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https://github.com/Z3Prover/z3
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add new option for adding tangent lemmas for integer monomials
Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
This commit is contained in:
Nikolaj Bjorner
parent
fc96f827a1
commit
2578218b6f
3 changed files with 80 additions and 29 deletions
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@ -161,13 +161,32 @@ struct solver::imp {
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}
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}
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polynomial::polynomial_ref sub(polynomial::polynomial *a, polynomial::polynomial *b) {
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return polynomial_ref(m_nlsat->pm().sub(a, b), m_nlsat->pm());
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}
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polynomial::polynomial_ref add(polynomial::polynomial *a, polynomial::polynomial *b) {
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return polynomial_ref(m_nlsat->pm().add(a, b), m_nlsat->pm());
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}
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polynomial::polynomial_ref mul(polynomial::polynomial *a, polynomial::polynomial *b) {
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return polynomial_ref(m_nlsat->pm().mul(a, b), m_nlsat->pm());
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}
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polynomial::polynomial_ref var(lp::lpvar v) {
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return polynomial_ref(m_nlsat->pm().mk_polynomial(lp2nl(v)), m_nlsat->pm());
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}
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polynomial::polynomial_ref constant(rational const& r) {
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return polynomial_ref(m_nlsat->pm().mk_const(r), m_nlsat->pm());
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}
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void add_tangent_lemmas() {
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for (auto const mi : m_nla_core.to_refine()) {
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auto const& m = m_nla_core.emon(mi);
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for (auto const xy : m_nla_core.to_refine()) {
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auto const& m = m_nla_core.emon(xy);
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if (m.size() != 2)
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continue;
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auto x = m.vars()[0];
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auto y = m.vars()[1];
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if (!m_nla_core.var_is_int(x) || !m_nla_core.var_is_int(y))
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continue;
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rational xv = m_nla_core.val(x);
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rational yv = m_nla_core.val(y);
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rational mv = m_nla_core.val(m.var());
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@ -177,7 +196,7 @@ struct solver::imp {
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// (x - a)(y - b) = 1
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// (x - a)(y - b) = xy - bx - ay + ab = 1
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// mv - bx - ay + ab < 1
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// lemma: x > a, y > b => xy - bx - ay + ab >= 1
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// lemma: x >= a + 1, y >= b + 1 => xy - bx - ay + ab >= 1
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//
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// mv > xv * yv
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// a := xv - 1, b := yv + 1
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@ -185,8 +204,45 @@ struct solver::imp {
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//
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// other lemmas around a < x, b < y and a < x, b > y
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//
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auto x_ge_xv = mk_literal(sub(var(x), constant(xv)), lp::lconstraint_kind::GE);
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auto y_ge_yv = mk_literal(sub(var(y), constant(yv)), lp::lconstraint_kind::GE);
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auto yv_ge_y = mk_literal(sub(constant(yv), var(y)), lp::lconstraint_kind::GE);
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auto xv_ge_x = mk_literal(sub(constant(xv), var(x)), lp::lconstraint_kind::GE);
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{
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auto ineq = mk_literal(
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sub(mul(sub(var(x), constant(xv - 1)), sub(var(y), constant(yv - 1))), constant(rational(1))),
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lp::lconstraint_kind::GE);
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nlsat::literal lits[3] = {~x_ge_xv, ~y_ge_yv, ineq};
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m_nlsat->mk_clause(3, lits, nullptr);
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}
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{
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// x >= a + 1, y <= b - 1 => (x - a)(y - b) = xy -bx - ay +ab <= -1
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auto ineq =
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mk_literal(sub(mul(sub(var(x), constant(xv - 1)), sub(var(y), constant(yv + 1))), constant(rational(-1))),
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lp::lconstraint_kind::LE);
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nlsat::literal lits[3] = {~x_ge_xv, ~yv_ge_y, ineq};
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m_nlsat->mk_clause(3, lits, nullptr);
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}
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{
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// x <= a - 1, y >= b + 1 => (x - a)(y - b) = xy - bx - ay + ab <= -1
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auto ineq = mk_literal(
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sub(mul(sub(var(x), constant(xv + 1)), sub(var(y), constant(yv - 1))), constant(rational(-1))),
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lp::lconstraint_kind::LE);
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nlsat::literal lits[3] = {~xv_ge_x, ~y_ge_yv, ineq};
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m_nlsat->mk_clause(3, lits, nullptr);
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}
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{
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// x <= a - 1, y <= b - 1 => (x - a)(y - b) = xy - bx - ay + ab >= 1
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auto ineq = mk_literal(
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sub(mul(sub(var(x), constant(xv + 1)), sub(var(y), constant(yv + 1))), constant(rational(1))),
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lp::lconstraint_kind::GE);
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nlsat::literal lits[3] = {~xv_ge_x, ~yv_ge_y, ineq};
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m_nlsat->mk_clause(3, lits, nullptr);
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}
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}
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}
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@ -216,6 +272,9 @@ struct solver::imp {
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if (p.arith_nl_nra_model_bounds())
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setup_model_bounds();
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if (p.arith_nl_nra_tangents())
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add_tangent_lemmas();
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TRACE(nra, m_nlsat->display(tout));
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if (p.arith_nl_log()) {
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@ -338,12 +397,25 @@ struct solver::imp {
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coeffs.push_back(mpz(1));
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coeffs.push_back(mpz(-1));
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polynomial::polynomial_ref p(pm.mk_polynomial(2, coeffs.data(), mls), pm);
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polynomial::polynomial* ps[1] = { p };
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bool even[1] = { false };
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nlsat::literal lit = m_nlsat->mk_ineq_literal(nlsat::atom::kind::EQ, 1, ps, even);
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auto lit = mk_literal(p.get(), lp::lconstraint_kind::EQ);
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nlsat::assumption a = nullptr;
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m_nlsat->mk_clause(1, &lit, nullptr);
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}
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nlsat::literal mk_literal(polynomial::polynomial *p, lp::lconstraint_kind k) {
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polynomial::polynomial *ps[1] = { p };
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bool is_even[1] = { false };
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switch (k) {
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case lp::lconstraint_kind::LE: return ~m_nlsat->mk_ineq_literal(nlsat::atom::kind::GT, 1, ps, is_even);
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case lp::lconstraint_kind::GE: return ~m_nlsat->mk_ineq_literal(nlsat::atom::kind::LT, 1, ps, is_even);
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case lp::lconstraint_kind::LT: return m_nlsat->mk_ineq_literal(nlsat::atom::kind::LT, 1, ps, is_even);
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case lp::lconstraint_kind::GT: return m_nlsat->mk_ineq_literal(nlsat::atom::kind::GT, 1, ps, is_even);
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case lp::lconstraint_kind::EQ: return m_nlsat->mk_ineq_literal(nlsat::atom::kind::EQ, 1, ps, is_even);
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default: UNREACHABLE(); // unreachable
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}
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throw default_exception("uexpected operator");
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}
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void add_constraint(unsigned idx) {
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m_max_constraint_index = std::max(m_max_constraint_index, idx);
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auto& c = lra.constraints()[idx];
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@ -364,29 +436,8 @@ struct solver::imp {
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}
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rhs *= den;
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polynomial::polynomial_ref p(pm.mk_linear(sz, coeffs.data(), vars.data(), -rhs), pm);
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polynomial::polynomial* ps[1] = { p };
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bool is_even[1] = { false };
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nlsat::literal lit;
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nlsat::literal lit = mk_literal(p.get(), k);
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nlsat::assumption a = this + idx;
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switch (k) {
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case lp::lconstraint_kind::LE:
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lit = ~m_nlsat->mk_ineq_literal(nlsat::atom::kind::GT, 1, ps, is_even);
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break;
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case lp::lconstraint_kind::GE:
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lit = ~m_nlsat->mk_ineq_literal(nlsat::atom::kind::LT, 1, ps, is_even);
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break;
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case lp::lconstraint_kind::LT:
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lit = m_nlsat->mk_ineq_literal(nlsat::atom::kind::LT, 1, ps, is_even);
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break;
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case lp::lconstraint_kind::GT:
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lit = m_nlsat->mk_ineq_literal(nlsat::atom::kind::GT, 1, ps, is_even);
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break;
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case lp::lconstraint_kind::EQ:
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lit = m_nlsat->mk_ineq_literal(nlsat::atom::kind::EQ, 1, ps, is_even);
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break;
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default:
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UNREACHABLE(); // unreachable
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}
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m_nlsat->mk_clause(1, &lit, a);
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}
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@ -1871,7 +1871,6 @@ namespace nlsat {
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lbool search_check() {
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verbose_stream() << "search check\n";
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lbool r = l_undef;
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m_stats.m_conflicts = 0;
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m_stats.m_restarts = 0;
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@ -93,6 +93,7 @@ def_module_params(module_name='smt',
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('arith.nl.linearize', BOOL, True, 'enable NL linearization'),
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('arith.nl.nra.poly', BOOL, False, 'use polynomial translation'),
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('arith.nl.nra.model_bounds', BOOL, False, 'use model-based integer bounds for nra solver'),
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('arith.nl.nra.tangents', BOOL, False, 'use tangent lemmas in nra solver'),
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('arith.nl.log', BOOL, False, 'Log lemmas sent to nra solver'),
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('arith.propagate_eqs', BOOL, True, 'propagate (cheap) equalities'),
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('arith.epsilon', DOUBLE, 1.0, 'initial value of epsilon used for model generation of infinitesimals'),
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