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/*++
Copyright (c) 2025 Microsoft Corporation
Module Name:
theory_finite_set_size.cpp
Abstract:
Theory solver for finite sets.
Implements axiom schemas for finite set operations.
--*/
#include "smt/theory_finite_set.h"
#include "smt/theory_finite_set_size.h"
#include "smt/smt_context.h"
#include "smt/smt_model_generator.h"
#include "smt/smt_arith_value.h"
#include "ast/ast_pp.h"
namespace smt {
theory_finite_set_size::theory_finite_set_size(theory_finite_set& th):
m(th.m), ctx(th.ctx), th(th), u(m), bs(m), m_assumption(m), m_slacks(m), m_pinned(m) {}
void theory_finite_set_size::add_set_size(func_decl* f) {
if (!m_set_size_decls.contains(f)) {
m_set_size_decls.push_back(f);
ctx.push_trail(push_back_trail(m_set_size_decls));
}
}
void theory_finite_set_size::initialize_solver() {
struct clear_solver : public trail {
theory_finite_set_size &s;
public:
clear_solver(theory_finite_set_size &s) : s(s) {}
void undo() override {
s.m_solver = nullptr;
s.n2b.reset();
s.m_assumptions.reset();
s.bs.reset();
s.m_slacks.reset();
s.m_slack_members.reset();
s.m_pinned.reset();
s.m_unique_values.reset();
}
};
ctx.push_trail(clear_solver(*this));
m_solver = alloc(context, m, ctx.get_fparams(), ctx.get_params());
// collect all expressons that use cardinality constraints
// collect cone of influence of sets terminals used in cardinality constraints
// for every visited uninterpreted set variable add a Boolean variable
// for every visited set expression add definitions as constraints to m_cardinality solver
// introduce fresh variables for set membership constraints
// assume that distinct enodes in set memberships produce different sets
// assert added disequalities
// assert equalities based on union-find equalities
// assert set membership constraints by in
enode_vector ns;
collect_subexpressions(ns);
//
// we now got all subexpressions from equalities, disequalities, set.in
//
// associate a Boolean variable with every set enode
for (auto n : ns) {
std::ostringstream strm;
strm << "|" << enode_pp(n, ctx) << "|";
symbol name = symbol(strm.str());
expr_ref b(m.mk_const(name, m.mk_bool_sort()), m);
bs.push_back(b);
n2b.insert(n, b);
TRACE(finite_set, tout << "assoc " << name << " to " << enode_pp(n, ctx) << " " << enode_pp(n->get_root(), ctx) << "\n";);
}
add_def_axioms(ns);
// b_{s u t} <-> b_{s} or b_{t},
// b_{s n t} <-> b_{s} and b_{t},
// b_{s\t} <-> b_{s} and not b_{t}
add_singleton_axioms(ns); // (set.in x s) -> b_{x} => b_s - for occurrences of (set.in x s)
add_diseq_axioms(ns); // s = t or |s\t| > 0 or |t\s| > 0 - for disqualities
add_eq_axioms(ns); // s = t -> b_{s} <=> b_{t} - for equalities
TRACE(finite_set, display(tout));
}
/**
* For every (supported) set expression ensure associated Boolean expressions follow semantics
*/
void theory_finite_set_size::add_def_axioms(enode_vector const& ns) {
for (auto n : ns) {
expr *e = n->get_expr();
if (u.is_union(e)) {
auto a = n2b[n->get_arg(0)];
auto b = n2b[n->get_arg(1)];
m_solver->assert_expr(m.mk_iff(n2b[n], m.mk_or(a, b)));
}
else if (u.is_intersect(e)) {
auto a = n2b[n->get_arg(0)];
auto b = n2b[n->get_arg(1)];
m_solver->assert_expr(m.mk_iff(n2b[n], m.mk_and(a, b)));
}
else if (u.is_difference(e)) {
auto a = n2b[n->get_arg(0)];
auto not_b = m.mk_not(n2b[n->get_arg(1)]);
m_solver->assert_expr(m.mk_iff(n2b[n], m.mk_and(a, not_b)));
}
}
}
enode* theory_finite_set_size::mk_singleton(enode* n) {
expr_ref s(u.mk_singleton(n->get_expr()), m);
ctx.ensure_internalized(s);
ctx.mark_as_relevant(s.get());
return ctx.get_enode(s);
}
enode* theory_finite_set_size::mk_diff(enode* a, enode* b) {
expr_ref d(u.mk_difference(a->get_expr(), b->get_expr()), m);
ctx.ensure_internalized(d);
ctx.mark_as_relevant(d.get());
return ctx.get_enode(d);
}
/**
* For every set membership (set.in x s) track propositional
* (set.in x S) => b_{x} => b_S
* ~(set.in x S) => b_{x} => not b_S
*
* Constrain singletons with cardinality constraints:
* |{x}| = 1
*/
void theory_finite_set_size::add_singleton_axioms(enode_vector const &ns) {
for (auto n : ns) {
for (auto p : enode::parents(n)) {
if (!u.is_in(p->get_expr()))
continue;
if (!ctx.is_relevant(p))
continue;
auto v = ctx.get_assignment(p);
if (v == l_undef)
continue;
auto e = p->get_arg(0)->get_root();
TRACE(finite_set, tout << "singleton axiom for " << enode_pp(e, ctx) << " in " << enode_pp(p, ctx)
<< " is " << v << "\n";);
auto s = mk_singleton(e);
SASSERT(n2b.contains(p->get_arg(1)));
SASSERT(n2b.contains(s));
auto X = n2b[s];
auto S = n2b[p->get_arg(1)];
if (v == l_false)
S = m.mk_not(S);
auto is_in = m.mk_implies(X, S);
in lit(p, v == l_true);
std::ostringstream strm;
strm << "|" << (v == l_false ? "~":"") << enode_pp(p, ctx) << "|";
symbol name = symbol(strm.str());
expr* t = m.mk_const(name, m.mk_bool_sort());
bs.push_back(t);
m_assumptions.insert(t, lit);
m_solver->assert_expr(m.mk_implies(t, is_in));
// add size axiom |s| = 1
arith_util a(m);
auto l = th.mk_literal(m.mk_eq(u.mk_size(s->get_expr()), a.mk_int(1)));
ctx.mk_th_axiom(th.get_id(), l);
}
}
}
/**
* For every asserted equality ensure equivalence
*/
void theory_finite_set_size::add_eq_axioms(enode_vector const &ns) {
for (auto [a, b] : th.m_eqs) {
auto x = th.get_enode(a);
auto y = th.get_enode(b);
if (n2b.contains(x) && n2b.contains(y)) {
eq e = {a, b};
std::ostringstream strm;
strm << "|" << enode_pp(x, ctx) << " == " << enode_pp(y, ctx) << "|";
symbol name = symbol(strm.str());
auto t = m.mk_const(name, m.mk_bool_sort());
bs.push_back(t);
m_assumptions.insert(t, e);
m_solver->assert_expr(m.mk_implies(t, m.mk_iff(n2b[x], n2b[y])));
}
}
}
/*
* For every disequality include the assertions x = y or |x\y| >= 1 or |y\z| >= 1
* The expressions x\y and y\x are created when ns is created.
*/
void theory_finite_set_size::add_diseq_axioms(enode_vector const &ns) {
for (auto [a, b] : th.m_diseqs) {
auto x = th.get_enode(a);
auto y = th.get_enode(b);
if (n2b.contains(x) && n2b.contains(y)) {
arith_util a(m);
auto d1 = mk_diff(x, y);
auto d2 = mk_diff(y, x);
expr_ref sz1(u.mk_size(d1->get_expr()), m);
expr_ref sz2(u.mk_size(d2->get_expr()), m);
literal l_eq = th.mk_literal(m.mk_eq(x->get_expr(), y->get_expr()));
literal l1 = th.mk_literal(a.mk_ge(sz1, a.mk_int(1)));
literal l2 = th.mk_literal(a.mk_ge(sz2, a.mk_int(1)));
ctx.mk_th_axiom(th.get_id(), l_eq, l1, l2);
}
}
}
/**
* Walk the cone of influence of expresions that depend on ns
*/
void theory_finite_set_size::collect_subexpressions(enode_vector &ns) {
// initialize disequality watch list
u_map<unsigned_vector> v2diseqs, v2eqs;
for (auto [a, b] : th.m_diseqs) {
v2diseqs.insert_if_not_there(a, unsigned_vector()).push_back(b);
v2diseqs.insert_if_not_there(b, unsigned_vector()).push_back(a);
}
for (auto [a, b] : th.m_eqs) {
v2eqs.insert_if_not_there(a, unsigned_vector()).push_back(b);
v2eqs.insert_if_not_there(b, unsigned_vector()).push_back(a);
}
auto add_expression = [&](enode *e) {
if (!ctx.is_relevant(e))
return;
if (e->is_marked())
return;
e->set_mark();
ns.push_back(e);
};
auto is_setop = [&](enode *n) {
auto e = n->get_expr();
return u.is_union(e) || u.is_intersect(e) || u.is_difference(e);
};
for (auto f : m_set_size_decls) {
for (auto n : ctx.enodes_of(f)) {
SASSERT(u.is_size(n->get_expr()));
add_expression(n->get_arg(0));
}
}
for (unsigned i = 0; i < ns.size(); ++i) {
auto n = ns[i];
// add children under set operations
if (is_setop(n))
for (auto arg : enode::args(n))
add_expression(arg);
// add parents that are operations and use n
for (auto p : enode::parents(n))
if (is_setop(p) && any_of(enode::args(p), [n](auto a) { return a == n; }))
add_expression(p);
// add equalities and disequalities
auto v = th.get_th_var(n);
if (v2eqs.contains(v)) {
auto const &other = v2eqs[v];
for (auto w : other)
add_expression(th.get_enode(w));
}
if (v2diseqs.contains(v)) {
auto const &other = v2diseqs[v];
for (auto w : other) {
auto n2 = th.get_enode(w);
add_expression(n2);
auto D1 = mk_diff(n, n2);
auto D2 = mk_diff(n2, n);
ctx.mark_as_relevant(D1->get_expr());
ctx.mark_as_relevant(D2->get_expr());
add_expression(D1);
add_expression(D2);
}
}
for (auto p : enode::parents(n)) {
if (!u.is_in(p->get_expr()))
continue;
if (!ctx.is_relevant(p))
continue;
auto x = p->get_arg(0)->get_root();
auto X = mk_singleton(x);
ctx.mark_as_relevant(X->get_expr());
add_expression(X);
}
}
for (auto n : ns)
n->unset_mark();
}
/**
* 1. Base implementation:
* Enumerate all satisfying assignments to m_solver for atoms based on |s|
* Extract Core from enumeration
* Assert Core => |s_i| = sum_ij n_ij for each |s_i| cardinality expression
* NB. Soundness of using Core has not been rigorously established.
* 2. We can check with theory_lra if slack_sums constraints are linear
* feasible. If they are we can possibly terminate by extracting a model
* If they are infeasible, temporarily strengthen m_solver using the negation of unsat core
* that comes from infeasible slack propositions. Then the next model releaxes at least one
* slack variable that is part of the infeasible subset.
*/
lbool theory_finite_set_size::run_solver() {
expr_ref_vector asms(m);
for (auto [k, v] : m_assumptions)
asms.push_back(k);
m_slacks.reset();
m_slack_members.reset();
expr_ref_vector slack_exprs(m);
obj_map<expr, expr *> slack_sums;
arith_util a(m);
expr_ref z(a.mk_int(0), m);
for (auto f : m_set_size_decls)
for (auto n : ctx.enodes_of(f))
slack_sums.insert(n->get_expr(), z);
while (true) {
lbool r = m_solver->check(asms.size(), asms.data());
if (r == l_false) {
auto const& core = m_solver->unsat_core();
literal_vector lits;
for (auto c : core) {
auto exp = m_assumptions[c];
if (std::holds_alternative<eq>(exp)) {
auto [a, b] = std::get<eq>(exp);
expr_ref eq(m.mk_eq(th.get_expr(a), th.get_expr(b)), m);
lits.push_back(~th.mk_literal(eq));
}
else if (std::holds_alternative<diseq>(exp)) {
auto [a, b] = std::get<diseq>(exp);
expr_ref eq(m.mk_eq(th.get_expr(a), th.get_expr(b)), m);
lits.push_back(th.mk_literal(eq));
}
else if (std::holds_alternative<in>(exp)) {
auto [n, is_pos] = std::get<in>(exp);
auto lit = th.mk_literal(n->get_expr());
lits.push_back(is_pos ? ~lit : lit);
}
}
for (auto f : m_set_size_decls) {
for (auto n : ctx.enodes_of(f)) {
expr_ref eq(m.mk_eq(n->get_expr(), slack_sums[n->get_expr()]), m);
auto lit = th.mk_literal(eq);
literal_vector lemma(lits);
lemma.push_back(lit);
TRACE(finite_set, tout << "Asserting cardinality lemma\n";
for (auto lit : lemma) tout << ctx.literal2expr(lit) << "\n";);
ctx.mk_th_axiom(th.get_id(), lemma);
}
}
ctx.push_trail(value_trail(m_solver_ran));
TRACE(finite_set, ctx.display(tout << "Core " << core << "\n"));
m_solver_ran = true;
return l_false;
}
if (r != l_true)
return r;
expr_ref slack(m.mk_fresh_const(symbol("slack"), a.mk_int()), m);
ctx.mk_th_axiom(th.get_id(), th.mk_literal(a.mk_ge(slack, a.mk_int(0)))); // slack is non-negative
model_ref mdl;
m_solver->get_model(mdl);
expr_ref_vector props(m);
for (auto f : m_set_size_decls) {
for (auto n : ctx.enodes_of(f)) {
auto arg = n->get_arg(0);
auto b = n2b[arg];
auto b_is_true = mdl->is_true(b);
props.push_back(b_is_true ? b : m.mk_not(b));
if (b_is_true) {
auto s = slack_sums[n->get_expr()];
s = s == z ? slack : a.mk_add(s, slack);
slack_exprs.push_back(s);
slack_sums.insert(n->get_expr(), s);
}
}
}
m_slacks.push_back(slack);
ptr_vector<expr> members;
for (auto [n, b] : n2b) {
expr *e = n->get_expr();
if (is_uninterp_const(e) && mdl->is_true(b))
members.push_back(e);
}
m_slack_members.push_back(members);
TRACE(finite_set, tout << *mdl << "\nPropositional model:\n" << props << "\n");
m_solver->assert_expr(m.mk_not(m.mk_and(props)));
}
return l_undef;
}
lbool theory_finite_set_size::final_check() {
if (m_set_size_decls.empty())
return l_true;
if (!m_solver) {
initialize_solver();
return l_false;
}
if (!m_solver_ran)
return run_solver();
//
// at this point we assume that
// cardinality constraints are satisfied
// by arithmetic solver.
//
// a refinement checks if this is really necessary
//
return l_true;
}
//
// construct model based on set variables that have cardinality constraints
// In this case the model construction is not explicit. It uses unique sets
// to represent sets of given cardinality.
//
void theory_finite_set_size::init_model(model_generator &mg) {
if (!m_solver || !m_solver_ran)
return;
TRACE(finite_set, tout << "Constructing model for finite set cardinalities\n";);
//
// construct model based on set variables that have cardinality constraints
// slack -> (set variable x truth assignment)*
// slack -> integer assignment from arithmetic solver
// u.mk_unique_set(unique_index, slack_value, type);
// add to model of set variables that are true for slack.
//
arith_value av(m);
av.init(&ctx);
rational value;
arith_util a(m);
SASSERT(m_slacks.size() == m_slack_members.size());
unsigned unique_index = 0;
for (unsigned i = 0; i < m_slacks.size(); ++i) {
auto s = m_slacks.get(i);
//
// slack s is equivalent to some integer value
// create a unique set corresponding to this slack value.
// The signature of the unique set is given by the sets that are
// satisfiable in the propositional assignment where the slack variable
// was introduced.
//
if (av.get_value_equiv(s, value)) {
if (value == 0)
continue;
if (m_slack_members[i].empty())
continue;
++unique_index;
for (auto e : m_slack_members[i]) {
app *unique_value = u.mk_unique_set(a.mk_int(unique_index), a.mk_int(value), e->get_sort());
if (m_unique_values.contains(e))
unique_value = u.mk_union(m_unique_values[e], unique_value);
m_unique_values.insert(e, unique_value);
m_pinned.push_back(unique_value);
}
}
}
}
std::ostream& theory_finite_set_size::display(std::ostream& out) const {
if (m_solver)
m_solver->display(out << "set.size-solver\n");
return out;
}
} // namespace smt