v0.1 of nla saturation

.clang-format

@@ -8,6 +8,7 @@ BasedOnStyle: LLVM
 IndentWidth: 4
 TabWidth: 4
 UseTab: Never
+IndentCaseLabels: false
 
 
 

src/math/lp/nla_core.h

@@ -128,6 +128,7 @@ public:
     const auto& monics_with_changed_bounds() const { return m_monics_with_changed_bounds; }
     lp::lar_solver& lra_solver() { return lra; }
     lp::lar_solver const & lra_solver() const { return lra; }
+
     indexed_uint_set const& to_refine() const { return m_to_refine; }
 
     void insert_to_refine(lpvar j);

src/math/lp/nla_mul_saturate.cpp

@@ -0,0 +1,383 @@
+/*++
+  Copyright (c) 2025 Microsoft Corporation
+
+  given a monic m = x * y * z ... with evaluation val(m) != val(x) * val(y) * val(z) ...
+
+  saturate constraints with respect to m
+  in other words, if a constraint contains x*y + p >= 0, 
+  then include the constraint z >= 0 => x*y*z + z*p >= 0
+  assuming current value of z is non-negative.
+  Check if the system with new constraints is LP feasible.
+  If it is not, then produce a lemma that explains the 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. 
+  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() {
+        local_solver = alloc(lp::lar_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) {
+            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 = local_solver->add_term(t.coeffs_as_vector(), v);
+            } 
+            else
+                w = local_solver->add_var(v, lra.var_is_int(v));
+
+            VERIFY(w == v);
+            if (lra.column_has_lower_bound(v)) {
+                auto lo_dep = lra.get_column_lower_bound_witness(v);
+                auto lo_bound = lra.get_lower_bound(v);
+                auto k = lo_bound.y > 0 ? lp::lconstraint_kind::GT : lp::lconstraint_kind::GE;
+                auto ci = local_solver->add_var_bound(v, k, lo_bound.x);
+            }
+            if (lra.column_has_upper_bound(v)) {
+                auto hi_dep = lra.get_column_upper_bound_witness(v);
+                auto hi_bound = lra.get_upper_bound(v);
+                auto k = hi_bound.y < 0 ? lp::lconstraint_kind::LT : lp::lconstraint_kind::LE;
+                auto ci = local_solver->add_var_bound(v, k, hi_bound.x);
+                m_ci2dep.setx(ci, hi_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;
+        for (auto p : ex1) {
+            lp::constraint_index src_ci;
+            if (!m_new_mul_constraints.find(p.ci(), src_ci))
+                src_ci = p.ci();
+            auto dep = m_ci2dep.get(src_ci, nullptr);
+            local_solver->push_explanation(dep, ex2);
+        }
+        for (auto [v, sign, dep] : m_var_signs) {            
+            if (!dep) {
+                verbose_stream() << "TODO explain non-implied bound\n";
+                continue;
+            }
+            local_solver->push_explanation(dep, ex2);
+        }
+        lemma_builder new_lemma(c(), "stellensatz");
+        new_lemma &= ex2;
+        IF_VERBOSE(1, verbose_stream() << "unsat \n" << new_lemma << "\n");
+        TRACE(arith, tout << "unsat\n" << new_lemma << "\n");
+    }
+
+    lbool mul_saturate::solve(lp::explanation& ex) {
+        for (auto const& [new_ci, old_ci] : m_new_mul_constraints)
+            local_solver->activate(new_ci);
+        auto st = local_solver->solve();
+        lbool r = l_undef;
+        if (st == lp::lp_status::INFEASIBLE) {
+            local_solver->get_infeasibility_explanation(ex);
+            r = l_false;
+        }
+        if (st == lp::lp_status::OPTIMAL || st == lp::lp_status::FEASIBLE) {
+            // TODO: check model just in case it got lucky.
+            IF_VERBOSE(1, verbose_stream() << "saturation is LP feasible, maybe it is a model of the NLA problem\n");
+        }
+        IF_VERBOSE(0, display(verbose_stream()));
+        return r;
+    }
+
+    // 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_seen_vars.reset();
+        m_var_signs.reset();
+        m_to_refine.reset();
+        vector<svector<lp::constraint_index>> var2cs;
+
+        for (auto ci : local_solver->constraints().indices()) {
+            auto const& con = local_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];
+            verbose_stream() << "refining " << j << " := " << m_mon2vars[j] << "\n";
+            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] : local_solver->constraints()[ci].coeffs()) {
+                        if (u == v)
+                            add_multiply_constraint(ci, j, v);
+                    }                    
+                }
+            }
+        }
+        IF_VERBOSE(0,
+                   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 = local_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;
+        svector<lpvar> vars;
+        bool first = true;
+        for (auto v : c().emon(mi).vars()) {
+            if (v != x || !first)
+                vars.push_back(v);
+            else
+                first = false;
+        }
+        // compute sign of vars
+        for (auto v : vars) {
+            if (m_seen_vars.contains(v))
+                continue;
+            m_seen_vars.insert(v);
+            // retrieve bounds of v
+            // if v has positive lower bound add as positive
+            // if v has negative upper bound add as negative
+            // otherwise, soft-fail (for now unsound)
+            // proper signs of variables from old tableau should be extracted using lra_solver()
+            // instead of local_solver.
+            // TODO is to also add case where lower or upper bound is zero and then the sign
+            // of the inequality is relaxed if it is strict.
+            if (lra.number_of_vars() > v && lra.column_has_lower_bound(v) && lra.get_lower_bound(v).is_pos()) {
+                m_var_signs.push_back({v, false, 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()) {
+                m_var_signs.push_back({v, true, lra.get_column_upper_bound_witness(v)});
+                sign = !sign;
+            } 
+            else if (m_values[v].is_neg()) {
+                m_var_signs.push_back({v, true, nullptr});
+                sign = !sign;
+            } 
+            else if (m_values[v].is_pos()) {
+                m_var_signs.push_back({v, false, nullptr});
+            } 
+            else {
+                IF_VERBOSE(0, verbose_stream() << "not separated from 0\n");
+                return;
+            }
+        }
+        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) {
+            switch (k) {
+            case lp::lconstraint_kind::LE:
+                k = lp::lconstraint_kind::GE;
+                break;
+            case lp::lconstraint_kind::LT:
+                k = lp::lconstraint_kind::GT;
+                break;
+            case lp::lconstraint_kind::GE:
+                k = lp::lconstraint_kind::LE;
+                break;
+            case lp::lconstraint_kind::GT:
+                k = lp::lconstraint_kind::LT;
+                break;
+            default:
+                break;
+            }
+        }
+        display_constraint(verbose_stream(), old_ci) << " -> ";
+        display_constraint(verbose_stream(), new_lhs.coeffs_as_vector(), k, new_rhs) << "\n";
+        auto ti = local_solver->add_term(new_lhs.coeffs_as_vector(), local_solver->number_of_vars());
+        auto new_ci = local_solver->add_var_bound(ti, k, new_rhs);
+        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, old_ci);
+    }
+
+    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 = local_solver->number_of_vars();
+        auto w = local_solver->add_var(v, is_int);
+        VERIFY(v == w);
+        return w;
+    }
+
+    void mul_saturate::insert_monomials_from_constraint(lp::constraint_index ci) {
+        if (constraint_is_true(ci))
+            return;
+        auto const& con = local_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 = local_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 {
+        local_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 = local_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;
+    }
+}

src/math/lp/nla_mul_saturate.h

@@ -0,0 +1,69 @@
+/*++
+  Copyright (c) 2025 Microsoft Corporation
+
+  --*/
+#pragma once
+
+#include "math/lp/nla_coi.h"
+
+namespace nla {
+
+    class core;
+    class lar_solver;    
+    class mul_saturate : common {
+
+        struct var_sign {
+            lpvar v = lp::null_lpvar;
+            bool is_neg = false;
+            u_dependency* dep = nullptr;
+        };
+        coi m_coi;
+        // source of multiplication constraint
+        u_map<lp::constraint_index> m_new_mul_constraints;
+        svector<var_sign> m_var_signs;
+        tracked_uint_set m_seen_vars;
+        indexed_uint_set m_to_refine;
+        scoped_ptr<lp::lar_solver> local_solver;
+        ptr_vector<u_dependency> m_ci2dep;
+        vector<rational> m_values;
+        struct eq {
+            bool operator()(unsigned_vector const& a, unsigned_vector const& b) const {
+                return a == b;
+            }
+        };
+        map<unsigned_vector, unsigned, svector_hash<unsigned_hash>, eq> m_vars2mon;
+        u_map<unsigned_vector> m_mon2vars;
+
+        // initialization
+        void init_solver();
+        void init_vars();
+        void init_monomial(unsigned mon_var);
+
+        bool constraint_is_true(lp::constraint_index ci);
+        void insert_monomials_from_constraint(lp::constraint_index ci);
+
+        // additional variables and monomials and constraints
+        lpvar add_monomial(svector<lp::lpvar> const& vars);
+        bool is_int(svector<lp::lpvar> const& vars) const;
+        lpvar add_var(bool is_int);
+        void add_multiply_constraints();
+        void add_multiply_constraint(lp::constraint_index con_id, lp::lpvar mi, lpvar x);
+
+        // solving
+        lbool solve(lp::explanation& ex);
+
+        // lemmas
+        void add_lemma(lp::explanation const& ex);
+
+        std::ostream& display(std::ostream& out) const;
+        std::ostream& display_product(std::ostream& out, svector<lpvar> const& vars) const;
+        std::ostream& display_constraint(std::ostream& out, lp::constraint_index ci) const;
+        std::ostream& display_constraint(std::ostream& out, vector<std::pair<rational, lpvar>> const& lhs,
+                                         lp::lconstraint_kind k, rational const& rhs) const;
+
+    public:
+        mul_saturate(core* core);
+        lbool saturate();
+    };
+
+}

src/smt/theory_lra.cpp

@@ -3475,7 +3475,8 @@ public:
         ++m_stats.m_conflicts;
         for (auto ev : m_explanation)
             set_evidence(ev.ci(), m_core, m_eqs);
-        if (all_of(m_core, [&](literal l) { return ctx().get_assignment(l) == l_false; }))
+
+        if (m_eqs.empty() && all_of(m_core, [&](literal l) { return ctx().get_assignment(l) == l_false; }))
             is_conflict = true;
         TRACE(arith_conflict,
               tout << "@" << ctx().get_scope_level() << (is_conflict ? " conflict":" lemma");