "brz";
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case seq_monadic::transition_mode::light_antimirov: return "light-ant";
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}
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UNREACHABLE();
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return "";
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}
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bool eval_guard(expr* guard, unsigned ch) {
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sort* elem_sort = nullptr;
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VERIFY(u.is_seq(m_str, elem_sort));
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expr_ref v0(m.mk_var(0, elem_sort), m);
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expr_safe_replace rep(m);
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rep.insert(v0, u.str.mk_char(ch));
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expr_ref instantiated(m), simplified(m);
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rep(guard, instantiated);
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th_rewriter rw(m);
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rw(instantiated, simplified);
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return m.is_true(simplified);
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}
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void check_ant_cofactors() {
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expr_ref a = word("a");
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expr_ref any = dotstar();
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expr_ref dot3 = loop(dot(), 3, 3);
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expr_ref R = cat(any, cat(a, dot3));
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expr_ref_pair_vector cof(m);
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m_rw.light_ant_derivative_cofactors(R, cof);
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bool found_R = false, found_dot3 = false, ok = cof.size() == 2;
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for (auto const& [guard, target] : cof) {
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if (target == R) {
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found_R = eval_guard(guard, 'a') && eval_guard(guard, 'b');
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}
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else if (target == dot3) {
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found_dot3 = eval_guard(guard, 'a') && !eval_guard(guard, 'b');
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}
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else {
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ok = false;
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}
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}
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ok = ok && found_R && found_dot3;
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if (!ok) ++m_fail;
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std::cout << (ok ? " OK " : " FAIL ")
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<< mode_name() << " cofactor construction\n";
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}
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void check(char const* name, expr* term, expr* R, lbool expected) {
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lbool got = m_mon.solve(term, R);
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bool ok = (got == expected);
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if (!ok) ++m_fail;
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std::cout << (ok ? " OK " : " FAIL ") << name
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<< " got=" << s(got) << " expected=" << s(expected) << "\n";
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}
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void check_extra(char const* name, expr* term, expr* R,
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obj_map<expr, expr*> const& ve, lbool expected) {
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lbool got = m_mon.solve(term, R, ve);
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bool ok = (got == expected);
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if (!ok) ++m_fail;
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std::cout << (ok ? " OK " : " FAIL ") << name
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<< " got=" << s(got) << " expected=" << s(expected) << "\n";
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}
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// flatten a ground sequence term into its element values.
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void flatten_seq(expr* seqv, ptr_vector<expr>& elems) {
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zstring zs;
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if (u.str.is_concat(seqv)) {
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for (expr* arg : *to_app(seqv)) flatten_seq(arg, elems);
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return;
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}
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if (u.str.is_empty(seqv))
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return;
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if (u.str.is_string(seqv, zs)) {
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for (unsigned i = 0; i < zs.length(); ++i) elems.push_back(u.str.mk_char(zs, i));
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return;
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}
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if (u.str.is_unit(seqv))
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elems.push_back(to_app(seqv)->get_arg(0));
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}
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// decide membership of a concrete word (list of element values) in R by folding
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// derivatives and testing nullability of the residual.
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bool word_in_re(ptr_vector<expr> const& elems, expr* R) {
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expr_ref cur(R, m);
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for (expr* e : elems) cur = m_rw.mk_derivative(e, cur);
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return m.is_true(m_rw.is_nullable(cur));
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}
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// solve for a model, then check the returned witness assignment actually makes
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// term a member of R (substitute var -> witness and re-decide by derivatives).
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void check_witness(char const* name, expr* term, expr* R,
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obj_map<expr, expr*> const& ve) {
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obj_map<expr, expr*> model;
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lbool got = m_mon.solve(term, R, ve, &model);
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bool ok = (got == l_true) && !model.empty();
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if (ok) {
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expr_safe_replace rep(m);
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for (auto const& kv : model) rep.insert(kv.m_key, kv.m_value);
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expr_ref g(m);
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rep(term, g);
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ptr_vector<expr> elems;
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flatten_seq(g, elems);
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ok = word_in_re(elems, R);
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}
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if (!ok) ++m_fail;
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std::cout << (ok ? " OK " : " FAIL ") << name
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<< " solve=" << s(got) << " witness-verified=" << (ok ? "yes" : "no") << "\n";
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}
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// decide a conjunction of memberships jointly (shared variables constrained together).
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void check_and(char const* name, vector<std::pair<expr*, expr*>> const& mems, lbool expected) {
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obj_map<expr, expr*> nove;
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lbool got = m_mon.solve_and(mems, nove, nullptr);
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bool ok = (got == expected);
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if (!ok) ++m_fail;
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std::cout << (ok ? " OK " : " FAIL ") << name
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<< " got=" << s(got) << " expected=" << s(expected) << "\n";
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}
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public:
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seq_monadic_test(seq_monadic::transition_mode mode) :
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m_reg(m), m_rw(m), m_mon(m_rw, mode), u(m), m_str(m), m_re(m), m_mode(mode) {
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m_str = u.str.mk_string_sort();
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m_re = re().mk_re(m_str);
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}
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void run() {
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std::cout << "=== seq_monadic mode: " << mode_name() << " ===\n";
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if (m_mode == seq_monadic::transition_mode::light_antimirov)
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check_ant_cofactors();
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expr_ref x = var("x");
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expr_ref a = word("a");
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expr_ref b = word("b");
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expr_ref ab = cat(a, b);
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expr_ref sig = dotstar(); // Sigma*
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expr_ref saas = cat(sig, cat(cat(a, a), sig)); // Sigma* a a Sigma*
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expr_ref sbbs = cat(sig, cat(cat(b, b), sig)); // Sigma* b b Sigma*
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std::cout << "=== seq_monadic: single-variable membership (x.a.x in R) ===\n";
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// sanity
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check("(a|b)* x.a.x", xwx(x, "a"), star(alt(a, b)), l_true);
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check("b* x.a.x", xwx(x, "a"), star(b), l_false);
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check("Sig*aaSig* x.a.x", xwx(x, "a"), saas, l_true);
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check("x in (a|b)* ", x, star(alt(a, b)), l_true);
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check("x in b* (x=aa) ", xwx(x, "a"), star(b), l_false);
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// ALT = (a|b)* & ~(Sig*aaSig*) & ~(Sig*bbSig*) (strictly alternating)
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expr_ref altre = inter(star(alt(a, b)), inter(comp(saas), comp(sbbs)));
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check("ALT x.a.x", xwx(x, "a"), altre, l_true);
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// R*.S complement family
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check("~(a*.b) x.a.x", xwx(x, "a"), comp(cat(star(a), b)), l_true);
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// L3-02 ~((ab)*.~((ab)*)) -> unsat (odd length)
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check("L3-02 x.a.x", xwx(x, "a"),
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comp(cat(star(ab), comp(star(ab)))), l_false);
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// L3-03 ~(a*.~(b*.~((ab)*))) -> sat
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check("L3-03 x.a.x", xwx(x, "a"),
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comp(cat(star(a), comp(cat(star(b), comp(star(ab)))))), l_true);
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std::cout << "=== seq_monadic: multi-variable ===\n";
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expr_ref y = var("y");
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check("(a|b)* x.a.y", xay(x, y), star(alt(a, b)), l_true);
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check("b* x.a.y", xay(x, y), star(b), l_false);
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check("L3-02 x.a.y", xay(x, y), comp(cat(star(ab), comp(star(ab)))), l_true);
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check("L3-03 x.a.y", xay(x, y),
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comp(cat(star(a), comp(cat(star(b), comp(star(ab)))))), l_true);
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check("empty ~Sig* x.y.x", xyx(x, y), comp(dotstar()), l_false);
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check("Sig* x.y.x", xyx(x, y), dotstar(), l_true);
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check("(a|b)* x.y.x", xyx(x, y), star(alt(a, b)), l_true);
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std::cout << "=== seq_monadic: per-variable constraints ===\n";
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expr_ref digitp = cat(rng('0', '9'), star(rng('0', '9'))); // [0-9]+
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obj_map<expr, expr*> ve;
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ve.insert(y, digitp);
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// y must be in the (a|b)* tail AND in [0-9]+ -> empty -> unsat
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check_extra("(a|b)* & y in[0-9]+ x.a.y", xay(x, y), star(alt(a, b)), ve, l_false);
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// y any digits, x/'a' anything -> sat
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check_extra("Sig* & y in[0-9]+ x.a.y", xay(x, y), dotstar(), ve, l_true);
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// Bounded loop (re.loop) with repeated variable -- exercises live_states on a
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// counted automaton (t04-exact benchmark family). Regression for a
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// reference-invalidation bug in live_states (succ[i].push_back(intern(t))).
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std::cout << "=== seq_monadic: bounded loop (t04-exact family) ===\n";
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expr_ref clsr = rng('0', '9'); // [0-9]
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expr_ref digitS = star(clsr); // [0-9]*
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expr_ref loop22 = loop(clsr, 2, 2); // [0-9]{2}
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check("[0-9]{2} x ", x, loop22, l_true); // x = "00"
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check("[0-9]{2} x.a.x", xwx(x, "a"), loop22, l_false); // 'a' not a digit
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obj_map<expr, expr*> ve2; ve2.insert(x, digitp); ve2.insert(y, digitS);
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// x.y.x in [0-9]{2}, x in [0-9]+, y in [0-9]* -> sat (x="0", y="")
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check_extra("[0-9]{2} & x[0-9]+ y[0-9]* x.y.x", xyx(x, y), loop22, ve2, l_true);
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obj_map<expr, expr*> ve3; ve3.insert(x, digitp);
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check_extra("[0-9]{2} & x[0-9]+ x.y.x", xyx(x, y), loop22, ve3, l_true);
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obj_map<expr, expr*> ve4; ve4.insert(x, digitp);
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// x.y.x in [0-9]{3}, x in [0-9]+ -> sat (x=1 digit, y=1 digit)
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check_extra("[0-9]{3} & x[0-9]+ x.y.x", xyx(x, y), loop(clsr, 3, 3), ve4, l_true);
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// ---- witness extraction: a produced witness must be a concrete SEQUENCE of
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// ---- elements that actually satisfies the membership (not a predicate).
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std::cout << "=== seq_monadic: witness extraction (char) ===\n";
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obj_map<expr, expr*> nove;
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check_witness("(a|b)* x.a.x", xwx(x, "a"), star(alt(a, b)), nove);
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check_witness("Sig*aaSig* x.a.x", xwx(x, "a"), saas, nove); // forces nonempty x
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check_witness("~(a*.b) x.a.x", xwx(x, "a"), comp(cat(star(a), b)), nove);
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check_witness("L3-03 x.a.x", xwx(x, "a"),
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comp(cat(star(a), comp(cat(star(b), comp(star(ab)))))), nove);
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check_witness("(a|b)* x.a.y", xay(x, y), star(alt(a, b)), nove);
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check_witness("Sig* x.y.x", xyx(x, y), dotstar(), nove);
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check_witness("Sig* & y[0-9]+ x.a.y", xay(x, y), dotstar(), ve); // ve: y in [0-9]+
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check_witness("[0-9]{2}&x[0-9]+ y[0-9]* x.y.x", xyx(x, y), loop22, ve2);
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// ---- generic element sort: sequences of Int exercise the non-character guard
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// ---- algebra (candidate-basis emptiness + witness), not seq::range_predicate.
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std::cout << "=== seq_monadic: generic element sort (Seq Int) ===\n";
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arith_util ar(m);
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sort_ref intS(ar.mk_int(), m);
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sort_ref seqI(u.str.mk_seq(intS), m);
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sort_ref reI(u.re.mk_re(seqI), m);
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expr_ref i1(ar.mk_numeral(rational(1), true), m);
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expr_ref i2(ar.mk_numeral(rational(2), true), m);
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expr_ref one_seq(u.str.mk_unit(i1), m); // [1] : (Seq Int)
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expr_ref re1(re().mk_to_re(u.str.mk_unit(i1)), m); // matches [1]
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expr_ref re2(re().mk_to_re(u.str.mk_unit(i2)), m); // matches [2]
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expr_ref re12s(star(alt(re1, re2)), m); // ([1]|[2])*
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expr_ref re2s(star(re2), m); // [2]*
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expr_ref xi(m.mk_const("xi", seqI), m);
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expr_ref yi(m.mk_const("yi", seqI), m);
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expr_ref xi1xi(sconcat(xi, sconcat(one_seq, xi)), m); // xi.[1].xi
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expr_ref xiyi(sconcat(xi, sconcat(one_seq, yi)), m); // xi.[1].yi
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obj_map<expr, expr*> nove2;
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check("([1]|[2])* xi.[1].xi", xi1xi, re12s, l_true);
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check("[2]* xi.[1].xi", xi1xi, re2s, l_false); // the middle [1] is not in [2]*
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check("([1]|[2])* xi ", xi, re12s, l_true);
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check("([1]|[2])* xi.[1].yi", xiyi, re12s, l_true);
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check_witness("([1]|[2])* xi.[1].xi", xi1xi, re12s, nove2);
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check_witness("([1]|[2])* xi ", xi, re12s, nove2);
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check_witness("([1]|[2])* xi.[1].yi", xiyi, re12s, nove2);
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// per-variable extra constraint over (Seq Int): yi must also be in [2]*
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obj_map<expr, expr*> veI; veI.insert(yi, re2s.get());
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check_extra("([1]|[2])* & yi[2]* xi.[1].yi", xiyi, re12s, veI, l_true);
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check_witness("([1]|[2])* & yi[2]* xi.[1].yi", xiyi, re12s, veI);
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// ---- conjunction of memberships (solve_and): a variable shared across memberships
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// ---- is constrained jointly. These are cases that are individually SAT but
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// ---- jointly UNSAT -- exactly what independent per-membership solving gets wrong.
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std::cout << "=== seq_monadic: conjunction of memberships (solve_and) ===\n";
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expr_ref aaS(star(cat(a, a)), m); // (aa)* : even number of a's
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expr_ref a_aaS(cat(a, star(cat(a, a))), m); // a(aa)* : odd number of a's
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expr_ref abS(star(ab), m); // (ab)*
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expr_ref sig2(dotstar(), m); // Sigma*
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// x in (aa)* /\ x in a(aa)* : even-and-odd length of a's -> unsat (each alone sat)
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vector<std::pair<expr*, expr*>> mUnsat1;
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mUnsat1.push_back(std::make_pair((expr*)x.get(), (expr*)aaS.get()));
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mUnsat1.push_back(std::make_pair((expr*)x.get(), (expr*)a_aaS.get()));
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check_and("x in (aa)* & x in a(aa)*", mUnsat1, l_false);
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// compound terms sharing x: x.a in (aa)* (x odd) /\ x.aa in (aa)* (x even) -> unsat
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expr_ref tXa(sconcat(x, sword("a")), m);
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expr_ref tXaa(sconcat(x, sword("aa")), m);
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vector<std::pair<expr*, expr*>> mUnsat2;
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mUnsat2.push_back(std::make_pair((expr*)tXa.get(), (expr*)aaS.get()));
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mUnsat2.push_back(std::make_pair((expr*)tXaa.get(), (expr*)aaS.get()));
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check_and("x.a in (aa)* & x.aa in (aa)*", mUnsat2, l_false);
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// consistent conjunction: x in (ab)* /\ x in Sigma* -> sat (x=eps or ab)
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vector<std::pair<expr*, expr*>> mSat;
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mSat.push_back(std::make_pair((expr*)x.get(), (expr*)abS.get()));
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mSat.push_back(std::make_pair((expr*)x.get(), (expr*)sig2.get()));
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check_and("x in (ab)* & x in Sigma*", mSat, l_true);
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// two variables, two memberships: x.a.y in (a|b)* /\ y.b.x in (a|b)* -> sat
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expr_ref tXaY(xay(x, y), m);
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expr_ref tYbX(sconcat(y, sconcat(sword("b"), x)), m);
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expr_ref abStar(star(alt(a, b)), m);
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vector<std::pair<expr*, expr*>> mSat2;
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mSat2.push_back(std::make_pair((expr*)tXaY.get(), (expr*)abStar.get()));
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mSat2.push_back(std::make_pair((expr*)tYbX.get(), (expr*)abStar.get()));
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check_and("x.a.y & y.b.x in (a|b)*", mSat2, l_true);
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|
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std::cout << "=== seq_monadic: " << (m_fail == 0 ? "ALL PASS" : "FAILURES") << " ("
|
|
<< m_fail << " fail) ===\n";
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|
ENSURE(m_fail == 0);
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|
}
|
|
};
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|
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}
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void tst_seq_monadic() {
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seq_monadic_test brz(seq_monadic::transition_mode::brzozowski);
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brz.run();
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seq_monadic_test light_ant(seq_monadic::transition_mode::light_antimirov);
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light_ant.run();
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}
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