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https://github.com/Z3Prover/z3
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402 lines
14 KiB
C++
402 lines
14 KiB
C++
/**
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Copyright (c) 2018 Microsoft Corporation
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Module Name:
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qe_solve.cpp
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Abstract:
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Light-weight variable solving plugin model for qe-lite and term_graph.
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Author:
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Nikolaj Bjorner (nbjorner), Arie Gurfinkel 2018-6-8
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Revision History:
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--*/
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#include "ast/arith_decl_plugin.h"
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#include "ast/datatype_decl_plugin.h"
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#include "ast/ast_util.h"
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#include "ast/ast_pp.h"
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#include "qe/qe_solve_plugin.h"
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namespace qe {
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expr_ref solve_plugin::operator()(expr* lit) {
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if (m.is_not(lit, lit))
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return solve(lit, false);
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else
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return solve(lit, true);
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}
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class arith_solve_plugin : public solve_plugin {
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arith_util a;
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public:
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arith_solve_plugin(ast_manager& m, is_variable_proc& is_var): solve_plugin(m, m.get_family_id("arith"), is_var), a(m) {}
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typedef std::pair<bool,expr*> signed_expr;
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/**
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*\brief
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* return r * (sum_{(sign,e) \in exprs} sign * e)
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*/
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expr_ref mk_term(bool is_int, rational const& r, bool sign, svector<signed_expr> const& exprs) {
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expr_ref_vector result(m);
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for (auto const& kv : exprs) {
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bool sign2 = kv.first;
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expr* e = kv.second;
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rational r2(r);
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if (sign == sign2) {
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r2.neg();
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}
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if (!r2.is_one()) {
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result.push_back(a.mk_mul(a.mk_numeral(r2, is_int), e));
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}
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else {
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result.push_back(e);
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}
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}
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return a.mk_add_simplify(result);
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}
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/**
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* \brief return true if lhs = rhs can be solved as v = t, where v is a variable.
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*/
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bool solve(expr* lhs, expr* rhs, expr_ref& v, expr_ref& t) {
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if (!a.is_int(lhs) && !a.is_real(rhs)) {
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return false;
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}
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rational a_val;
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bool is_int = a.is_int(lhs);
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svector<signed_expr> todo, done;
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todo.push_back(std::make_pair(true, lhs));
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todo.push_back(std::make_pair(false, rhs));
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while (!todo.empty()) {
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expr* e = todo.back().second;
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bool sign = todo.back().first;
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todo.pop_back();
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if (a.is_add(e)) {
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for (expr* arg : *to_app(e)) {
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todo.push_back(std::make_pair(sign, arg));
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}
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}
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else if (a.is_sub(e)) {
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app* sub = to_app(e);
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todo.push_back(std::make_pair(sign, sub->get_arg(0)));
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for (unsigned i = 1; i < sub->get_num_args(); ++i) {
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todo.push_back(std::make_pair(!sign, sub->get_arg(i)));
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}
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}
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else if (a.is_uminus(e)) {
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todo.push_back(std::make_pair(!sign, to_app(e)->get_arg(0)));
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}
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else if (is_invertible_mul(is_int, e, a_val)) {
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done.append(todo);
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v = e;
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a_val = rational(1)/a_val;
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t = mk_term(is_int, a_val, sign, done);
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TRACE("qe", tout << mk_pp(lhs, m) << " " << mk_pp(rhs, m) << " " << mk_pp(e, m) << " := " << t << "\n";);
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return true;
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}
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else {
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done.push_back(std::make_pair(sign, e));
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}
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}
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return false;
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}
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// is x a constant and can we safely divide by x to solve for a variable?
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bool is_invertible_const(bool is_int, expr* x, rational& a_val) {
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expr* y;
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if (a.is_uminus(x, y) && is_invertible_const(is_int, y, a_val)) {
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a_val.neg();
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return true;
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}
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else if (a.is_numeral(x, a_val) && !a_val.is_zero()) {
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if (!is_int || a_val.is_one() || a_val.is_minus_one()) {
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return true;
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}
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}
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return false;
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}
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// is arg of the form a_val * v, where a_val
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// is a constant that we can safely divide by.
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bool is_invertible_mul(bool is_int, expr*& arg, rational& a_val) {
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if (is_variable(arg)) {
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a_val = rational(1);
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return true;
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}
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expr* x, *y;
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if (a.is_mul(arg, x, y)) {
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if (is_variable(x) && is_invertible_const(is_int, y, a_val)) {
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arg = x;
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return true;
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}
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if (is_variable(y) && is_invertible_const(is_int, x, a_val)) {
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arg = y;
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return true;
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}
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}
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return false;
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}
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expr_ref mk_eq_core (expr *e1, expr *e2) {
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expr_ref v(m), t(m);
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if (solve(e1, e2, v, t)) {
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return expr_ref(m.mk_eq(v, t), m);
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}
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if (a.is_zero(e1)) {
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std::swap(e1, e2);
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}
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// y + -1*x == 0 --> y = x
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expr *a0 = nullptr, *a1 = nullptr, *x = nullptr;
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if (a.is_zero(e2) && a.is_add(e1, a0, a1)) {
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if (a.is_times_minus_one(a1, x)) {
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e1 = a0;
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e2 = x;
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}
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else if (a.is_times_minus_one(a0, x)) {
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e1 = a1;
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e2 = x;
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}
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}
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return expr_ref(m.mk_eq(e1, e2), m);
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}
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app* mk_le_zero(expr *arg) {
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expr *e1, *e2, *e3;
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if (a.is_add(arg, e1, e2)) {
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// e1-e2<=0 --> e1<=e2
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if (a.is_times_minus_one(e2, e3)) {
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return a.mk_le(e1, e3);
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}
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// -e1+e2<=0 --> e2<=e1
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else if (a.is_times_minus_one(e1, e3)) {
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return a.mk_le(e2, e3);
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}
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}
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return a.mk_le(arg, mk_zero());
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}
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app* mk_ge_zero(expr *arg) {
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expr *e1, *e2, *e3;
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if (a.is_add(arg, e1, e2)) {
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// e1-e2>=0 --> e1>=e2
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if (a.is_times_minus_one(e2, e3)) {
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return a.mk_ge(e1, e3);
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}
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// -e1+e2>=0 --> e2>=e1
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else if (a.is_times_minus_one(e1, e3)) {
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return a.mk_ge(e2, e3);
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}
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}
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return a.mk_ge(arg, mk_zero());
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}
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bool mk_le_core (expr *arg1, expr * arg2, expr_ref &result) {
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// t <= -1 ==> t < 0 ==> ! (t >= 0)
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rational n;
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if (a.is_int (arg1) && a.is_minus_one (arg2)) {
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result = m.mk_not (mk_ge_zero (arg1));
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return true;
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}
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else if (a.is_zero(arg2)) {
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result = mk_le_zero(arg1);
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return true;
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}
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else if (a.is_int(arg1) && a.is_numeral(arg2, n) && n < 0) {
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// t <= n ==> t < n + 1 ==> ! (t >= n + 1)
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result = m.mk_not(a.mk_ge(arg1, a.mk_numeral(n+1, true)));
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return true;
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}
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return false;
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}
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expr * mk_zero () {
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return a.mk_numeral (rational (0), true);
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}
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bool is_one (expr const * n) const {
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rational val;
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return a.is_numeral (n, val) && val.is_one ();
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}
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bool mk_ge_core (expr * arg1, expr * arg2, expr_ref &result) {
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// t >= 1 ==> t > 0 ==> ! (t <= 0)
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rational n;
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if (a.is_int (arg1) && is_one (arg2)) {
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result = m.mk_not (mk_le_zero (arg1));
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return true;
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}
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else if (a.is_zero(arg2)) {
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result = mk_ge_zero(arg1);
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return true;
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}
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else if (a.is_int(arg1) && a.is_numeral(arg2, n) && n > 0) {
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// t >= n ==> t > n - 1 ==> ! (t <= n - 1)
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result = m.mk_not(a.mk_le(arg1, a.mk_numeral(n-1, true)));
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return true;
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}
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return false;
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}
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expr_ref solve(expr* atom, bool is_pos) override {
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expr *e1, *e2;
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expr_ref res(atom, m);
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if (m.is_eq (atom, e1, e2)) {
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expr_ref v(m), t(m);
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v = e1; t = e2;
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// -- attempt to solve using arithmetic
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solve(e1, e2, v, t);
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// -- normalize equality
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res = mk_eq_core(v, t);
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}
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else if (a.is_le(atom, e1, e2)) {
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mk_le_core(e1, e2, res);
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}
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else if (a.is_ge(atom, e1, e2)) {
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mk_ge_core(e1, e2, res);
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}
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// restore negation
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if (!is_pos) {
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res = mk_not(m, res);
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}
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return res;
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}
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};
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class basic_solve_plugin : public solve_plugin {
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public:
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basic_solve_plugin(ast_manager& m, is_variable_proc& is_var):
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solve_plugin(m, m.get_basic_family_id(), is_var) {}
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expr_ref solve(expr *atom, bool is_pos) override {
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expr_ref res(atom, m);
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expr* lhs = nullptr, *rhs = nullptr, *n = nullptr;
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if (m.is_eq(atom, lhs, rhs)) {
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if (m.is_not(lhs, n) && is_variable(n)) {
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res = m.mk_eq(n, mk_not(m, rhs));
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}
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else if (m.is_not(rhs, n) && is_variable(n)) {
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res = m.mk_eq(n, mk_not(m, lhs));
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}
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else if (is_variable(rhs) && !is_variable(lhs)) {
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res = m.mk_eq(rhs, lhs);
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}
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}
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// (ite cond (= VAR t) (= VAR t2)) case
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expr* cond = nullptr, *th = nullptr, *el = nullptr;
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if (m.is_ite(atom, cond, th, el)) {
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expr_ref r1 = solve(th, true);
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expr_ref r2 = solve(el, true);
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expr* v1 = nullptr, *t1 = nullptr, *v2 = nullptr, *t2 = nullptr;
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if (m.is_eq(r1, v1, t1) && m.is_eq(r2, v2, t2) && v1 == v2) {
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res = m.mk_eq(v1, m.mk_ite(cond, t1, t2));
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}
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}
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if (is_variable(atom) && m.is_bool(atom)) {
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res = m.mk_eq(atom, m.mk_bool_val(is_pos));
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return res;
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}
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return is_pos ? res : mk_not(res);
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}
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};
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class dt_solve_plugin : public solve_plugin {
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datatype_util dt;
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public:
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dt_solve_plugin(ast_manager& m, is_variable_proc& is_var):
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solve_plugin(m, m.get_family_id("datatype"), is_var),
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dt(m) {}
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expr_ref solve(expr *atom, bool is_pos) override {
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expr_ref res(atom, m);
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expr* lhs = nullptr, *rhs = nullptr;
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if (m.is_eq(atom, lhs, rhs)) {
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if (dt.is_constructor(rhs)) {
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std::swap(lhs, rhs);
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}
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if (dt.is_constructor(lhs) && dt.is_constructor(rhs)) {
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app* l = to_app(lhs), *r = to_app(rhs);
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if (l->get_decl() == r->get_decl()) {
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expr_ref_vector eqs(m);
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for (unsigned i = 0, sz = l->get_num_args(); i < sz; ++i) {
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eqs.push_back(m.mk_eq(l->get_arg(i), r->get_arg(i)));
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}
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res = mk_and(eqs);
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}
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else {
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res = m.mk_false();
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}
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}
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else if (dt.is_constructor(lhs)) {
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app* l = to_app(lhs);
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func_decl* d = l->get_decl();
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expr_ref_vector conjs(m);
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conjs.push_back(dt.mk_is(d, rhs));
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ptr_vector<func_decl> const& acc = *dt.get_constructor_accessors(d);
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for (unsigned i = 0; i < acc.size(); ++i) {
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conjs.push_back(m.mk_eq(l->get_arg(i), m.mk_app(acc[i], rhs)));
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}
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res = mk_and(conjs);
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}
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}
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// TBD: can also solve for is_nil(x) by x = nil
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//
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return is_pos ? res : mk_not(res);
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}
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};
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class bv_solve_plugin : public solve_plugin {
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public:
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bv_solve_plugin(ast_manager& m, is_variable_proc& is_var): solve_plugin(m, m.get_family_id("bv"), is_var) {}
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expr_ref solve(expr *atom, bool is_pos) override {
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expr_ref res(atom, m);
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return is_pos ? res : mk_not(res);
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}
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};
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class array_solve_plugin : public solve_plugin {
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public:
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array_solve_plugin(ast_manager& m, is_variable_proc& is_var): solve_plugin(m, m.get_family_id("array"), is_var) {}
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expr_ref solve(expr *atom, bool is_pos) override {
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expr_ref res(atom, m);
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return is_pos ? res : mk_not(res);
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}
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};
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solve_plugin* mk_basic_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
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return alloc(basic_solve_plugin, m, is_var);
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}
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solve_plugin* mk_arith_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
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return alloc(arith_solve_plugin, m, is_var);
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}
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solve_plugin* mk_dt_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
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return alloc(dt_solve_plugin, m, is_var);
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}
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#if 0
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solve_plugin* mk_bv_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
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return alloc(bv_solve_plugin, m, is_var);
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}
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solve_plugin* mk_array_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
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return alloc(array_solve_plugin, m, is_var);
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}
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#endif
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}
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