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Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
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Nikolaj Bjorner 2025-09-27 16:59:22 +03:00
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src/math/lp/nla_mul_saturate.cpp
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/*++
Copyright (c) 2025 Microsoft Corporation
given a monic m = x * y * z ... used in a constraint that is false under the current evaluation of x,y,z
saturate constraints with respect to m
in other words, if a constraint contains x*y + p >= 0,
then include the constraint x*y*z + z*p >= 0
assuming current value of z is non-negative.
Check if the system with new constraints is LP (and MIP) feasible.
If it is not, then produce a lemma that explains infeasibility.
Strategy 1: The lemma is in terms of the original constraints and bounds.
Strategy 2: Attempt to eliminate new monomials from the lemma by relying on Farkas multipliers.
If it succeeds to eliminate new monomials we have a lemma that is a linear
combination of existing variables.
The idea is that a conflict over the new system is a set of multipliers lambda, such
that lambda * A is separated from b for the constraints Axy >= b.
The coefficients in lambda are non-negative.
They correspond to variables x, y where x are variables from the input constraints
and y are new variables introduced for new monomials.
We can test if lambda allows eliminating y by taking a subset of lambda indices where
A has rows containing y_i for some fresh y_i. Replace those rows r_j by the partial sum of
of rows multiplied by lambda_j. The sum r_j1 * lambda_j1 + .. + r_jk * lambda_jk does not
contain y_i. Repeat the same process with other variables y_i'. If a sum is
generated without any y, it is a linear consequence of the new constraints but not
necessarily derivable with the old constraints.
Strategy 3: The lemma uses the new constraints.
--*/
#include "math/lp/nla_core.h"
#include "math/lp/nla_coi.h"
#include "math/lp/nla_mul_saturate.h"
namespace nla {
mul_saturate::mul_saturate(core* core) : common(core), m_coi(*core) {}
lbool mul_saturate::saturate() {
lp::explanation ex;
init_solver();
add_multiply_constraints();
lbool r = solve(ex);
if (r == l_false)
add_lemma(ex);
return r;
}
void mul_saturate::init_solver() {
lra_solver = alloc(lp::lar_solver);
int_solver = alloc(lp::int_solver, *lra_solver);
m_vars2mon.reset();
m_mon2vars.reset();
m_values.reset();
m_coi.init();
init_vars();
}
void mul_saturate::init_vars() {
auto const& lra = c().lra_solver();
auto sz = lra.number_of_vars();
for (unsigned v = 0; v < sz; ++v) {
// Declare v into lra_solver
unsigned w;
if (m_coi.mons().contains(v))
init_monomial(v);
else
m_values.push_back(c().val(v));
if (m_coi.terms().contains(v)) {
auto const& t = lra.get_term(v);
// Assumption: variables in coefficients are always declared before term variable.
SASSERT(all_of(t, [&](auto p) { return p.j() < v; }));
w = lra_solver->add_term(t.coeffs_as_vector(), v);
}
else
w = lra_solver->add_var(v, lra.var_is_int(v));
// assert bounds on v in the new solver.
VERIFY(w == v);
if (lra.column_has_lower_bound(v)) {
auto lo_bound = lra.get_lower_bound(v);
auto k = lo_bound.y > 0 ? lp::lconstraint_kind::GT : lp::lconstraint_kind::GE;
auto rhs = lo_bound.x;
auto dep = lra.get_column_lower_bound_witness(v);
auto ci = lra_solver->add_var_bound(v, k, rhs);
m_ci2dep.setx(ci, dep, nullptr);
}
if (lra.column_has_upper_bound(v)) {
auto hi_bound = lra.get_upper_bound(v);
auto k = hi_bound.y < 0 ? lp::lconstraint_kind::LT : lp::lconstraint_kind::LE;
auto rhs = hi_bound.x;
auto dep = lra.get_column_upper_bound_witness(v);
auto ci = lra_solver->add_var_bound(v, k, rhs);
m_ci2dep.setx(ci, dep, nullptr);
}
}
}
void mul_saturate::init_monomial(unsigned mon_var) {
auto& mon = c().emons()[mon_var];
svector<lpvar> vars(mon.vars());
std::sort(vars.begin(), vars.end());
m_vars2mon.insert(vars, mon_var);
m_mon2vars.insert(mon_var, vars);
rational p(1);
for (auto v : vars)
p *= m_values[v];
m_values.push_back(p);
}
void mul_saturate::add_lemma(lp::explanation const& ex1) {
auto& lra = c().lra_solver();
lp::explanation ex2;
lemma_builder new_lemma(c(), "stellensatz");
for (auto p : ex1) {
auto dep = m_ci2dep.get(p.ci(), nullptr);
lra_solver->push_explanation(dep, ex2);
if (!m_new_mul_constraints.contains(p.ci()))
continue;
auto const& bounds = m_new_mul_constraints[p.ci()];
for (auto const& b : bounds) {
if (std::holds_alternative<u_dependency*>(b)) {
auto dep = *std::get_if<u_dependency*>(&b);
lra_solver->push_explanation(dep, ex2);
}
else {
auto const &[v, k, rhs] = *std::get_if<bound>(&b);
new_lemma |= ineq(v, negate(k), rhs);
}
}
}
new_lemma &= ex2;
IF_VERBOSE(5, verbose_stream() << "unsat \n" << new_lemma << "\n");
TRACE(arith, tout << "unsat\n" << new_lemma << "\n");
c().lra_solver().settings().stats().m_nla_stellensatz++;
}
lbool mul_saturate::solve(lp::explanation& ex) {
lbool r = solve_lra(ex);
if (r != l_true)
return r;
r = solve_lia(ex);
if (r != l_true)
return r;
// if (r == l_true) check if solution satisfies constraints
// variables outside the slice have values from the outer solver.
return l_undef;
}
lbool mul_saturate::solve_lra(lp::explanation& ex) {
auto st = lra_solver->solve();
if (st == lp::lp_status::INFEASIBLE) {
lra_solver->get_infeasibility_explanation(ex);
return l_false;
}
else if (lra_solver->is_feasible())
return l_true;
else
return l_undef;
}
lbool mul_saturate::solve_lia(lp::explanation& ex) {
switch (int_solver->check(&ex)) {
case lp::lia_move::sat:
return l_true;
case lp::lia_move::conflict:
return l_false;
default: // TODO: an option is to perform (bounded) search here to get an LIA verdict.
return l_undef;
}
return l_undef;
}
// record new monomials that are created and recursively down-saturate with respect to these.
// this is a simplistic pass
void mul_saturate::add_multiply_constraints() {
m_new_mul_constraints.reset();
m_to_refine.reset();
vector<svector<lp::constraint_index>> var2cs;
// current approach: only resolve against var2cs, which is initialized
// with monomials in the input.
for (auto ci : lra_solver->constraints().indices()) {
auto const& con = lra_solver->constraints()[ci];
for (auto [coeff, v] : con.coeffs()) {
if (v >= var2cs.size())
var2cs.resize(v + 1);
var2cs[v].push_back(ci);
}
// insert monomials to be refined
insert_monomials_from_constraint(ci);
}
for (unsigned it = 0; it < m_to_refine.size(); ++it) {
auto j = m_to_refine[it];
auto vars = m_mon2vars[j];
for (auto v : vars) {
if (v >= var2cs.size())
continue;
auto cs = var2cs[v];
for (auto ci : cs) {
for (auto [coeff, u] : lra_solver->constraints()[ci].coeffs()) {
if (u == v)
add_multiply_constraint(ci, j, v);
}
}
}
}
IF_VERBOSE(5,
c().lra_solver().display(verbose_stream() << "original\n");
c().display(verbose_stream());
display(verbose_stream() << "saturated\n"));
}
// multiply by remaining vars
void mul_saturate::add_multiply_constraint(lp::constraint_index old_ci, lp::lpvar mi, lpvar x) {
lp::lar_base_constraint const& con = lra_solver->constraints()[old_ci];
auto &lra = c().lra_solver();
auto const& lhs = con.coeffs();
auto const& rhs = con.rhs();
auto k = con.kind();
if (k == lp::lconstraint_kind::NE || k == lp::lconstraint_kind::EQ)
return; // not supported
auto sign = false, strict = true;
svector<lpvar> vars;
bool first = true;
for (auto v : m_mon2vars[mi]) {
if (v != x || !first)
vars.push_back(v);
else
first = false;
}
SASSERT(!first); // v was a member and was removed
vector<bound_justification> bounds;
// compute bounds constraints and sign of vars
auto add_bound = [&](lpvar v, lp::lconstraint_kind k, int r) {
bound b(v, k, rational(r));
bounds.push_back(b);
};
for (auto v : vars) {
// retrieve bounds of v
// if v has positive lower bound add as positive
// if v has negative upper bound add as negative
// otherwise look at the current value of v and add bounds assumption based on current sign.
if (lra.number_of_vars() > v && lra.column_has_lower_bound(v) && lra.get_lower_bound(v).is_pos()) {
bounds.push_back(lra.get_column_lower_bound_witness(v));
}
else if (lra.number_of_vars() > v && lra.column_has_upper_bound(v) && lra.get_upper_bound(v).is_neg()) {
bounds.push_back(lra.get_column_upper_bound_witness(v));
sign = !sign;
}
else if (m_values[v].is_neg()) {
if (lra.var_is_int(v))
add_bound(v, lp::lconstraint_kind::LE, -1);
else
add_bound(v, lp::lconstraint_kind::LT, 0);
sign = !sign;
}
else if (m_values[v].is_pos()) {
if (lra.var_is_int(v))
add_bound(v, lp::lconstraint_kind::GE, 1);
else
add_bound(v, lp::lconstraint_kind::GT, 0);
}
else {
SASSERT(m_values[v] == 0);
strict = false;
add_bound(v, lp::lconstraint_kind::GE, 0);
}
}
lp::lar_term new_lhs;
rational new_rhs(rhs), term_value(0);
for (auto const & [coeff, v] : lhs) {
unsigned old_sz = vars.size();
if (m_mon2vars.contains(v))
vars.append(m_mon2vars[v]);
else
vars.push_back(v);
lpvar new_monic_var = add_monomial(vars);
new_lhs.add_monomial(coeff, new_monic_var);
term_value += coeff * m_values[new_monic_var];
vars.shrink(old_sz);
}
if (rhs != 0) {
lpvar new_monic_var = add_monomial(vars);
new_lhs.add_monomial(-rhs, new_monic_var);
term_value -= rhs * m_values[new_monic_var];
new_rhs = 0;
}
if (sign)
k = swap_side(k);
if (!strict) {
switch (k) {
case lp::lconstraint_kind::GT:
k = lp::lconstraint_kind::GE;
break;
case lp::lconstraint_kind::LT:
k = lp::lconstraint_kind::LE;
break;
}
}
auto ti = lra_solver->add_term(new_lhs.coeffs_as_vector(), lra_solver->number_of_vars());
auto new_ci = lra_solver->add_var_bound(ti, k, new_rhs);
IF_VERBOSE(4, display_constraint(verbose_stream(), old_ci) << " -> ";
display_constraint(verbose_stream(), new_lhs.coeffs_as_vector(), k, new_rhs) << "\n");
insert_monomials_from_constraint(new_ci);
m_values.push_back(term_value);
SASSERT(m_values.size() - 1 == ti);
m_new_mul_constraints.insert(new_ci, bounds);
m_ci2dep.setx(new_ci, m_ci2dep.get(old_ci, nullptr), nullptr);
}
// Insert a (new) monomial representing product of vars.
// if the length of vars is 1 then the new monomial is vars[0]
// if the same monomial was previous defined, return the previously defined monomial.
// otherwise create a fresh variable representing the monomial.
lpvar mul_saturate::add_monomial(svector<lpvar> const& vars) {
lpvar v;
if (vars.size() == 1)
return vars[0];
svector<lpvar> _vars(vars);
std::sort(_vars.begin(), _vars.end());
if (m_vars2mon.find(_vars, v))
return v;
v = add_var(is_int(vars));
m_vars2mon.insert(_vars, v);
m_mon2vars.insert(v, _vars);
rational p(1);
for (auto v : vars)
p *= m_values[v];
m_values.push_back(p);
SASSERT(m_values.size() - 1 == v);
return v;
}
bool mul_saturate::is_int(svector<lp::lpvar> const& vars) const {
auto const& lra = m_core.lra;
return all_of(vars, [&](lpvar v) { return lra.var_is_int(v); });
}
lpvar mul_saturate::add_var(bool is_int) {
auto v = lra_solver->number_of_vars();
auto w = lra_solver->add_var(v, is_int);
SASSERT(v == w);
return w;
}
// if a constraint is false, then insert _all_ monomials from that constraint.
// other approach: use a lex ordering on monomials and insert the lex leading monomial.
void mul_saturate::insert_monomials_from_constraint(lp::constraint_index ci) {
if (constraint_is_true(ci))
return;
auto const& con = lra_solver->constraints()[ci];
for (auto [coeff, v] : con.coeffs())
if (c().is_monic_var(v))
m_to_refine.insert(v);
}
bool mul_saturate::constraint_is_true(lp::constraint_index ci) {
auto const& con = lra_solver->constraints()[ci];
auto lhs = -con.rhs();
for (auto const& [coeff, v] : con.coeffs())
lhs += coeff * m_values[v];
switch (con.kind()) {
case lp::lconstraint_kind::GT:
return lhs > 0;
case lp::lconstraint_kind::GE:
return lhs >= 0;
case lp::lconstraint_kind::LE:
return lhs <= 0;
case lp::lconstraint_kind::LT:
return lhs < 0;
case lp::lconstraint_kind::EQ:
return lhs == 0;
case lp::lconstraint_kind::NE:
return lhs != 0;
default:
UNREACHABLE();
return false;
}
}
std::ostream& mul_saturate::display(std::ostream& out) const {
lra_solver->display(out);
for (auto const& [vars, v] : m_vars2mon) {
out << "j" << v << " := ";
display_product(out, vars);
out << "\n";
}
return out;
}
std::ostream& mul_saturate::display_product(std::ostream& out, svector<lpvar> const& vars) const {
bool first = true;
for (auto v : vars) {
if (first)
first = false;
else
out << " * ";
out << "j" << v;
}
return out;
}
std::ostream& mul_saturate::display_constraint(std::ostream& out, lp::constraint_index ci) const {
auto const& con = lra_solver->constraints()[ci];
return display_constraint(out, con.coeffs(), con.kind(), con.rhs());
}
std::ostream& mul_saturate::display_constraint(std::ostream& out,
vector<std::pair<rational, lpvar>> const& lhs,
lp::lconstraint_kind k, rational const& rhs) const {
bool first = true;
for (auto [coeff, v] : lhs) {
c().display_coeff(out, first, coeff);
first = false;
if (m_mon2vars.contains(v))
display_product(out, m_mon2vars[v]);
else
out << "j" << v;
}
return out << " " << k << " " << rhs;
}
}