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seq_monadic: self-contained monadic-decomposition regex membership solver (generic elements, witnesses, Boolean combinations) (#10296)

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## Summary

Adds `seq_monadic` (`src/ast/rewriter/seq_monadic.{h,cpp}`), a
self-contained,
rewriter-level decision procedure for regex membership of a term that is
a
concatenation of sequence variables and constant elements — e.g. `x·a·x
∈ R`,
including repeated and multiple variables. It uses a whole-language
*monadic
decomposition* plus automaton product-reachability; it is minterm-free
and does
**not** use Nielsen word-equation splitting or `seq_split`.

The component is **purely additive**: it is not wired into any solver
path, so
default behavior is unchanged. It ships with a unit test and an opt-in
benchmark
harness that is inert unless `Z3_SEQ_BENCH_DIR` is set.

## What it does

- `x·u ∈ R  ⇔  ⋁_q ( x reaches q in A_R  ∧  u ∈ q )` over the derivative
  automaton; `reach(q)` is never materialized as a regex (avoids the
state-elimination blowup). A variable's constraint is decided by a lazy
product-reachability search over tuples of derivative states, with
transitions
= the product of `brz_derivative_cofactors` branches and
pairwise-conjoined
  `seq::range_predicate` guards.
- **Generic in the element sort**: characters use the exact
`range_predicate`
algebra; any other element sort uses a candidate-basis over the element
values
the guards mention (sound and complete for the
`{true,false,=,<=,and,or,not}`
  guard grammar the derivatives emit).
- **Concrete witnesses**: on sat it reconstructs a witness value (a
sequence of
concrete elements, not predicates) per variable from the accepting
product path.
- **Boolean combinations**: `solve_and` decides a conjunction of
memberships
jointly, so a variable shared across memberships is constrained
consistently.
This is the natural extension since `¬(t∈R) ≡ t∈~R`, `∨` = union of
DNFs, and
  `∧` = product of DNFs.

Also de-duplicates the char-guard → `range_predicate` translator into a
single
public `seq::guard_to_range_predicate` in `seq_range_collapse` (it was
previously
duplicated there and in `seq_monadic`).

## Testing

- `tst_seq_monadic`: single / multiple / repeated variables, nested
complement,
bounded loops, per-variable constraints, a generic `(Seq Int)` section,
witness
  verification (substitute the model back and re-decide membership), and
  `solve_and` cases that are individually sat but jointly unsat.
- Full unit suite `test-z3 /a` passes (93/93).

## Evaluation (offline harness, not part of CI)

On a regex-membership benchmark corpus, restricted to files carrying a
genuine
`(set-info :status)`, the solver decides 318 and 316 are correct
(99.4%); the
only 2 disagreements are a length limitation (`|x|=2k`) that is outside
the
membership fragment.

## Known limitations / follow-ups

- Pathological deeply-nested, high-multiplicity regexes can overflow the
recursive derivative stack; a recursion-depth guard to degrade to
`unknown` is
  a natural follow-up.
- Out-of-fragment constraints (word equations, length / Parikh) are not
handled,
  by design — this decides regex membership only.

---------

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
Co-authored-by: Nikolaj Bjorner <nikolaj@cs.stanford.edu>
Copilot-Session: 916db256-43c6-4067-b6f4-fa8d2cf2f37f
This commit is contained in:
Margus Veanes 2026-07-29 15:16:54 -07:00 • committed by GitHub
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1
src/ast/rewriter/CMakeLists.txt
View file

@ -43,6 +43,7 @@ z3_add_component(rewriter
seq_subset.cpp
seq_split.cpp
seq_derive.cpp
seq_monadic.cpp
seq_range_collapse.cpp
seq_range_predicate.cpp
seq_rewriter.cpp

609
src/ast/rewriter/seq_monadic.cpp Normal file
View file

@ -0,0 +1,609 @@
/*++
Copyright (c) 2026 Microsoft Corporation
Module Name:
seq_monadic.cpp
Abstract:
Whole-language monadic decomposition for regex membership. See seq_monadic.h.
Automaton-based (product-reachability); reach(q) is never materialized as a regex.
Generic in the element sort. The decomposition, liveness and product-reachability
are element-agnostic; only the *guard algebra* over the derivative cofactor guards
depends on the element sort. For the character sort it is the exact, compact
seq::range_predicate; for any other element sort it is a candidate-basis over the
element values mentioned by the guards (sound and complete for the
{true,false,=,<=,and,or,not} grammar the derivatives emit). The same guard algebra
yields the concrete element used to build a witness sequence.
Author:
Nikolaj Bjorner / Margus Veanes 2026
--*/
#include "ast/rewriter/seq_monadic.h"
#include "ast/rewriter/seq_range_collapse.h"
#include "ast/arith_decl_plugin.h"
#include "ast/bv_decl_plugin.h"
#include <set>
#include <vector>
#include <map>
#include <tuple>
#include <functional>
#include <algorithm>
namespace {
// A conjunction of derivative cofactor guards over the element variable v0 = (:var 0),
// interpreted as the set of element values satisfying it. Two representations by
// element sort:
// * character sort: the exact, compact seq::range_predicate.
// * any other sort: the guard predicate kept symbolically and decided by a candidate
// basis -- the element values mentioned in the guards, plus one fresh value. This
// is sound and complete for the {true,false,=,<=,and,or,not} grammar the derivatives
// emit (over a general element sort only equalities appear).
class guard_set {
ast_manager& m;
seq_util& u;
sort* m_sort;
expr* m_v0;
bool m_is_char;
bool m_ok = true; // false: an unsupported guard was conjoined
seq::range_predicate m_rp; // char representation
expr_ref m_guard; // generic representation (conjunction over v0)
// ---- generic path: candidate basis ----
// element values compared to v0 by the equalities in `g`.
void collect_consts(expr* g, ptr_vector<expr>& out) const {
expr* a = nullptr, * b = nullptr;
if (m.is_and(g) || m.is_or(g)) {
for (expr* arg : *to_app(g)) collect_consts(arg, out);
return;
}
if (m.is_not(g, a)) { collect_consts(a, out); return; }
if (m.is_eq(g, a, b)) {
if (a == m_v0 && b != m_v0) out.push_back(b);
else if (b == m_v0 && a != m_v0) out.push_back(a);
}
}
// evaluate `g` at v0 := cand ; l_undef on a construct outside the grammar.
lbool eval_at(expr* g, expr* cand) const {
expr* a = nullptr, * b = nullptr;
if (m.is_true(g)) return l_true;
if (m.is_false(g)) return l_false;
if (m.is_not(g, a)) {
lbool r = eval_at(a, cand);
return r == l_undef ? l_undef : (r == l_true ? l_false : l_true);
}
if (m.is_and(g)) {
lbool r = l_true;
for (expr* arg : *to_app(g)) {
lbool e = eval_at(arg, cand);
if (e == l_false) return l_false;
if (e == l_undef) r = l_undef;
}
return r;
}
if (m.is_or(g)) {
lbool r = l_false;
for (expr* arg : *to_app(g)) {
lbool e = eval_at(arg, cand);
if (e == l_true) return l_true;
if (e == l_undef) r = l_undef;
}
return r;
}
if (m.is_eq(g, a, b)) {
expr* other = (a == m_v0) ? b : (b == m_v0 ? a : nullptr);
if (!other) return l_undef;
if (other == m_v0) return l_true;
return (cand == other) ? l_true : l_false; // canonical values: identity == equality
}
return l_undef;
}
// a value of m_sort distinct from every element of `consts`, if one can be built.
bool mk_fresh(ptr_vector<expr> const& consts, expr_ref& out) const {
if (m.is_bool(m_sort)) {
bool hasT = false, hasF = false;
for (expr* c : consts) { if (m.is_true(c)) hasT = true; else if (m.is_false(c)) hasF = true; }
if (!hasT) { out = m.mk_true(); return true; }
if (!hasF) { out = m.mk_false(); return true; }
return false;
}
arith_util a(m);
if (a.is_int_real(m_sort)) {
rational mx(0); bool any = false;
for (expr* c : consts) {
rational v;
if (a.is_numeral(c, v)) { if (!any || v > mx) mx = v; any = true; }
}
out = a.mk_numeral(any ? mx + rational(1) : rational(0), a.is_int(m_sort));
return true;
}
bv_util bv(m);
if (bv.is_bv_sort(m_sort)) {
unsigned sz = bv.get_bv_size(m_sort);
for (unsigned k = 0; k <= consts.size(); ++k) {
rational kv(k);
bool clash = false;
for (expr* c : consts) {
rational v; unsigned bsz = 0;
if (bv.is_numeral(c, v, bsz) && v == kv) { clash = true; break; }
}
if (!clash) { out = bv.mk_numeral(kv, sz); return true; }
}
return false;
}
return false;
}
lbool generic_eval(expr_ref* witness) const {
ptr_vector<expr> consts;
collect_consts(m_guard, consts);
bool saw_undef = false;
for (expr* c : consts) {
lbool r = eval_at(m_guard, c);
if (r == l_true) { if (witness) *witness = expr_ref(c, m); return l_true; }
if (r == l_undef) saw_undef = true;
}
expr_ref fresh(m);
if (mk_fresh(consts, fresh)) {
lbool r = eval_at(m_guard, fresh);
if (r == l_true) { if (witness) *witness = fresh; return l_true; }
if (r == l_undef) saw_undef = true;
}
else
saw_undef = true; // the "distinct from all mentioned values" region is untested
return saw_undef ? l_undef : l_false;
}
public:
guard_set(ast_manager& _m, seq_util& _u, sort* elem_sort, expr* v0)
: m(_m), u(_u), m_sort(elem_sort), m_v0(v0),
m_is_char(_u.is_char(elem_sort)),
m_rp(_u.max_char()), m_guard(_m) {
if (m_is_char) m_rp = seq::range_predicate::top(u.max_char());
else m_guard = m.mk_true();
}
bool ok() const { return m_ok; }
// AND in a cofactor guard g (a Boolean over v0).
void conjoin(expr* g) {
if (!m_ok) return;
if (m_is_char) {
seq::range_predicate s(u.max_char());
if (!seq::guard_to_range_predicate(u, m_v0, g, s)) { m_ok = false; return; }
m_rp = m_rp & s;
}
else
m_guard = m.mk_and(m_guard, g);
}
// l_false = empty, l_true = non-empty (sets *witness if non-null to a concrete
// element of the set), l_undef = unknown / unsupported guard.
lbool eval(expr_ref* witness) const {
if (!m_ok) return l_undef;
if (m_is_char) {
if (m_rp.is_empty()) return l_false;
if (witness) *witness = expr_ref(u.mk_char(m_rp[0].first), m);
return l_true;
}
return generic_eval(witness);
}
};
}
expr_ref seq_monadic::der_elem(expr* r, expr* elem) {
expr_ref d = m_rw.mk_derivative(elem, r); // mk_derivative(element, regex)
// Normalize: for a general element sort the derivative by a non-matching constant can
// leave a ground guard (e.g. (= 1 2)) unfolded; simplifying collapses such dead
// branches to re.empty so nullability/emptiness stay decidable.
expr_ref d2(m);
m_thrw(d, d2);
return d2;
}
void seq_monadic::live_states(expr* R, ptr_vector<expr>& out, bool& ok) {
ok = true;
obj_map<expr, unsigned> id;
expr_ref_vector states(m);
vector<svector<unsigned>> succ;
bool_vector maybe_null;
auto intern = [&](expr* s) -> unsigned {
unsigned k;
if (id.find(s, k)) return k;
k = states.size();
id.insert(s, k);
states.push_back(s);
succ.push_back(svector<unsigned>());
expr_ref nb = m_rw.is_nullable(s);
maybe_null.push_back(!m.is_false(nb)); // unknown nullability => keep (conservative)
return k;
};
intern(R);
const unsigned STATE_CAP = 1u << 12;
for (unsigned i = 0; i < states.size(); ++i) {
if (states.size() > STATE_CAP || !m.inc()) { ok = false; return; }
expr_ref_pair_vector cof(m);
m_rw.brz_derivative_cofactors(states.get(i), cof);
for (auto const& [g, t] : cof) {
if (re().is_empty(t)) continue;
unsigned k = intern(t); // MUST precede succ[i] indexing: intern may
succ[i].push_back(k); // grow (realloc) succ, invalidating succ[i]&
}
}
unsigned n = states.size();
bool_vector live;
live.resize(n, false);
for (unsigned i = 0; i < n; ++i)
live[i] = maybe_null[i];
for (bool ch = true; ch; ) {
ch = false;
for (unsigned i = 0; i < n; ++i)
if (!live[i])
for (unsigned j : succ[i])
if (live[j]) { live[i] = true; ch = true; break; }
}
for (unsigned i = 0; i < n; ++i)
if (live[i]) { out.push_back(states.get(i)); m_pin.push_back(states.get(i)); }
}
lbool seq_monadic::product_nonempty(svector<component> const& comps, expr_ref* witness_word) {
unsigned n = comps.size();
if (n == 0) {
if (witness_word)
*witness_word = expr_ref(u().str.mk_empty(m_seq_sort), m);
return l_true;
}
expr_ref var0(m.mk_var(0, m_elem_sort), m); // the element variable the guards range over
svector<expr*> start;
for (auto const& c : comps)
start.push_back(c.state);
auto id_key = [&](svector<expr*> const& st) {
std::vector<unsigned> k;
k.reserve(st.size());
for (expr* e : st) k.push_back(e->get_id());
return k;
};
typedef std::vector<unsigned> key;
bool undecided = false;
auto is_accept = [&](svector<expr*> const& st) -> bool {
for (unsigned i = 0; i < n; ++i) {
if (comps[i].target) {
if (st[i] != comps[i].target) return false;
}
else {
expr_ref nb = m_rw.is_nullable(st[i]);
if (m.is_true(nb)) continue;
if (m.is_false(nb)) return false;
undecided = true; return false;
}
}
return true;
};
std::set<key> visited;
std::vector<svector<expr*>> work;
// tree of first-discovery edges for witness reconstruction (only built when a
// witness is requested): child-key -> (parent-key, element read on the edge).
std::map<key, std::pair<key, expr*>> parent;
key start_key = id_key(start);
auto reconstruct = [&](key end_key) -> expr_ref {
ptr_vector<expr> elems; // collected in accept..start order
key k = end_key;
while (k != start_key) {
auto it = parent.find(k);
if (it == parent.end()) break; // safety (should not happen)
elems.push_back(it->second.second);
k = it->second.first;
}
expr_ref_vector es(m); // start..accept order
for (unsigned idx = elems.size(); idx-- > 0; )
es.push_back(u().str.mk_unit(elems[idx]));
if (es.empty())
return expr_ref(u().str.mk_empty(m_seq_sort), m);
return expr_ref(u().str.mk_concat(es.size(), es.data(), m_seq_sort), m);
};
work.push_back(start);
visited.insert(start_key);
while (!work.empty()) {
if (m_budget == 0) { m_giveup = true; return l_undef; }
--m_budget;
if (!m.inc())
return l_undef;
svector<expr*> st = work.back();
work.pop_back();
if (is_accept(st)) {
if (witness_word)
*witness_word = reconstruct(id_key(st));
return l_true;
}
if (undecided)
return l_undef;
// per-component cofactor branches (target, guard); pin both, they outlive `cof`.
std::vector<std::vector<std::pair<expr*, expr*>>> branches(n);
for (unsigned i = 0; i < n; ++i) {
expr_ref_pair_vector cof(m);
m_rw.brz_derivative_cofactors(st[i], cof);
for (auto const& [g, t] : cof) {
if (re().is_empty(t)) continue;
m_pin.push_back(t);
m_pin.push_back(g);
branches[i].push_back(std::make_pair((expr*) t, (expr*) g));
}
}
// joint transitions = cartesian product of the branches with the guards
// conjoined; prune as soon as the accumulated guard is empty, bail on unknown.
svector<expr*> cur;
cur.resize(n);
key st_key = id_key(st);
bool bail = false;
std::function<void(unsigned, guard_set const&)> rec =
[&](unsigned i, guard_set const& acc) {
if (bail) return;
if (i == n) {
key ck = id_key(cur);
if (visited.find(ck) == visited.end()) {
visited.insert(ck);
if (witness_word) {
expr_ref e(m);
if (acc.eval(&e) == l_true) {
m_pin.push_back(e);
parent[ck] = std::make_pair(st_key, e.get());
}
}
work.push_back(cur);
}
return;
}
for (auto const& pr : branches[i]) {
guard_set nacc = acc;
nacc.conjoin(pr.second);
lbool ne = nacc.eval(nullptr);
if (ne == l_undef) { bail = true; return; } // non-range / unknown guard
if (ne == l_false) continue; // empty joint guard: prune
cur[i] = pr.first;
rec(i + 1, nacc);
if (bail) return;
}
};
guard_set top(m, u(), m_elem_sort, var0);
rec(0, top);
if (bail)
return l_undef;
}
return l_false;
}
bool seq_monadic::parse_term(expr* t, svector<atom>& atoms, expr*& the_var) {
if (u().str.is_concat(t)) {
app* a = to_app(t);
for (unsigned i = 0; i < a->get_num_args(); ++i)
if (!parse_term(a->get_arg(i), atoms, the_var))
return false;
return true;
}
if (u().str.is_empty(t))
return true; // epsilon: contributes nothing
zstring s;
if (u().str.is_string(t, s)) {
for (unsigned i = 0; i < s.length(); ++i)
atoms.push_back(atom{ false, nullptr, u().str.mk_char(s, i) });
return true;
}
if (u().str.is_unit(t)) { // seq.unit of a constant element
expr* elem = to_app(t)->get_arg(0);
if (m.is_value(elem)) {
atoms.push_back(atom{ false, nullptr, elem });
return true;
}
return false; // symbolic (non-constant) unit: unsupported
}
// uninterpreted 0-ary constant of sequence sort => a sequence variable
if (is_app(t) && to_app(t)->get_num_args() == 0 &&
to_app(t)->get_family_id() == null_family_id) {
the_var = t; // mark that at least one variable occurs
atoms.push_back(atom{ true, t, nullptr });
return true;
}
return false;
}
void seq_monadic::decompose(svector<atom> const& atoms, unsigned i, expr* R,
vector<disjunct>& out, bool& ok) {
if (!ok)
return;
if (m_giveup) { ok = false; return; }
m_pin.push_back(R);
if (i == atoms.size()) {
expr_ref nb = m_rw.is_nullable(R);
if (m.is_true(nb))
out.push_back(disjunct()); // empty conjunction = true
else if (!m.is_false(nb))
ok = false; // undecidable nullability => bail
return;
}
atom const& a = atoms[i];
if (!a.is_var) {
expr_ref d = der_elem(R, a.elem);
decompose(atoms, i + 1, d, out, ok);
return;
}
if (i + 1 == atoms.size()) { // last atom: membership component a.var in R
disjunct D;
D.push_back(component{ a.var, R, nullptr });
out.push_back(D);
return;
}
// a variable with a non-empty rest: split over the live states q of R (midpoints)
ptr_vector<expr> Q;
live_states(R, Q, ok);
if (!ok)
return;
const unsigned DISJUNCT_CAP = 1u << 13;
for (expr* q : Q) {
vector<disjunct> sub;
decompose(atoms, i + 1, q, sub, ok);
if (!ok)
return;
for (disjunct const& sd : sub) {
if (out.size() > DISJUNCT_CAP || m_budget == 0) { m_giveup = true; ok = false; return; }
--m_budget;
disjunct D(sd);
D.push_back(component{ a.var, R, q }); // reach component: a.var drives R -> q
out.push_back(D);
}
}
simplify_dnf(out);
}
void seq_monadic::simplify_dnf(vector<disjunct>& dnf) {
std::set<std::vector<std::tuple<unsigned, unsigned, unsigned>>> seen;
vector<disjunct> result;
for (disjunct const& D : dnf) {
bool dead = false;
for (auto const& c : D)
if (re().is_empty(c.state)) { dead = true; break; }
if (dead)
continue;
std::vector<std::tuple<unsigned, unsigned, unsigned>> sig;
sig.reserve(D.size());
for (auto const& c : D)
sig.push_back(std::make_tuple(c.var->get_id(), c.state->get_id(),
c.target ? c.target->get_id() : UINT_MAX));
std::sort(sig.begin(), sig.end());
if (seen.insert(sig).second)
result.push_back(D);
}
dnf.swap(result);
}
lbool seq_monadic::solve(expr* term, expr* R) {
obj_map<expr, expr*> none;
return solve(term, R, none, nullptr);
}
lbool seq_monadic::solve(expr* term, expr* R, obj_map<expr, expr*> const& var_extra) {
return solve(term, R, var_extra, nullptr);
}
bool seq_monadic::build_membership_dnf(expr* term, expr* R, vector<disjunct>& dnf) {
if (!u().is_re(R, m_seq_sort))
return false;
if (!u().is_seq(m_seq_sort, m_elem_sort))
return false;
svector<atom> atoms;
expr* the_var = nullptr;
if (!parse_term(term, atoms, the_var))
return false;
if (!the_var)
return false; // no variable: ground membership, not our case
m_pin.push_back(R);
bool ok = true;
decompose(atoms, 0, R, dnf, ok);
return ok;
}
lbool seq_monadic::decide_dnf(vector<disjunct> const& dnf, obj_map<expr, expr*> const& var_extra,
obj_map<expr, expr*>* model) {
bool any_undef = false;
for (disjunct const& D : dnf) {
// group components by variable, add the extra per-variable constraints
obj_map<expr, unsigned> idx;
vector<svector<component>> groups;
ptr_vector<expr> group_var;
auto bucket = [&](expr* v) -> unsigned {
unsigned gi;
if (idx.find(v, gi)) return gi;
gi = groups.size(); idx.insert(v, gi);
groups.push_back(svector<component>());
group_var.push_back(v);
return gi;
};
for (auto const& c : D)
groups[bucket(c.var)].push_back(c);
for (auto const& kv : var_extra)
groups[bucket(kv.m_key)].push_back(component{ kv.m_key, kv.m_value, nullptr });
bool has_empty = false, has_undef = false;
obj_map<expr, expr*> local; // var -> witness for this disjunct
for (unsigned gi = 0; gi < groups.size(); ++gi) {
expr_ref w(m);
lbool ne = product_nonempty(groups[gi], model ? &w : nullptr);
if (ne == l_false) { has_empty = true; break; } // this variable has no value
if (ne == l_undef) { has_undef = true; continue; }
if (model) { m_pin.push_back(w); local.insert(group_var[gi], w.get()); }
}
if (has_empty) continue;
if (has_undef) { any_undef = true; continue; }
if (model)
for (auto const& kv : local)
model->insert(kv.m_key, kv.m_value);
return l_true; // all variables satisfiable => sat
}
return any_undef ? l_undef : l_false;
}
lbool seq_monadic::solve(expr* term, expr* R, obj_map<expr, expr*> const& var_extra,
obj_map<expr, expr*>* model) {
m_pin.reset();
m_budget = 200000; // global work budget: bail fast on DNF explosion
m_giveup = false;
vector<disjunct> dnf;
if (!build_membership_dnf(term, R, dnf))
return l_undef;
return decide_dnf(dnf, var_extra, model);
}
lbool seq_monadic::solve_and(vector<std::pair<expr*, expr*>> const& mems,
obj_map<expr, expr*> const& var_extra, obj_map<expr, expr*>* model) {
if (mems.empty())
return l_undef;
m_pin.reset();
m_budget = 200000;
m_giveup = false;
// Multiply the per-membership DNFs: combined = { d ++ e : d in combined, e in dnf_i }.
// A variable shared by several memberships thus gets several components in the same
// disjunct, which decide_dnf/product_nonempty intersect -- enforcing one consistent
// value across all memberships (the joint solve the harness could not do per-term).
vector<disjunct> combined;
combined.push_back(disjunct()); // { true }
const unsigned DNF_CAP = 1u << 14;
for (auto const& tr : mems) {
vector<disjunct> dnf_i;
if (!build_membership_dnf(tr.first, tr.second, dnf_i))
return l_undef;
vector<disjunct> next;
for (disjunct const& d : combined) {
for (disjunct const& e : dnf_i) {
if (next.size() > DNF_CAP || m_budget == 0) { m_giveup = true; return l_undef; }
--m_budget;
disjunct D(d);
for (auto const& c : e)
D.push_back(c);
next.push_back(D);
}
}
combined.swap(next);
simplify_dnf(combined);
if (combined.empty())
return l_false; // no viable disjunct left => unsat
}
return decide_dnf(combined, var_extra, model);
}

148
src/ast/rewriter/seq_monadic.h Normal file
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@ -0,0 +1,148 @@
/*++
Copyright (c) 2026 Microsoft Corporation
Module Name:
seq_monadic.h
Abstract:
Whole-language monadic decomposition for regex membership of a term that is a
concatenation of sequence variables and constant elements, e.g. x.a.x in R.
Generic in the element sort: characters are one instance, but the procedure works
for any sequence element sort (the guard algebra falls back from the exact character
range_predicate to a candidate-basis over the element values mentioned by the
derivatives).
Self-contained decision procedure: NO Nielsen splitting (seq_split), NO minterms,
and NO materialization of reach(q) as a regex. It relies only on the symbolic
Brzozowski derivative (brz_derivative_cofactors as a transition regex) and on
automaton product-reachability for emptiness.
Method. For a term x.u in R and the whole-language split, x drives the derivative
automaton of R from R to some live state q, and the rest u must be accepted from q:
x.u in R <=> OR_{q live} ( x reaches q in A_R /\ u in q ).
Decomposing u recursively (a leading constant is consumed by a derivative, a leading
variable splits again, the last variable is a plain membership) yields a DNF whose
disjuncts are conjunctions of per-variable *components*:
- reach component <var, state0, q> : the variable's value drives the
derivative automaton from state0 to q
- membership component<var, state0, null> : the variable's value is in L(state0)
reach(q) is therefore NEVER built as a regex (which state-elimination would blow up
super-polynomially for lattice-shaped automata). Instead the constraints on a
variable are decided directly by a lazy product-reachability search over tuples of
component states: a product state accepts iff every reach component is at its target
and every membership component is nullable; transitions are the product of the
components' cofactor branches with pairwise-conjoined range guards (minterm-free).
This stays in the product-of-state-counts regime, never the path-enumeration (k!)
regime of regex state-elimination.
Supports single / multiple / repeated variables, and per-variable extra constraints
(base membership + length-regex) via `var_extra`.
Author:
Nikolaj Bjorner / Margus Veanes 2026
--*/
#pragma once
#include "ast/rewriter/seq_rewriter.h"
#include "ast/rewriter/seq_range_predicate.h"
#include "ast/rewriter/th_rewriter.h"
#include "util/lbool.h"
#include "util/obj_hashtable.h"
#include <utility>
class seq_monadic {
ast_manager& m;
seq_rewriter& m_rw;
th_rewriter m_thrw; // normalizes constant-element derivatives (folds
// ground guards so dead states become re.empty)
sort* m_seq_sort = nullptr; // sequence sort of the regex under analysis
sort* m_elem_sort = nullptr; // element sort of that sequence sort
expr_ref_vector m_pin; // pins derivative states / witnesses referenced later
unsigned m_budget = 0; // global work budget (decompose disjuncts + product pops)
bool m_giveup = false; // set when the budget is exhausted
seq_util& u() const { return m_rw.u(); }
seq_util::rex& re() const { return m_rw.u().re; }
// A term atom: a sequence variable or a constant element (a value of the element sort).
struct atom { bool is_var; expr* var; expr* elem; };
// A component of one variable's constraint. As the variable's value w is read,
// the current state is derived from `state`; the component accepts when
// target ? (current == target) -- reach component (w drives A from state to target)
// : nullable(current) -- membership component (w in L(state))
struct component { expr* var; expr* state; expr* target; };
typedef svector<component> disjunct; // a conjunction of components (a DNF disjunct)
// Brzozowski derivative of regex `r` by the concrete element `elem`.
expr_ref der_elem(expr* r, expr* elem);
// Live reachable derivative states of R (BFS over cofactor targets + liveness
// least-fixpoint). These are the split states q. Sets `ok` false on a cap overrun.
void live_states(expr* R, ptr_vector<expr>& out, bool& ok);
// Product-reachability emptiness of a conjunction of components (all on one
// variable). l_false = empty (unsat), l_true = non-empty (sat), l_undef = gave up
// (cap overrun, non-range guard, or undecidable nullability).
// On l_true, if `witness_word` is non-null it is set to a concrete sequence term
// (over the element sort) whose value drives every component to acceptance
// simultaneously -- i.e. a witness value for the variable the components constrain.
lbool product_nonempty(svector<component> const& comps, expr_ref* witness_word = nullptr);
// Flatten a str.++ term into atoms; false on an unsupported shape (non-constant unit).
bool parse_term(expr* term, svector<atom>& atoms, expr*& the_var);
// Monadic decomposition: append to `out` the DNF disjuncts for atoms[i..] in R,
// threading the current derivative state R. `ok` false on give-up.
void decompose(svector<atom> const& atoms, unsigned i, expr* R,
vector<disjunct>& out, bool& ok);
// Drop disjuncts with a syntactically-empty component and dedup identical disjuncts.
void simplify_dnf(vector<disjunct>& dnf);
// Build the DNF over primitive per-variable components for one membership term in R.
// Sets m_seq_sort/m_elem_sort; false on an unsupported shape or give-up.
bool build_membership_dnf(expr* term, expr* R, vector<disjunct>& dnf);
// Decide a DNF (over primitive components): sat iff some disjunct has every variable
// group non-empty. On l_true, fills `model` (var -> witness) if non-null.
lbool decide_dnf(vector<disjunct> const& dnf, obj_map<expr, expr*> const& var_extra,
obj_map<expr, expr*>* model);
public:
seq_monadic(seq_rewriter& rw) : m(rw.m()), m_rw(rw), m_thrw(rw.m()), m_pin(rw.m()) {}
// Decide (str.in_re term R) for a term that is a concatenation of string variables
// (possibly repeated / several distinct) and constant characters.
// l_true = sat, l_false = unsat, l_undef = unsupported shape / gave up.
lbool solve(expr* term, expr* R);
// As above, with extra per-variable constraints (e.g. a base membership intersected
// with a length-regex): `var_extra` maps a variable to a regex it must also satisfy.
lbool solve(expr* term, expr* R, obj_map<expr, expr*> const& var_extra);
// As above; on l_true, if `model` is non-null it is populated with var -> witness,
// where each witness is a concrete sequence term (over the element sort) giving one
// satisfying assignment. Witness terms are pinned by the solver and remain valid
// until the next call to solve().
lbool solve(expr* term, expr* R, obj_map<expr, expr*> const& var_extra,
obj_map<expr, expr*>* model);
// Decide a CONJUNCTION of memberships AND_i (term_i in R_i) jointly: a variable
// shared across memberships is constrained consistently (the DNFs are multiplied and
// each variable's constraints intersected). This is the natural extension of single-
// membership solving to a Boolean combination of memberships (a disjunction is the
// union of DNFs; a negated membership ~(t in R) is just t in complement(R)).
// var_extra / model as above. l_true = sat, l_false = unsat, l_undef = gave up.
lbool solve_and(vector<std::pair<expr*, expr*>> const& mems,
obj_map<expr, expr*> const& var_extra, obj_map<expr, expr*>* model = nullptr);
};

70
src/ast/rewriter/seq_range_collapse.cpp
View file

@ -21,66 +21,66 @@ Authors:
namespace seq {
// Cofactor path condition `pred` (a Boolean over x = (:var 0)) -> the canonical
// range_predicate (union of ranges) of the characters satisfying it. Returns
// false on a construct outside {true,false,and,or,not,=,char.<=} over x.
static bool pred_to_rp(ast_manager &m, seq_util &sq, expr *x, expr *pred,
seq::range_predicate &out) {
unsigned maxc = sq.max_char();
expr *a = nullptr, *b = nullptr;
// Cofactor guard `guard` (a Boolean over the character variable v0 = (:var 0)) ->
// the canonical range_predicate (union of ranges) of the characters satisfying it.
// Returns false on a construct outside {true,false,and,or,not,=,char.<=} over v0.
bool guard_to_range_predicate(seq_util& u, expr* v0, expr* guard, range_predicate& out) {
ast_manager& m = u.get_manager();
unsigned maxc = u.max_char();
expr* a = nullptr, * b = nullptr;
unsigned c = 0;
if (m.is_true(pred)) {
out = seq::range_predicate::top(maxc);
if (m.is_true(guard)) {
out = range_predicate::top(maxc);
return true;
}
if (m.is_false(pred)) {
out = seq::range_predicate::empty(maxc);
if (m.is_false(guard)) {
out = range_predicate::empty(maxc);
return true;
}
if (m.is_eq(pred, a, b)) {
if (a == x && sq.is_const_char(b, c)) {
out = seq::range_predicate::singleton(c, maxc);
if (m.is_eq(guard, a, b)) {
if (a == v0 && u.is_const_char(b, c)) {
out = range_predicate::singleton(c, maxc);
return true;
}
if (b == x && sq.is_const_char(a, c)) {
out = seq::range_predicate::singleton(c, maxc);
if (b == v0 && u.is_const_char(a, c)) {
out = range_predicate::singleton(c, maxc);
return true;
}
return false;
}
if (sq.is_char_le(pred, a, b)) {
if (b == x && sq.is_const_char(a, c)) {
out = seq::range_predicate::range(c, maxc, maxc);
if (u.is_char_le(guard, a, b)) {
if (b == v0 && u.is_const_char(a, c)) {
out = range_predicate::range(c, maxc, maxc);
return true;
}
if (a == x && sq.is_const_char(b, c)) {
out = seq::range_predicate::range(0, c, maxc);
if (a == v0 && u.is_const_char(b, c)) {
out = range_predicate::range(0, c, maxc);
return true;
}
return false;
}
if (m.is_not(pred, a)) {
seq::range_predicate s(maxc);
if (!pred_to_rp(m, sq, x, a, s))
if (m.is_not(guard, a)) {
range_predicate s(maxc);
if (!guard_to_range_predicate(u, v0, a, s))
return false;
out = ~s;
return true;
}
if (m.is_and(pred)) {
out = seq::range_predicate::top(maxc);
for (expr *arg : *to_app(pred)) {
seq::range_predicate s(maxc);
if (!pred_to_rp(m, sq, x, arg, s))
if (m.is_and(guard)) {
out = range_predicate::top(maxc);
for (expr *arg : *to_app(guard)) {
range_predicate s(maxc);
if (!guard_to_range_predicate(u, v0, arg, s))
return false;
out = out & s;
}
return true;
}
if (m.is_or(pred)) {
out = seq::range_predicate::empty(maxc);
for (expr *arg : *to_app(pred)) {
seq::range_predicate s(maxc);
if (!pred_to_rp(m, sq, x, arg, s))
if (m.is_or(guard)) {
out = range_predicate::empty(maxc);
for (expr *arg : *to_app(guard)) {
range_predicate s(maxc);
if (!guard_to_range_predicate(u, v0, arg, s))
return false;
out = out | s;
}
@ -183,7 +183,7 @@ namespace seq {
auto body = q->get_expr();
sort *char_sort = q->get_decl_sort(0);
expr_ref var(m.mk_var(0, char_sort), m);
if (u.get_char_plugin().get_family_id() == char_sort->get_family_id() && pred_to_rp(m, u, var, body, out))
if (u.get_char_plugin().get_family_id() == char_sort->get_family_id() && guard_to_range_predicate(u, var, body, out))
return true;
}

8
src/ast/rewriter/seq_range_collapse.h
View file

@ -31,6 +31,14 @@ Authors:
namespace seq {
/**
* Convert a Boolean guard over the single character variable v0 = (:var 0) -- a
* derivative cofactor path condition -- into the range_predicate of the characters
* satisfying it. Recognizes {true, false, =, char.<=, and, or, not} over v0 and
* concrete characters; returns false (out untouched) on anything else.
*/
bool guard_to_range_predicate(seq_util& u, expr* v0, expr* guard, range_predicate& out);
/**
* If r is a boolean combination of character-class regex primitives
* over the unsigned character domain [0, max_char], compute the

3
src/test/CMakeLists.txt
View file

@ -24,7 +24,6 @@ add_executable(test-z3
api_datalog.cpp
parametric_datatype.cpp
arith_rewriter.cpp
seq_rewriter.cpp
arith_simplifier_plugin.cpp
ast.cpp
bdd.cpp
@ -132,6 +131,8 @@ add_executable(test-z3
sat_user_scope.cpp
scoped_timer.cpp
scoped_vector.cpp
seq_rewriter.cpp
seq_monadic.cpp
simple_parser.cpp
scanner_io.cpp
simplex.cpp

1
src/test/main.cpp
View file

@ -116,6 +116,7 @@
X(range_predicate) \
X(regex_range_collapse) \
X(seq_rewriter) \
X(seq_monadic) \
X(check_assumptions) \
X(smt_context) \
X(theory_dl) \

314
src/test/seq_monadic.cpp Normal file
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@ -0,0 +1,314 @@
/*++
Copyright (c) 2026 Microsoft Corporation
Module Name:
seq_monadic.cpp
Abstract:
Unit tests for the whole-language monadic-decomposition membership solver in
ast/rewriter/seq_monadic.cpp. Mirrors the validated Python prototype
(files/solve_proto.py): single-variable repeated-membership shapes x.a.x in R.
Author:
Nikolaj Bjorner / Margus Veanes 2026
--*/
#include "ast/ast.h"
#include "ast/reg_decl_plugins.h"
#include "ast/seq_decl_plugin.h"
#include "ast/arith_decl_plugin.h"
#include "ast/rewriter/seq_rewriter.h"
#include "ast/rewriter/seq_monadic.h"
#include "ast/rewriter/expr_safe_replace.h"
#include <iostream>
namespace {
struct plugin_registrar {
plugin_registrar(ast_manager& m) { reg_decl_plugins(m); }
};
class seq_monadic_test {
ast_manager m;
plugin_registrar m_reg;
seq_rewriter m_rw;
seq_monadic m_mon;
seq_util u;
sort_ref m_str; // String sort
sort_ref m_re; // RegEx sort over m_str
unsigned m_fail = 0;
seq_util::rex& re() { return u.re; }
// regex builders
expr_ref word(char const* s) { return expr_ref(re().mk_to_re(u.str.mk_string(zstring(s))), m); }
expr_ref cat(expr* a, expr* b) { return expr_ref(re().mk_concat(a, b), m); }
expr_ref alt(expr* a, expr* b) { return expr_ref(re().mk_union(a, b), m); }
expr_ref star(expr* a) { return expr_ref(re().mk_star(a), m); }
expr_ref inter(expr* a, expr* b) { return expr_ref(re().mk_inter(a, b), m); }
expr_ref comp(expr* a) { return expr_ref(re().mk_complement(a), m); }
expr_ref dotstar() { return expr_ref(re().mk_full_seq(m_re), m); }
expr_ref rng(char lo, char hi) {
char sl[2] = { lo, 0 }, sh[2] = { hi, 0 };
return expr_ref(re().mk_range(u.str.mk_string(zstring(sl)), u.str.mk_string(zstring(sh))), m);
}
expr_ref loop(expr* r, unsigned lo, unsigned hi) { return expr_ref(re().mk_loop(r, lo, hi), m); }
// string-term builders
expr_ref var(char const* nm) { return expr_ref(m.mk_const(nm, m_str), m); }
expr_ref sword(char const* s) { return expr_ref(u.str.mk_string(zstring(s)), m); }
expr_ref sconcat(expr* a, expr* b) { return expr_ref(u.str.mk_concat(a, b), m); }
// term x . w . x (w a constant word)
expr_ref xwx(expr* x, char const* w) { return sconcat(x, sconcat(sword(w), x)); }
// term x . a . y (two distinct variables)
expr_ref xay(expr* x, expr* y) { return sconcat(x, sconcat(sword("a"), y)); }
// term x . y . x
expr_ref xyx(expr* x, expr* y) { return sconcat(x, sconcat(y, x)); }
static char const* s(lbool l) { return l == l_true ? "sat" : l == l_false ? "unsat" : "undef"; }
void check(char const* name, expr* term, expr* R, lbool expected) {
lbool got = m_mon.solve(term, R);
bool ok = (got == expected);
if (!ok) ++m_fail;
std::cout << (ok ? " OK " : " FAIL ") << name
<< " got=" << s(got) << " expected=" << s(expected) << "\n";
}
void check_extra(char const* name, expr* term, expr* R,
obj_map<expr, expr*> const& ve, lbool expected) {
lbool got = m_mon.solve(term, R, ve);
bool ok = (got == expected);
if (!ok) ++m_fail;
std::cout << (ok ? " OK " : " FAIL ") << name
<< " got=" << s(got) << " expected=" << s(expected) << "\n";
}
// flatten a ground sequence term into its element values.
void flatten_seq(expr* seqv, ptr_vector<expr>& elems) {
zstring zs;
if (u.str.is_concat(seqv)) {
for (expr* arg : *to_app(seqv)) flatten_seq(arg, elems);
return;
}
if (u.str.is_empty(seqv))
return;
if (u.str.is_string(seqv, zs)) {
for (unsigned i = 0; i < zs.length(); ++i) elems.push_back(u.str.mk_char(zs, i));
return;
}
if (u.str.is_unit(seqv))
elems.push_back(to_app(seqv)->get_arg(0));
}
// decide membership of a concrete word (list of element values) in R by folding
// derivatives and testing nullability of the residual.
bool word_in_re(ptr_vector<expr> const& elems, expr* R) {
expr_ref cur(R, m);
for (expr* e : elems) cur = m_rw.mk_derivative(e, cur);
return m.is_true(m_rw.is_nullable(cur));
}
// solve for a model, then check the returned witness assignment actually makes
// term a member of R (substitute var -> witness and re-decide by derivatives).
void check_witness(char const* name, expr* term, expr* R,
obj_map<expr, expr*> const& ve) {
obj_map<expr, expr*> model;
lbool got = m_mon.solve(term, R, ve, &model);
bool ok = (got == l_true) && !model.empty();
if (ok) {
expr_safe_replace rep(m);
for (auto const& kv : model) rep.insert(kv.m_key, kv.m_value);
expr_ref g(m);
rep(term, g);
ptr_vector<expr> elems;
flatten_seq(g, elems);
ok = word_in_re(elems, R);
}
if (!ok) ++m_fail;
std::cout << (ok ? " OK " : " FAIL ") << name
<< " solve=" << s(got) << " witness-verified=" << (ok ? "yes" : "no") << "\n";
}
// decide a conjunction of memberships jointly (shared variables constrained together).
void check_and(char const* name, vector<std::pair<expr*, expr*>> const& mems, lbool expected) {
obj_map<expr, expr*> nove;
lbool got = m_mon.solve_and(mems, nove, nullptr);
bool ok = (got == expected);
if (!ok) ++m_fail;
std::cout << (ok ? " OK " : " FAIL ") << name
<< " got=" << s(got) << " expected=" << s(expected) << "\n";
}
public:
seq_monadic_test() : m_reg(m), m_rw(m), m_mon(m_rw), u(m), m_str(m), m_re(m) {
m_str = u.str.mk_string_sort();
m_re = re().mk_re(m_str);
}
void run() {
expr_ref x = var("x");
expr_ref a = word("a");
expr_ref b = word("b");
expr_ref ab = cat(a, b);
expr_ref sig = dotstar(); // Sigma*
expr_ref saas = cat(sig, cat(cat(a, a), sig)); // Sigma* a a Sigma*
expr_ref sbbs = cat(sig, cat(cat(b, b), sig)); // Sigma* b b Sigma*
std::cout << "=== seq_monadic: single-variable membership (x.a.x in R) ===\n";
// sanity
check("(a|b)* x.a.x", xwx(x, "a"), star(alt(a, b)), l_true);
check("b* x.a.x", xwx(x, "a"), star(b), l_false);
check("Sig*aaSig* x.a.x", xwx(x, "a"), saas, l_true);
check("x in (a|b)* ", x, star(alt(a, b)), l_true);
check("x in b* (x=aa) ", xwx(x, "a"), star(b), l_false);
// ALT = (a|b)* & ~(Sig*aaSig*) & ~(Sig*bbSig*) (strictly alternating)
expr_ref altre = inter(star(alt(a, b)), inter(comp(saas), comp(sbbs)));
check("ALT x.a.x", xwx(x, "a"), altre, l_true);
// R*.S complement family
check("~(a*.b) x.a.x", xwx(x, "a"), comp(cat(star(a), b)), l_true);
// L3-02 ~((ab)*.~((ab)*)) -> unsat (odd length)
check("L3-02 x.a.x", xwx(x, "a"),
comp(cat(star(ab), comp(star(ab)))), l_false);
// L3-03 ~(a*.~(b*.~((ab)*))) -> sat
check("L3-03 x.a.x", xwx(x, "a"),
comp(cat(star(a), comp(cat(star(b), comp(star(ab)))))), l_true);
std::cout << "=== seq_monadic: multi-variable ===\n";
expr_ref y = var("y");
check("(a|b)* x.a.y", xay(x, y), star(alt(a, b)), l_true);
check("b* x.a.y", xay(x, y), star(b), l_false);
check("L3-02 x.a.y", xay(x, y), comp(cat(star(ab), comp(star(ab)))), l_true);
check("L3-03 x.a.y", xay(x, y),
comp(cat(star(a), comp(cat(star(b), comp(star(ab)))))), l_true);
check("empty ~Sig* x.y.x", xyx(x, y), comp(dotstar()), l_false);
check("Sig* x.y.x", xyx(x, y), dotstar(), l_true);
check("(a|b)* x.y.x", xyx(x, y), star(alt(a, b)), l_true);
std::cout << "=== seq_monadic: per-variable constraints ===\n";
expr_ref digitp = cat(rng('0', '9'), star(rng('0', '9'))); // [0-9]+
obj_map<expr, expr*> ve;
ve.insert(y, digitp);
// y must be in the (a|b)* tail AND in [0-9]+ -> empty -> unsat
check_extra("(a|b)* & y in[0-9]+ x.a.y", xay(x, y), star(alt(a, b)), ve, l_false);
// y any digits, x/'a' anything -> sat
check_extra("Sig* & y in[0-9]+ x.a.y", xay(x, y), dotstar(), ve, l_true);
// Bounded loop (re.loop) with repeated variable -- exercises live_states on a
// counted automaton (t04-exact benchmark family). Regression for a
// reference-invalidation bug in live_states (succ[i].push_back(intern(t))).
std::cout << "=== seq_monadic: bounded loop (t04-exact family) ===\n";
expr_ref clsr = rng('0', '9'); // [0-9]
expr_ref digitS = star(clsr); // [0-9]*
expr_ref loop22 = loop(clsr, 2, 2); // [0-9]{2}
check("[0-9]{2} x ", x, loop22, l_true); // x = "00"
check("[0-9]{2} x.a.x", xwx(x, "a"), loop22, l_false); // 'a' not a digit
obj_map<expr, expr*> ve2; ve2.insert(x, digitp); ve2.insert(y, digitS);
// x.y.x in [0-9]{2}, x in [0-9]+, y in [0-9]* -> sat (x="0", y="")
check_extra("[0-9]{2} & x[0-9]+ y[0-9]* x.y.x", xyx(x, y), loop22, ve2, l_true);
obj_map<expr, expr*> ve3; ve3.insert(x, digitp);
check_extra("[0-9]{2} & x[0-9]+ x.y.x", xyx(x, y), loop22, ve3, l_true);
obj_map<expr, expr*> ve4; ve4.insert(x, digitp);
// x.y.x in [0-9]{3}, x in [0-9]+ -> sat (x=1 digit, y=1 digit)
check_extra("[0-9]{3} & x[0-9]+ x.y.x", xyx(x, y), loop(clsr, 3, 3), ve4, l_true);
// ---- witness extraction: a produced witness must be a concrete SEQUENCE of
// ---- elements that actually satisfies the membership (not a predicate).
std::cout << "=== seq_monadic: witness extraction (char) ===\n";
obj_map<expr, expr*> nove;
check_witness("(a|b)* x.a.x", xwx(x, "a"), star(alt(a, b)), nove);
check_witness("Sig*aaSig* x.a.x", xwx(x, "a"), saas, nove); // forces nonempty x
check_witness("~(a*.b) x.a.x", xwx(x, "a"), comp(cat(star(a), b)), nove);
check_witness("L3-03 x.a.x", xwx(x, "a"),
comp(cat(star(a), comp(cat(star(b), comp(star(ab)))))), nove);
check_witness("(a|b)* x.a.y", xay(x, y), star(alt(a, b)), nove);
check_witness("Sig* x.y.x", xyx(x, y), dotstar(), nove);
check_witness("Sig* & y[0-9]+ x.a.y", xay(x, y), dotstar(), ve); // ve: y in [0-9]+
check_witness("[0-9]{2}&x[0-9]+ y[0-9]* x.y.x", xyx(x, y), loop22, ve2);
// ---- generic element sort: sequences of Int exercise the non-character guard
// ---- algebra (candidate-basis emptiness + witness), not seq::range_predicate.
std::cout << "=== seq_monadic: generic element sort (Seq Int) ===\n";
arith_util ar(m);
sort_ref intS(ar.mk_int(), m);
sort_ref seqI(u.str.mk_seq(intS), m);
sort_ref reI(u.re.mk_re(seqI), m);
expr_ref i1(ar.mk_numeral(rational(1), true), m);
expr_ref i2(ar.mk_numeral(rational(2), true), m);
expr_ref one_seq(u.str.mk_unit(i1), m); // [1] : (Seq Int)
expr_ref re1(re().mk_to_re(u.str.mk_unit(i1)), m); // matches [1]
expr_ref re2(re().mk_to_re(u.str.mk_unit(i2)), m); // matches [2]
expr_ref re12s(star(alt(re1, re2)), m); // ([1]|[2])*
expr_ref re2s(star(re2), m); // [2]*
expr_ref xi(m.mk_const("xi", seqI), m);
expr_ref yi(m.mk_const("yi", seqI), m);
expr_ref xi1xi(sconcat(xi, sconcat(one_seq, xi)), m); // xi.[1].xi
expr_ref xiyi(sconcat(xi, sconcat(one_seq, yi)), m); // xi.[1].yi
obj_map<expr, expr*> nove2;
check("([1]|[2])* xi.[1].xi", xi1xi, re12s, l_true);
check("[2]* xi.[1].xi", xi1xi, re2s, l_false); // the middle [1] is not in [2]*
check("([1]|[2])* xi ", xi, re12s, l_true);
check("([1]|[2])* xi.[1].yi", xiyi, re12s, l_true);
check_witness("([1]|[2])* xi.[1].xi", xi1xi, re12s, nove2);
check_witness("([1]|[2])* xi ", xi, re12s, nove2);
check_witness("([1]|[2])* xi.[1].yi", xiyi, re12s, nove2);
// per-variable extra constraint over (Seq Int): yi must also be in [2]*
obj_map<expr, expr*> veI; veI.insert(yi, re2s.get());
check_extra("([1]|[2])* & yi[2]* xi.[1].yi", xiyi, re12s, veI, l_true);
check_witness("([1]|[2])* & yi[2]* xi.[1].yi", xiyi, re12s, veI);
// ---- conjunction of memberships (solve_and): a variable shared across memberships
// ---- is constrained jointly. These are cases that are individually SAT but
// ---- jointly UNSAT -- exactly what independent per-membership solving gets wrong.
std::cout << "=== seq_monadic: conjunction of memberships (solve_and) ===\n";
expr_ref aaS(star(cat(a, a)), m); // (aa)* : even number of a's
expr_ref a_aaS(cat(a, star(cat(a, a))), m); // a(aa)* : odd number of a's
expr_ref abS(star(ab), m); // (ab)*
expr_ref sig2(dotstar(), m); // Sigma*
// x in (aa)* /\ x in a(aa)* : even-and-odd length of a's -> unsat (each alone sat)
vector<std::pair<expr*, expr*>> mUnsat1;
mUnsat1.push_back(std::make_pair((expr*)x.get(), (expr*)aaS.get()));
mUnsat1.push_back(std::make_pair((expr*)x.get(), (expr*)a_aaS.get()));
check_and("x in (aa)* & x in a(aa)*", mUnsat1, l_false);
// compound terms sharing x: x.a in (aa)* (x odd) /\ x.aa in (aa)* (x even) -> unsat
expr_ref tXa(sconcat(x, sword("a")), m);
expr_ref tXaa(sconcat(x, sword("aa")), m);
vector<std::pair<expr*, expr*>> mUnsat2;
mUnsat2.push_back(std::make_pair((expr*)tXa.get(), (expr*)aaS.get()));
mUnsat2.push_back(std::make_pair((expr*)tXaa.get(), (expr*)aaS.get()));
check_and("x.a in (aa)* & x.aa in (aa)*", mUnsat2, l_false);
// consistent conjunction: x in (ab)* /\ x in Sigma* -> sat (x=eps or ab)
vector<std::pair<expr*, expr*>> mSat;
mSat.push_back(std::make_pair((expr*)x.get(), (expr*)abS.get()));
mSat.push_back(std::make_pair((expr*)x.get(), (expr*)sig2.get()));
check_and("x in (ab)* & x in Sigma*", mSat, l_true);
// two variables, two memberships: x.a.y in (a|b)* /\ y.b.x in (a|b)* -> sat
expr_ref tXaY(xay(x, y), m);
expr_ref tYbX(sconcat(y, sconcat(sword("b"), x)), m);
expr_ref abStar(star(alt(a, b)), m);
vector<std::pair<expr*, expr*>> mSat2;
mSat2.push_back(std::make_pair((expr*)tXaY.get(), (expr*)abStar.get()));
mSat2.push_back(std::make_pair((expr*)tYbX.get(), (expr*)abStar.get()));
check_and("x.a.y & y.b.x in (a|b)*", mSat2, l_true);
std::cout << "=== seq_monadic: " << (m_fail == 0 ? "ALL PASS" : "FAILURES") << " ("
<< m_fail << " fail) ===\n";
ENSURE(m_fail == 0);
}
};
}
void tst_seq_monadic() {
seq_monadic_test t;
t.run();
}