z3/src/sat/smt/array_axioms.cpp

/*++
Copyright (c) 2020 Microsoft Corporation

Module Name:

    array_axioms.cpp

Abstract:

    Routines for instantiating array axioms

Author:

    Nikolaj Bjorner (nbjorner) 2020-09-08

--*/

#include "ast/ast_trail.h"
#include "ast/ast_ll_pp.h"
#include "sat/smt/array_solver.h"
#include "sat/smt/euf_solver.h"

namespace array {

    void solver::push_axiom(axiom_record const& r) { 
        unsigned idx = m_axiom_trail.size();
        m_axiom_trail.push_back(r); 
        if (m_axioms.contains(idx))
            m_axiom_trail.pop_back();
        else
            ctx.push(push_back_vector<euf::solver, svector<axiom_record>>(m_axiom_trail));
    }

    bool solver::assert_axiom(unsigned idx) {
        axiom_record const& r = m_axiom_trail[idx];
        if (m_axioms.contains(idx))
            return false;
        m_axioms.insert(idx);
        ctx.push(insert_map<euf::solver, axiom_table_t, unsigned>(m_axioms, idx));
        expr* child = r.n->get_expr();
        app* select;
        switch (r.m_kind) {
        case axiom_record::kind_t::is_store:
            TRACE("array", tout << "store-axiom: " << mk_bounded_pp(child, m, 2) << "\n";);
            return assert_store_axiom(to_app(child));
        case axiom_record::kind_t::is_select:
            select = r.select->get_app();
            SASSERT(a.is_select(select));
            SASSERT(can_beta_reduce(r.n));
            TRACE("array", tout << "select-axiom: " << mk_bounded_pp(select, m, 2) << " " << mk_bounded_pp(child, m, 2) << "\n";);
            if (r.select->get_arg(0)->get_root() != r.n->get_root()) {
                IF_VERBOSE(0, verbose_stream() << "could delay " << mk_pp(select, m) << " " << mk_pp(child, m) << "\n");
            }
            if (a.is_const(child))
                return assert_select_const_axiom(select, to_app(child));
            else if (a.is_as_array(child))
                return assert_select_as_array_axiom(select, to_app(child));
            else if (a.is_store(child))
                return assert_select_store_axiom(select, to_app(child));
            else if (a.is_map(child))
                return assert_select_map_axiom(select, to_app(child));
            else if (is_lambda(child))
                return assert_select_lambda_axiom(select, child);
            else
                UNREACHABLE();
            break;
        case axiom_record::kind_t::is_default:
            SASSERT(can_beta_reduce(r.n));
            TRACE("array", tout << "default-axiom: " << mk_bounded_pp(child, m, 2) << "\n";);
            if (a.is_const(child))
                return assert_default_const_axiom(to_app(child));
            else if (a.is_store(child))
                return assert_default_store_axiom(to_app(child));
            else if (a.is_map(child))
                return assert_default_map_axiom(to_app(child));
            else
                return true;                
            break;
        case axiom_record::kind_t::is_extensionality:
            TRACE("array", tout << "extensionality-axiom: " << mk_bounded_pp(child, m, 2) << "\n";);
            return assert_extensionality(r.n->get_arg(0)->get_expr(), r.n->get_arg(1)->get_expr());
        case axiom_record::kind_t::is_congruence:
            TRACE("array", tout << "congruence-axiom: " << mk_bounded_pp(child, m, 2) << " " << mk_bounded_pp(r.select->get_expr(), m, 2) << "\n";);
            return assert_congruent_axiom(child, r.select->get_expr());
        default:
            UNREACHABLE();
            break;
        }
        return false;
    }

    /**
     * Assert
     *    select(n, i) = v
     * Where
     *    n := store(a, i, v)
     */
    bool solver::assert_store_axiom(app* e) {
        ++m_stats.m_num_store_axiom;
        SASSERT(a.is_store(e));
        unsigned num_args = e->get_num_args();
        ptr_vector<expr> sel_args(num_args - 1, e->get_args());
        sel_args[0] = e;
        expr_ref sel(a.mk_select(sel_args), m);
        euf::enode* n1 = e_internalize(sel);
        euf::enode* n2 = expr2enode(e->get_arg(num_args - 1));
        return ctx.propagate(n1, n2, array_axiom());
    }


    /**
     * Assert
     *   i_k = j_k or select(store(a, i, v), j) = select(a, j)
     * where i = (i_1, ..., i_n), j = (j_1, .., j_n), k in 1..n
     */
    bool solver::assert_select_store_axiom(app* select, app* store) {
        ++m_stats.m_num_select_store_axiom;
        SASSERT(a.is_store(store));
        SASSERT(a.is_select(select));
        SASSERT(store->get_num_args() == 1 + select->get_num_args());
        ptr_buffer<expr> sel1_args, sel2_args;
        unsigned num_args = select->get_num_args();
        sel1_args.push_back(store);
        sel2_args.push_back(store->get_arg(0));

        for (unsigned i = 1; i < num_args; i++) {
            sel1_args.push_back(select->get_arg(i));
            sel2_args.push_back(select->get_arg(i));
        }

        expr_ref sel1(a.mk_select(sel1_args), m);
        expr_ref sel2(a.mk_select(sel2_args), m);
        expr_ref sel_eq_e(m.mk_eq(sel1, sel2), m);
        euf::enode* s1 = e_internalize(sel1);
        euf::enode* s2 = e_internalize(sel2);
        if (s1->get_root() == s2->get_root())
            return false;
        sat::literal sel_eq = b_internalize(sel_eq_e);
        if (s().value(sel_eq) == l_true)
            return false;
        
        bool new_prop = false;
        for (unsigned i = 1; i < num_args; i++) {
            expr* idx1 = store->get_arg(i);
            expr* idx2 = select->get_arg(i);
            euf::enode* r1 = expr2enode(idx1)->get_root();
            euf::enode* r2 = expr2enode(idx2)->get_root();
            if (r1 == r2)
                continue;
            if (m.are_distinct(r1->get_expr(), r2->get_expr())) {
                new_prop = true;
                add_clause(sel_eq);
                break;
            }
            sat::literal idx_eq = b_internalize(m.mk_eq(idx1, idx2));
            if (add_clause(idx_eq, sel_eq))
                new_prop = true;
        }
        return new_prop;
    }

    /**
     * Assert
     *    select(const(v), i) = v
     */
    bool solver::assert_select_const_axiom(app* select, app* cnst) {
        ++m_stats.m_num_select_const_axiom;
        expr* val = nullptr;
        VERIFY(a.is_const(cnst, val));
        SASSERT(a.is_select(select));
        unsigned num_args = select->get_num_args();
        ptr_vector<expr> sel_args(num_args, select->get_args());
        sel_args[0] = cnst;
        expr_ref sel(a.mk_select(sel_args), m);
        euf::enode* n1 = e_internalize(sel);
        euf::enode* n2 = expr2enode(val);
        return ctx.propagate(n1, n2, array_axiom());
    }


    /**
     * e1 = e2 or select(e1, diff(e1,e2)) != select(e2, diff(e1, e2))
     */
    bool solver::assert_extensionality(expr* e1, expr* e2) {
        ++m_stats.m_num_extensionality_axiom;
        func_decl_ref_vector* funcs = nullptr;
        VERIFY(m_sort2diff.find(m.get_sort(e1), funcs));
        expr_ref_vector args1(m), args2(m);
        args1.push_back(e1);
        args2.push_back(e2);
        for (func_decl* f : *funcs) {
            expr* k = m.mk_app(f, e1, e2);
            args1.push_back(k);
            args2.push_back(k);
        }
        expr_ref sel1(a.mk_select(args1), m);
        expr_ref sel2(a.mk_select(args2), m);
        expr_ref n1_eq_n2(m.mk_eq(e1, e2), m);
        expr_ref sel1_eq_sel2(m.mk_eq(sel1, sel2), m);
        literal lit1 = b_internalize(n1_eq_n2);
        literal lit2 = b_internalize(sel1_eq_sel2);
        return add_clause(lit1, ~lit2);
    }

    /**
    * Assert axiom:
    * select(map[f](a, ... d), i) = f(select(a,i),...,select(d,i))
    */
    bool solver::assert_select_map_axiom(app* select, app* map) {
        ++m_stats.m_num_select_map_axiom;
        SASSERT(a.is_map(map));
        SASSERT(a.is_select(select));
        SASSERT(map->get_num_args() > 0);
        func_decl* f = a.get_map_func_decl(map);
        unsigned num_args = select->get_num_args();
        ptr_buffer<expr> args1, args2;
        vector<ptr_vector<expr> > args2l;
        args1.push_back(map);
        for (expr* ar : *map) {
            ptr_vector<expr> arg;
            arg.push_back(ar);
            args2l.push_back(arg);
        }
        for (unsigned i = 1; i < num_args; ++i) {
            expr* arg = select->get_arg(i);
            for (auto& args : args2l)
                args.push_back(arg);
            args1.push_back(arg);
        }
        for (auto const& args : args2l)
            args2.push_back(a.mk_select(args));

        expr_ref sel1(m), sel2(m);
        sel1 = a.mk_select(args1);
        sel2 = m.mk_app(f, args2);
        rewrite(sel2); 
        euf::enode* n1 = e_internalize(sel1);
        euf::enode* n2 = e_internalize(sel2);
        return ctx.propagate(n1, n2, array_axiom());
   }


    /**
     * Assert axiom:
     * select(as-array f, i_1, ..., i_n) = (f i_1 ... i_n)
     */
    bool solver::assert_select_as_array_axiom(app* select, app* arr) {
        ++m_stats.m_num_select_as_array_axiom;
        SASSERT(a.is_as_array(arr));
        SASSERT(a.is_select(select));
        unsigned num_args = select->get_num_args();
        func_decl* f = a.get_as_array_func_decl(arr);
        ptr_vector<expr> sel_args(num_args, select->get_args());
        sel_args[0] = arr;
        expr_ref sel(a.mk_select(sel_args), m);
        expr_ref val(m.mk_app(f, sel_args.size() - 1, sel_args.c_ptr() + 1), m);
        euf::enode* n1 = e_internalize(sel);
        euf::enode* n2 = e_internalize(val);
        return ctx.propagate(n1, n2, array_axiom());
    }

    /**
     * Assert:
     *    default(map[f](a,..,d)) = f(default(a),..,default(d))
     */
    bool solver::assert_default_map_axiom(app* map) {
        ++m_stats.m_num_default_map_axiom;
        SASSERT(a.is_map(map));
        func_decl* f = a.get_map_func_decl(map);
        SASSERT(map->get_num_args() == f->get_arity());
        expr_ref_vector args2(m);
        for (expr* arg : *map)
            args2.push_back(a.mk_default(arg));
        expr_ref def1(a.mk_default(map), m);
        expr_ref def2(m.mk_app(f, args2), m);
        rewrite(def2);
        return ctx.propagate(e_internalize(def1), e_internalize(def2), array_axiom());
    }

    /**
     * Assert:
     *    default(const(e)) = e
     */
    bool solver::assert_default_const_axiom(app* cnst) {
        ++m_stats.m_num_default_const_axiom;
        expr* val = nullptr;
        VERIFY(a.is_const(cnst, val));
        expr_ref def(a.mk_default(cnst), m);
        return ctx.propagate(expr2enode(val), e_internalize(def), array_axiom());
    }


    /**
     * let n := store(a, i, v)
     * Assert:
     * - when sort(n) has exactly one element:
     *      default(n) = v
     * - for small domains:
     *     default(n) = ite(epsilon1 = i, v, default(a))
           n[diag(i)] = a[diag(i)]
     * - for large domains:
     *     default(n) = default(a)
     */
    bool solver::assert_default_store_axiom(app* store) {
        ++m_stats.m_num_default_store_axiom;
        SASSERT(a.is_store(store));
        SASSERT(store->get_num_args() >= 3);
        expr_ref def1(m), def2(m);
        bool prop = false;
        unsigned num_args = store->get_num_args();
        def1 = a.mk_default(store);
        def2 = a.mk_default(store->get_arg(0));

        if (has_unitary_domain(store)) {
            def2 = store->get_arg(num_args - 1);
        }
        else if (!has_large_domain(store)) {
            //
            // let A = store(B, i, v)
            // 
            // Add:
            //   default(A) = ite(epsilon1 = i, v, default(B))
            //   A[diag(i)] = B[diag(i)]
            // 
            expr_ref_vector eqs(m);
            expr_ref_vector args1(m), args2(m);
            args1.push_back(store->get_arg(0));
            args2.push_back(store);

            for (unsigned i = 1; i + 1 < num_args; ++i) {
                expr* arg = store->get_arg(i);
                sort* srt = m.get_sort(arg);
                auto ep = mk_epsilon(srt);
                eqs.push_back(m.mk_eq(ep.first, arg));
                args1.push_back(m.mk_app(ep.second, arg));
                args2.push_back(m.mk_app(ep.second, arg));
            }
            expr_ref eq(m.mk_and(eqs), m);
            def2 = m.mk_ite(eq, store->get_arg(num_args - 1), def2);
            app_ref sel1(m), sel2(m);
            sel1 = a.mk_select(args1);
            sel2 = a.mk_select(args2);
            if (ctx.propagate(e_internalize(sel1), e_internalize(sel2), array_axiom()))
                prop = true;
        }
        if (ctx.propagate(e_internalize(def1), e_internalize(def2), array_axiom()))
            prop = true;
        return prop;
    }

    /**
    * Assert select(lambda xs . M, N1,.., Nk) -> M[N1/x1, ..., Nk/xk]
    */
    bool solver::assert_select_lambda_axiom(app* select, expr* lambda) {
        ++m_stats.m_num_select_lambda_axiom;
        SASSERT(is_lambda(lambda));
        SASSERT(a.is_select(select));
        SASSERT(m.get_sort(lambda) == m.get_sort(select->get_arg(0)));
        ptr_vector<expr> args(select->get_num_args(), select->get_args());
        args[0] = lambda;
        expr_ref alpha(a.mk_select(args), m);
        expr_ref beta(alpha);
        rewrite(beta);
        return ctx.propagate(e_internalize(alpha), e_internalize(beta), array_axiom());
    }

    /**
       \brief assert n1 = n2 => forall vars . (n1 vars) = (n2 vars)
     */
    bool solver::assert_congruent_axiom(expr* e1, expr* e2) {
        ++m_stats.m_num_congruence_axiom;
        sort* srt         = m.get_sort(e1);
        unsigned dimension = get_array_arity(srt);
        expr_ref n1_eq_n2(m.mk_eq(e1, e2), m);
        expr_ref_vector args1(m), args2(m);
        args1.push_back(e1);
        args2.push_back(e2);
        svector<symbol> names;
        sort_ref_vector sorts(m);
        for (unsigned i = 0; i < dimension; i++) {
            sort * asrt = get_array_domain(srt, i);
            sorts.push_back(asrt);
            names.push_back(symbol(i));
            expr * k = m.mk_var(dimension - i - 1, asrt);
            args1.push_back(k);
            args2.push_back(k);            
        }
        expr * sel1 = a.mk_select(dimension+1, args1.c_ptr());
        expr * sel2 = a.mk_select(dimension+1, args2.c_ptr());
        expr * eq = m.mk_eq(sel1, sel2);
        expr_ref q(m.mk_forall(dimension, sorts.c_ptr(), names.c_ptr(), eq), m);
        rewrite(q);
        sat::literal fa_eq = b_internalize(q);
        sat::literal neq = b_internalize(n1_eq_n2);        
        return add_clause(~neq, fa_eq);
    }

    bool solver::has_unitary_domain(app* array_term) {
        SASSERT(a.is_array(array_term));
        sort* s = m.get_sort(array_term);
        unsigned dim = get_array_arity(s);
        for (unsigned i = 0; i < dim; ++i) {
            sort* d = get_array_domain(s, i);
            if (d->is_infinite() || d->is_very_big() || 1 != d->get_num_elements().size())
                return false;
        }
        return true;
    }

    bool solver::has_large_domain(expr* array_term) {
        SASSERT(a.is_array(array_term));
        sort* s = m.get_sort(array_term);
        unsigned dim = get_array_arity(s);
        rational sz(1);
        for (unsigned i = 0; i < dim; ++i) {
            sort* d = get_array_domain(s, i);
            if (d->is_infinite() || d->is_very_big()) {
                return true;
            }
            sz *= rational(d->get_num_elements().size(), rational::ui64());
            if (sz >= rational(1 << 14)) {
                return true;
            }
        }
        return false;
    }

    std::pair<app*, func_decl*> solver::mk_epsilon(sort* s) {
        app* eps = nullptr;
        func_decl* diag = nullptr;
        if (!m_sort2epsilon.find(s, eps)) {
            eps = m.mk_fresh_const("epsilon", s);
            ctx.push(ast2ast_trail<euf::solver, sort, app>(m_sort2epsilon, s, eps));
        }
        if (!m_sort2diag.find(s, diag)) {
            diag = m.mk_fresh_func_decl("diag", 1, &s, s);
            ctx.push(ast2ast_trail<euf::solver, sort, func_decl>(m_sort2diag, s, diag));
        }
        return std::make_pair(eps, diag);
    }

    bool solver::add_delayed_axioms() {
        if (!get_config().m_array_delay_exp_axiom)
            return false;
        unsigned num_vars = get_num_vars();
        for (unsigned v = 0; v < num_vars; v++) {
            propagate_parent_select_axioms(v);
            auto& d = get_var_data(v);
            if (d.m_prop_upward) 
                propagate_parent_default(v);            
        }
        return unit_propagate();
    }

    bool solver::add_interface_equalities() {
        sbuffer<theory_var> roots;
        collect_shared_vars(roots);
        bool prop = false;
        for (unsigned i = roots.size(); i-- > 0; ) {
            theory_var v1 = roots[i];
            expr* e1 = var2expr(v1);
            for (unsigned j = i; j-- > 0; ) {
                theory_var v2 = roots[j];
                expr* e2 = var2expr(v2);
                if (m.get_sort(e1) != m.get_sort(e2))
                    continue;
                if (have_different_model_values(v1, v2))
                    continue;
                expr_ref eq(m.mk_eq(e1, e2), m);
                sat::literal lit = b_internalize(eq);
                if (s().value(lit) == l_undef)
                    prop = true;
            }
        }
        return prop;
    }

    void solver::collect_shared_vars(sbuffer<theory_var>& roots) {
        ptr_buffer<euf::enode> to_unmark;
        unsigned num_vars = get_num_vars();
        for (unsigned i = 0; i < num_vars; i++) {
            euf::enode * n = var2enode(i);
            if (!a.is_array(n->get_expr())) {
                continue;
            }
            euf::enode * r = n->get_root();
            if (r->is_marked1()) {
                continue;
            }
            // arrays used as indices in other arrays have to be treated as shared issue #3532, #3529            
            if (ctx.is_shared(r) || is_select_arg(r)) 
                roots.push_back(r->get_th_var(get_id()));
            
            r->mark1();
            to_unmark.push_back(r);            
        }
        TRACE("array", tout << "collecting shared vars...\n" << unsigned_vector(roots.size(), (unsigned*)roots.c_ptr())  << "\n";);
        for (auto* n : to_unmark)
            n->unmark1();
    }

    bool solver::is_select_arg(euf::enode* r) {
        SASSERT(r->is_root());
        for (euf::enode* n : euf::enode_parents(r)) 
            if (a.is_select(n->get_expr())) 
                for (unsigned i = 1; i < n->num_args(); ++i) 
                    if (r == n->get_arg(i)->get_root()) 
                        return true;
        return false;
    }

}