z3-z3-4.13.0.src.qe.mbp.mbp_solve_plugin.cpp Maven / Gradle / Ivy
The newest version!
/**
Copyright (c) 2018 Microsoft Corporation
Module Name:
qe_solve.cpp
Abstract:
Light-weight variable solving plugin model for qe-lite and term_graph.
Author:
Nikolaj Bjorner (nbjorner), Arie Gurfinkel 2018-6-8
Revision History:
--*/
#include "ast/arith_decl_plugin.h"
#include "ast/bv_decl_plugin.h"
#include "ast/datatype_decl_plugin.h"
#include "ast/ast_util.h"
#include "ast/ast_pp.h"
#include "qe/mbp/mbp_solve_plugin.h"
namespace mbp {
expr_ref solve_plugin::operator()(expr* lit) {
if (m.is_not(lit, lit))
return solve(lit, false);
else
return solve(lit, true);
}
class arith_solve_plugin : public solve_plugin {
arith_util a;
public:
arith_solve_plugin(ast_manager& m, is_variable_proc& is_var): solve_plugin(m, m.get_family_id("arith"), is_var), a(m) {}
typedef std::pair signed_expr;
/**
*\brief
* return r * (sum_{(sign,e) \in exprs} sign * e)
*/
expr_ref mk_term(bool is_int, rational const& r, bool sign, svector const& exprs) {
expr_ref_vector result(m);
for (auto const& kv : exprs) {
bool sign2 = kv.first;
expr* e = kv.second;
rational r2(r);
if (sign == sign2) {
r2.neg();
}
if (!r2.is_one()) {
result.push_back(a.mk_mul(a.mk_numeral(r2, is_int), e));
}
else {
result.push_back(e);
}
}
return a.mk_add_simplify(result);
}
/**
* \brief return true if lhs = rhs can be solved as v = t, where v is a variable.
*/
bool solve(expr* lhs, expr* rhs, expr_ref& v, expr_ref& t) {
if (!a.is_int(lhs) && !a.is_real(rhs)) {
return false;
}
rational a_val;
bool is_int = a.is_int(lhs);
svector todo, done;
todo.push_back(std::make_pair(true, lhs));
todo.push_back(std::make_pair(false, rhs));
while (!todo.empty()) {
expr* e = todo.back().second;
bool sign = todo.back().first;
todo.pop_back();
if (a.is_add(e)) {
for (expr* arg : *to_app(e)) {
todo.push_back(std::make_pair(sign, arg));
}
}
else if (a.is_sub(e)) {
app* sub = to_app(e);
todo.push_back(std::make_pair(sign, sub->get_arg(0)));
for (unsigned i = 1; i < sub->get_num_args(); ++i) {
todo.push_back(std::make_pair(!sign, sub->get_arg(i)));
}
}
else if (a.is_uminus(e)) {
todo.push_back(std::make_pair(!sign, to_app(e)->get_arg(0)));
}
else if (is_invertible_mul(is_int, e, a_val)) {
done.append(todo);
v = e;
a_val = rational(1)/a_val;
t = mk_term(is_int, a_val, sign, done);
TRACE("qe", tout << mk_pp(e, m) << " := " << t << "\n";);
return true;
}
else {
done.push_back(std::make_pair(sign, e));
}
}
return false;
}
// is x a constant and can we safely divide by x to solve for a variable?
bool is_invertible_const(bool is_int, expr* x, rational& a_val) {
expr* y;
if (a.is_uminus(x, y) && is_invertible_const(is_int, y, a_val)) {
a_val.neg();
return true;
}
else if (a.is_numeral(x, a_val) && !a_val.is_zero()) {
if (!is_int || a_val.is_one() || a_val.is_minus_one()) {
return true;
}
}
return false;
}
// is arg of the form a_val * v, where a_val
// is a constant that we can safely divide by.
bool is_invertible_mul(bool is_int, expr*& arg, rational& a_val) {
if (is_variable(arg)) {
a_val = rational(1);
return true;
}
expr* x, *y;
if (a.is_mul(arg, x, y)) {
if (is_variable(x) && is_invertible_const(is_int, y, a_val)) {
arg = x;
return true;
}
if (is_variable(y) && is_invertible_const(is_int, x, a_val)) {
arg = y;
return true;
}
}
return false;
}
expr_ref mk_eq_core(expr *e1, expr *e2) {
expr_ref v(m), t(m);
if (solve(e1, e2, v, t)) {
return expr_ref(m.mk_eq(v, t), m);
}
if (a.is_zero(e1)) {
std::swap(e1, e2);
}
// y + -1*x == 0 --> y = x
expr *a0 = nullptr, *a1 = nullptr, *x = nullptr;
if (a.is_zero(e2) && a.is_add(e1, a0, a1)) {
if (a.is_times_minus_one(a1, x)) {
e1 = a0;
e2 = x;
}
else if (a.is_times_minus_one(a0, x)) {
e1 = a1;
e2 = x;
}
}
return expr_ref(m.mk_eq(e1, e2), m);
}
app* mk_le_zero(expr *arg) {
expr *e1, *e2, *e3;
if (a.is_add(arg, e1, e2)) {
// e1-e2<=0 --> e1<=e2
if (a.is_times_minus_one(e2, e3)) {
return a.mk_le(e1, e3);
}
// -e1+e2<=0 --> e2<=e1
else if (a.is_times_minus_one(e1, e3)) {
return a.mk_le(e2, e3);
}
}
return a.mk_le(arg, mk_zero());
}
app* mk_ge_zero(expr *arg) {
expr *e1, *e2, *e3;
if (a.is_add(arg, e1, e2)) {
// e1-e2>=0 --> e1>=e2
if (a.is_times_minus_one(e2, e3)) {
return a.mk_ge(e1, e3);
}
// -e1+e2>=0 --> e2>=e1
else if (a.is_times_minus_one(e1, e3)) {
return a.mk_ge(e2, e3);
}
}
return a.mk_ge(arg, mk_zero());
}
bool mk_le_core(expr *arg1, expr * arg2, expr_ref &result) {
// t <= -1 ==> t < 0 ==> ! (t >= 0)
rational n;
if (a.is_int (arg1) && a.is_minus_one (arg2)) {
result = m.mk_not (mk_ge_zero (arg1));
return true;
}
else if (a.is_zero(arg2)) {
result = mk_le_zero(arg1);
return true;
}
else if (a.is_int(arg1) && a.is_numeral(arg2, n) && n < 0) {
// t <= n ==> t < n + 1 ==> ! (t >= n + 1)
result = m.mk_not(a.mk_ge(arg1, a.mk_numeral(n+1, true)));
return true;
}
return false;
}
expr * mk_zero() {
return a.mk_numeral (rational (0), true);
}
bool is_one(expr const * n) const {
rational val;
return a.is_numeral (n, val) && val.is_one ();
}
bool mk_ge_core(expr * arg1, expr * arg2, expr_ref &result) {
// t >= 1 ==> t > 0 ==> ! (t <= 0)
rational n;
if (a.is_int (arg1) && is_one (arg2)) {
result = m.mk_not (mk_le_zero (arg1));
return true;
}
else if (a.is_zero(arg2)) {
result = mk_ge_zero(arg1);
return true;
}
else if (a.is_int(arg1) && a.is_numeral(arg2, n) && n > 0) {
// t >= n ==> t > n - 1 ==> ! (t <= n - 1)
result = m.mk_not(a.mk_le(arg1, a.mk_numeral(n-1, true)));
return true;
}
return false;
}
// returns `true` if a rewriting happened
bool try_int_mul_solve(expr *atom, bool is_pos, expr_ref &res) {
if (!is_pos)
return false; // negation of multiplication is not a cube for
// integers
// we want k*y == x -----> y = x div k && x mod k == 0
expr *lhs = nullptr, *rhs = nullptr;
if (!m.is_eq(atom, lhs, rhs)) return false;
if (!a.is_int(lhs)) { return false; }
if (!a.is_mul(rhs)) {
if (a.is_mul(lhs))
std::swap(lhs, rhs);
else
return false; // no muls
}
// of the form v = k*expr
expr *first = nullptr, *second = nullptr;
if (!a.is_mul(rhs, first, second)) return false;
if (!(is_app(first) && a.plugin().is_value(to_app(first)))) {
if (is_app(second) && a.plugin().is_value(to_app(second))) {
std::swap(first, second);
} else {
return false;
}
}
if (a.is_zero(first)) {
// SASSERT(a.is_int(lhs));
res = m.mk_eq(lhs, a.mk_int(0));
return true;
};
// `first` is a value, different from 0
res = m.mk_and(m.mk_eq(second, a.mk_idiv(lhs, first)),
m.mk_eq(a.mk_int(0), a.mk_mod(lhs, first)));
return true;
}
expr_ref solve(expr* atom, bool is_pos) override {
expr_ref res(atom, m);
if (try_int_mul_solve(atom, is_pos, res)) return res;
expr *e1, *e2;
if (m.is_eq(atom, e1, e2)) {
expr_ref v(m), t(m);
v = e1;
t = e2;
// -- attempt to solve using arithmetic
solve(e1, e2, v, t);
// -- normalize equality
res = mk_eq_core(v, t);
}
else if (a.is_le(atom, e1, e2)) {
mk_le_core(e1, e2, res);
}
else if (a.is_ge(atom, e1, e2)) {
mk_ge_core(e1, e2, res);
}
// restore negation
if (!is_pos) {
res = mk_not(m, res);
}
return res;
}
};
class basic_solve_plugin : public solve_plugin {
public:
basic_solve_plugin(ast_manager& m, is_variable_proc& is_var):
solve_plugin(m, m.get_basic_family_id(), is_var) {}
expr_ref solve(expr *atom, bool is_pos) override {
expr_ref res(atom, m);
expr* lhs = nullptr, *rhs = nullptr, *n = nullptr;
if (m.is_eq(atom, lhs, rhs)) {
if (m.is_not(lhs, n) && is_variable(n)) {
res = m.mk_eq(n, mk_not(m, rhs));
}
else if (m.is_not(rhs, n) && is_variable(n)) {
res = m.mk_eq(n, mk_not(m, lhs));
}
else if (is_variable(rhs) && !is_variable(lhs)) {
res = m.mk_eq(rhs, lhs);
}
}
// (ite cond (= VAR t) (= VAR t2)) case
expr* cond = nullptr, *th = nullptr, *el = nullptr;
if (m.is_ite(atom, cond, th, el)) {
expr_ref r1 = solve(th, true);
expr_ref r2 = solve(el, true);
expr* v1 = nullptr, *t1 = nullptr, *v2 = nullptr, *t2 = nullptr;
if (m.is_eq(r1, v1, t1) && m.is_eq(r2, v2, t2) && v1 == v2) {
res = m.mk_eq(v1, m.mk_ite(cond, t1, t2));
}
}
if (is_variable(atom) && m.is_bool(atom)) {
res = m.mk_eq(atom, m.mk_bool_val(is_pos));
return res;
}
return is_pos ? res : mk_not(res);
}
};
class dt_solve_plugin : public solve_plugin {
datatype_util dt;
public:
dt_solve_plugin(ast_manager& m, is_variable_proc& is_var):
solve_plugin(m, m.get_family_id("datatype"), is_var),
dt(m) {}
expr_ref solve(expr *atom, bool is_pos) override {
expr_ref res(atom, m);
expr* lhs = nullptr, *rhs = nullptr;
if (m.is_eq(atom, lhs, rhs)) {
if (dt.is_constructor(rhs)) {
std::swap(lhs, rhs);
}
if (dt.is_constructor(lhs) && dt.is_constructor(rhs)) {
app* l = to_app(lhs), *r = to_app(rhs);
if (l->get_decl() == r->get_decl()) {
expr_ref_vector eqs(m);
for (unsigned i = 0, sz = l->get_num_args(); i < sz; ++i) {
eqs.push_back(m.mk_eq(l->get_arg(i), r->get_arg(i)));
}
res = mk_and(eqs);
}
else {
res = m.mk_false();
}
}
else if (dt.is_constructor(lhs)) {
app* l = to_app(lhs);
func_decl* d = l->get_decl();
expr_ref_vector conjs(m);
conjs.push_back(dt.mk_is(d, rhs));
ptr_vector const& acc = *dt.get_constructor_accessors(d);
for (unsigned i = 0; i < acc.size(); ++i) {
conjs.push_back(m.mk_eq(l->get_arg(i), m.mk_app(acc[i], rhs)));
}
res = mk_and(conjs);
}
}
// TBD: can also solve for is_nil(x) by x = nil
//
return is_pos ? res : mk_not(res);
}
};
class bv_solve_plugin : public solve_plugin {
bv_util m_bv;
bool solve_eq(expr*& lhs, expr*& rhs) {
unsigned lo, hi;
expr* e = nullptr;
if (m_bv.is_extract(lhs, lo, hi, e) && is_variable(e)) {
lhs = e;
unsigned sz = m_bv.get_bv_size(e);
if (lo > 0 && hi + 1 < sz) {
expr* args[3] = { m_bv.mk_extract(sz-1, hi + 1, e), rhs, m_bv.mk_extract(lo - 1, 0, e)};
rhs = m_bv.mk_concat(3, args);
}
else if (lo == 0 && hi + 1 < sz) {
expr* args[2] = { m_bv.mk_extract(sz-1, hi + 1, e), rhs };
rhs = m_bv.mk_concat(2, args);
}
else if (lo > 0 && hi + 1 == sz) {
expr* args[2] = { rhs, m_bv.mk_extract(lo - 1, 0, e) };
rhs = m_bv.mk_concat(2, args);
}
else {
return false;
}
return true;
}
#if 0
expr *f = nullptr
// this one is for Nuno to find and generalize for invertible expressions
if (m_bv.is_bv_add(lhs, e, f) && is_variable(e)) {
lhs = e;
rhs = m_bv.mk_bv_sub(rhs, f);
return true;
}
#endif
return false;
}
public:
bv_solve_plugin(ast_manager& m, is_variable_proc& is_var):
solve_plugin(m, m.get_family_id("bv"), is_var), m_bv(m) {}
expr_ref solve(expr *atom, bool is_pos) override {
expr_ref res(atom, m);
expr* lhs = nullptr, *rhs = nullptr;
if (is_pos && m.is_eq(atom, lhs, rhs) && solve_eq(lhs, rhs)) {
res = m.mk_eq(lhs, rhs);
return res;
}
if (is_pos && m.is_eq(atom, lhs, rhs) && solve_eq(rhs, lhs)) {
res = m.mk_eq(rhs, lhs);
return res;
}
return is_pos ? res : mk_not(res);
}
};
class array_solve_plugin : public solve_plugin {
public:
array_solve_plugin(ast_manager& m, is_variable_proc& is_var): solve_plugin(m, m.get_family_id("array"), is_var) {}
expr_ref solve(expr *atom, bool is_pos) override {
expr_ref res(atom, m);
return is_pos ? res : mk_not(res);
}
};
solve_plugin* mk_basic_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
return alloc(basic_solve_plugin, m, is_var);
}
solve_plugin* mk_arith_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
return alloc(arith_solve_plugin, m, is_var);
}
solve_plugin* mk_dt_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
return alloc(dt_solve_plugin, m, is_var);
}
solve_plugin* mk_bv_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
return alloc(bv_solve_plugin, m, is_var);
}
#if 0
solve_plugin* mk_array_solve_plugin(ast_manager& m, is_variable_proc& is_var) {
return alloc(array_solve_plugin, m, is_var);
}
#endif
}