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z3/src/qe/qe_arith.cpp
Nikolaj Bjorner 7fc294d329 move arithmetical mbp functionality to model_based_opt
Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
2016-06-26 14:30:35 -07:00

612 lines
23 KiB
C++

/*++
Copyright (c) 2015 Microsoft Corporation
Module Name:
qe_arith.cpp
Abstract:
Simple projection function for real arithmetic based on Loos-W.
Author:
Nikolaj Bjorner (nbjorner) 2013-09-12
Revision History:
Moved projection functionality to model_based_opt module. 2016-06-26
--*/
#include "qe_arith.h"
#include "qe_mbp.h"
#include "ast_util.h"
#include "arith_decl_plugin.h"
#include "ast_pp.h"
#include "model_v2_pp.h"
#include "th_rewriter.h"
#include "expr_functors.h"
#include "expr_safe_replace.h"
#include "model_based_opt.h"
#include "model_evaluator.h"
namespace qe {
bool is_divides(arith_util& a, expr* e1, expr* e2, rational& k, expr_ref& p) {
expr* t1, *t2;
if (a.is_mod(e2, t1, t2) &&
a.is_numeral(e1, k) &&
k.is_zero() &&
a.is_numeral(t2, k)) {
p = t1;
return true;
}
return false;
}
bool is_divides(arith_util& a, expr* e, rational& k, expr_ref& t) {
expr* e1, *e2;
if (!a.get_manager().is_eq(e, e1, e2)) {
return false;
}
return is_divides(a, e1, e2, k, t) || is_divides(a, e2, e1, k, t);
}
struct arith_project_plugin::imp {
ast_manager& m;
arith_util a;
th_rewriter m_rw;
expr_ref_vector m_trail;
void insert_mul(expr* x, rational const& v, obj_map<expr, rational>& ts) {
TRACE("qe", tout << "Adding variable " << mk_pp(x, m) << " " << v << "\n";);
rational w;
if (ts.find(x, w)) {
ts.insert(x, w + v);
}
else {
ts.insert(x, v);
}
}
//
// extract linear inequalities from literal 'lit' into the model-based optimization manager 'mbo'.
// It uses the current model to choose values for conditionals and it primes mbo with the current
// interpretation of sub-expressions that are treated as variables for mbo.
//
bool linearize(opt::model_based_opt& mbo, model_evaluator& eval, expr* lit, expr_ref_vector& fmls, obj_map<expr, unsigned>& tids) {
obj_map<expr, rational> ts;
rational c(0), mul(1);
expr_ref t(m);
opt::ineq_type ty = opt::t_le;
expr* e1, *e2;
DEBUG_CODE(expr_ref val(m);
eval(lit, val);
CTRACE("qe", !m.is_true(val), tout << mk_pp(lit, m) << " := " << val << "\n";);
SASSERT(m.is_true(val)););
bool is_not = m.is_not(lit, lit);
if (is_not) {
mul.neg();
}
SASSERT(!m.is_not(lit));
if ((a.is_le(lit, e1, e2) || a.is_ge(lit, e2, e1))) {
linearize(mbo, eval, mul, e1, c, fmls, ts, tids);
linearize(mbo, eval, -mul, e2, c, fmls, ts, tids);
ty = is_not ? opt::t_lt : opt::t_le;
}
else if ((a.is_lt(lit, e1, e2) || a.is_gt(lit, e2, e1))) {
linearize(mbo, eval, mul, e1, c, fmls, ts, tids);
linearize(mbo, eval, -mul, e2, c, fmls, ts, tids);
ty = is_not ? opt::t_le: opt::t_lt;
}
else if (m.is_eq(lit, e1, e2) && !is_not && is_arith(e1)) {
linearize(mbo, eval, mul, e1, c, fmls, ts, tids);
linearize(mbo, eval, -mul, e2, c, fmls, ts, tids);
ty = opt::t_eq;
}
else if (m.is_eq(lit, e1, e2) && is_not && is_arith(e1)) {
expr_ref val1(m), val2(m);
rational r1, r2;
eval(e1, val1); eval(e2, val2);
VERIFY(a.is_numeral(val1, r1));
VERIFY(a.is_numeral(val2, r2));
SASSERT(r1 != r2);
if (r1 < r2) {
std::swap(e1, e2);
}
ty = opt::t_lt;
linearize(mbo, eval, mul, e1, c, fmls, ts, tids);
linearize(mbo, eval, -mul, e2, c, fmls, ts, tids);
}
else if (m.is_distinct(lit) && !is_not && is_arith(to_app(lit)->get_arg(0))) {
expr_ref val(m);
rational r;
app* alit = to_app(lit);
vector<std::pair<expr*,rational> > nums;
for (unsigned i = 0; i < alit->get_num_args(); ++i) {
eval(alit->get_arg(i), val);
VERIFY(a.is_numeral(val, r));
nums.push_back(std::make_pair(alit->get_arg(i), r));
}
std::sort(nums.begin(), nums.end(), compare_second());
for (unsigned i = 0; i + 1 < nums.size(); ++i) {
SASSERT(nums[i].second < nums[i+1].second);
expr_ref fml(a.mk_lt(nums[i].first, nums[i+1].first), m);
if (!linearize(mbo, eval, fml, fmls, tids)) {
return false;
}
}
return true;
}
else if (m.is_distinct(lit) && is_not && is_arith(to_app(lit)->get_arg(0))) {
// find the two arguments that are equal.
// linearize these.
map<rational, expr*, rational::hash_proc, rational::eq_proc> values;
bool found_eq = false;
for (unsigned i = 0; !found_eq && i < to_app(lit)->get_num_args(); ++i) {
expr* arg1 = to_app(lit)->get_arg(i), *arg2 = 0;
expr_ref val(m);
rational r;
eval(arg1, val);
VERIFY(a.is_numeral(val, r));
if (values.find(r, arg2)) {
ty = opt::t_eq;
linearize(mbo, eval, mul, arg1, c, fmls, ts, tids);
linearize(mbo, eval, -mul, arg2, c, fmls, ts, tids);
found_eq = true;
}
else {
values.insert(r, arg1);
}
}
SASSERT(found_eq);
}
else {
TRACE("qe", tout << "Skipping " << mk_pp(lit, m) << "\n";);
return false;
}
vars coeffs;
extract_coefficients(mbo, eval, ts, tids, coeffs);
mbo.add_constraint(coeffs, c, ty);
return true;
}
//
// convert linear arithmetic term into an inequality for mbo.
//
void linearize(opt::model_based_opt& mbo, model_evaluator& eval, rational const& mul, expr* t, rational& c,
expr_ref_vector& fmls, obj_map<expr, rational>& ts, obj_map<expr, unsigned>& tids) {
expr* t1, *t2, *t3;
rational mul1;
expr_ref val(m);
if (a.is_mul(t, t1, t2) && is_numeral(eval, t1, mul1)) {
linearize(mbo, eval, mul* mul1, t2, c, fmls, ts, tids);
}
else if (a.is_mul(t, t1, t2) && is_numeral(eval, t2, mul1)) {
linearize(mbo, eval, mul* mul1, t1, c, fmls, ts, tids);
}
else if (a.is_add(t)) {
app* ap = to_app(t);
for (unsigned i = 0; i < ap->get_num_args(); ++i) {
linearize(mbo, eval, mul, ap->get_arg(i), c, fmls, ts, tids);
}
}
else if (a.is_sub(t, t1, t2)) {
linearize(mbo, eval, mul, t1, c, fmls, ts, tids);
linearize(mbo, eval, -mul, t2, c, fmls, ts, tids);
}
else if (a.is_uminus(t, t1)) {
linearize(mbo, eval, -mul, t1, c, fmls, ts, tids);
}
else if (a.is_numeral(t, mul1)) {
c += mul*mul1;
}
else if (m.is_ite(t, t1, t2, t3)) {
eval(t1, val);
SASSERT(m.is_true(val) || m.is_false(val));
TRACE("qe", tout << mk_pp(t1, m) << " := " << val << "\n";);
if (m.is_true(val)) {
linearize(mbo, eval, mul, t2, c, fmls, ts, tids);
fmls.push_back(t1);
}
else {
expr_ref not_t1(mk_not(m, t1), m);
fmls.push_back(not_t1);
linearize(mbo, eval, mul, t3, c, fmls, ts, tids);
}
}
else if (a.is_mod(t, t1, t2) && is_numeral(eval, t2, mul1)) {
rational r;
eval(t, val);
VERIFY(a.is_numeral(val, r));
c += mul*r;
// t1 mod mul1 == r
rational c0(-r), mul0(1);
obj_map<expr, rational> ts0;
linearize(mbo, eval, mul0, t1, c0, fmls, ts0, tids);
vars coeffs;
extract_coefficients(mbo, eval, ts0, tids, coeffs);
mbo.add_divides(coeffs, c0, mul1);
}
else {
insert_mul(t, mul, ts);
}
}
bool is_numeral(model_evaluator& eval, expr* t, rational& r) {
expr* t1, *t2, *t3;
rational r1, r2;
expr_ref val(m);
if (a.is_numeral(t, r)) return true;
if (a.is_uminus(t, t1) && is_numeral(eval, t1, r)) {
r.neg();
return true;
}
else if (a.is_mul(t, t1, t2) && is_numeral(eval, t1, r1) && is_numeral(eval, t2, r2)) {
r = r1*r2;
return true;
}
else if (a.is_add(t)) {
app* ap = to_app(t);
r = rational(1);
for (unsigned i = 0; i < ap->get_num_args(); ++i) {
if (!is_numeral(eval, ap->get_arg(i), r1)) return false;
r *= r1;
}
return true;
}
else if (m.is_ite(t, t1, t2, t3)) {
eval(t1, val);
if (m.is_true(val)) {
return is_numeral(eval, t1, r);
}
else {
return is_numeral(eval, t2, r);
}
}
else if (a.is_sub(t, t1, t2) && is_numeral(eval, t1, r1) && is_numeral(eval, t2, r2)) {
r = r1 - r2;
return true;
}
return false;
}
struct compare_second {
bool operator()(std::pair<expr*, rational> const& a,
std::pair<expr*, rational> const& b) const {
return a.second < b.second;
}
};
bool is_arith(expr* e) {
return a.is_int(e) || a.is_real(e);
}
rational n_sign(rational const& b) {
return rational(b.is_pos()?-1:1);
}
imp(ast_manager& m):
m(m), a(m), m_rw(m), m_trail(m) {
params_ref params;
params.set_bool("gcd_rouding", true);
m_rw.updt_params(params);
}
~imp() {
}
bool solve(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
return false;
}
bool operator()(model& model, app* v, app_ref_vector& vars, expr_ref_vector& lits) {
app_ref_vector vs(m);
vs.push_back(v);
(*this)(model, vs, lits);
return vs.empty();
}
typedef opt::model_based_opt::var var;
typedef opt::model_based_opt::row row;
typedef vector<var> vars;
void operator()(model& model, app_ref_vector& vars, expr_ref_vector& fmls) {
bool has_arith = false;
for (unsigned i = 0; !has_arith && i < vars.size(); ++i) {
expr* v = vars[i].get();
has_arith |= is_arith(v);
}
if (!has_arith) {
return;
}
model_evaluator eval(model);
// eval.set_model_completion(true);
opt::model_based_opt mbo;
obj_map<expr, unsigned> tids;
m_trail.reset();
unsigned j = 0;
for (unsigned i = 0; i < fmls.size(); ++i) {
expr* fml = fmls[i].get();
if (!linearize(mbo, eval, fml, fmls, tids)) {
if (i != j) {
fmls[j] = fmls[i].get();
}
++j;
}
else {
TRACE("qe", tout << mk_pp(fml, m) << "\n";);
}
}
fmls.resize(j);
// fmls holds residue,
// mbo holds linear inequalities that are in scope
// collect variables in residue an in tids.
// filter variables that are absent from residue.
// project those.
// collect result of projection
// return those to fmls.
expr_mark var_mark, fmls_mark;
for (unsigned i = 0; i < vars.size(); ++i) {
app* v = vars[i].get();
var_mark.mark(v);
if (is_arith(v) && !tids.contains(v)) {
expr_ref val(m);
rational r;
eval(v, val);
a.is_numeral(val, r);
TRACE("qe", tout << mk_pp(v, m) << " " << val << "\n";);
tids.insert(v, mbo.add_var(r, a.is_int(v)));
}
}
for (unsigned i = 0; i < fmls.size(); ++i) {
fmls_mark.mark(fmls[i].get());
}
obj_map<expr, unsigned>::iterator it = tids.begin(), end = tids.end();
ptr_vector<expr> index2expr;
for (; it != end; ++it) {
expr* e = it->m_key;
if (!var_mark.is_marked(e)) {
mark_rec(fmls_mark, e);
}
index2expr.setx(it->m_value, e, 0);
}
j = 0;
unsigned_vector real_vars;
for (unsigned i = 0; i < vars.size(); ++i) {
app* v = vars[i].get();
if (is_arith(v) && !fmls_mark.is_marked(v)) {
real_vars.push_back(tids.find(v));
}
else {
if (i != j) {
vars[j] = v;
}
++j;
}
}
vars.resize(j);
TRACE("qe", tout << "remaining vars: " << vars << "\n";
for (unsigned i = 0; i < real_vars.size(); ++i) {
unsigned v = real_vars[i];
tout << "v" << v << " " << mk_pp(index2expr[v], m) << "\n";
}
mbo.display(tout););
mbo.project(real_vars.size(), real_vars.c_ptr());
TRACE("qe", mbo.display(tout););
vector<row> rows;
mbo.get_live_rows(rows);
for (unsigned i = 0; i < rows.size(); ++i) {
expr_ref_vector ts(m);
expr_ref t(m), s(m), val(m);
row const& r = rows[i];
if (r.m_vars.size() == 0) {
continue;
}
if (r.m_vars.size() == 1 && r.m_vars[0].m_coeff.is_neg() && r.m_type != opt::t_mod) {
var const& v = r.m_vars[0];
t = index2expr[v.m_id];
if (!v.m_coeff.is_minus_one()) {
t = a.mk_mul(a.mk_numeral(-v.m_coeff, a.is_int(t)), t);
}
s = a.mk_numeral(r.m_coeff, a.is_int(t));
switch (r.m_type) {
case opt::t_lt: t = a.mk_gt(t, s); break;
case opt::t_le: t = a.mk_ge(t, s); break;
case opt::t_eq: t = a.mk_eq(t, s); break;
default: UNREACHABLE();
}
fmls.push_back(t);
VERIFY(model.eval(t, val));
CTRACE("qe", !m.is_true(val), tout << "Evaluated unit " << t << " to " << val << "\n";);
continue;
}
for (j = 0; j < r.m_vars.size(); ++j) {
var const& v = r.m_vars[j];
t = index2expr[v.m_id];
if (!v.m_coeff.is_one()) {
t = a.mk_mul(a.mk_numeral(v.m_coeff, a.is_int(t)), t);
}
ts.push_back(t);
}
s = a.mk_numeral(-r.m_coeff, a.is_int(t));
if (ts.size() == 1) {
t = ts[0].get();
}
else {
t = a.mk_add(ts.size(), ts.c_ptr());
}
switch (r.m_type) {
case opt::t_lt: t = a.mk_lt(t, s); break;
case opt::t_le: t = a.mk_le(t, s); break;
case opt::t_eq: t = a.mk_eq(t, s); break;
case opt::t_mod: {
rational sval;
if (!a.is_numeral(s, sval) || !sval.is_zero()) {
t = a.mk_sub(t, s);
}
t = a.mk_eq(a.mk_mod(t, a.mk_numeral(r.m_mod, true)), a.mk_int(0));
break;
}
}
fmls.push_back(t);
VERIFY(model.eval(t, val));
CTRACE("qe", !m.is_true(val), tout << "Evaluated " << t << " to " << val << "\n";);
}
}
opt::inf_eps maximize(expr_ref_vector const& fmls0, model& mdl, app* t, expr_ref& ge, expr_ref& gt) {
validate_model(mdl, fmls0);
m_trail.reset();
SASSERT(a.is_real(t));
expr_ref_vector fmls(fmls0);
opt::model_based_opt mbo;
opt::inf_eps value;
obj_map<expr, rational> ts;
obj_map<expr, unsigned> tids;
model_evaluator eval(mdl);
// extract objective function.
vars coeffs;
rational c(0), mul(1);
linearize(mbo, eval, mul, t, c, fmls, ts, tids);
extract_coefficients(mbo, eval, ts, tids, coeffs);
mbo.set_objective(coeffs, c);
// extract linear constraints
for (unsigned i = 0; i < fmls.size(); ++i) {
linearize(mbo, eval, fmls[i].get(), fmls, tids);
}
// find optimal value
value = mbo.maximize();
// update model to use new values that satisfy optimality
ptr_vector<expr> vars;
obj_map<expr, unsigned>::iterator it = tids.begin(), end = tids.end();
for (; it != end; ++it) {
expr* e = it->m_key;
if (is_uninterp_const(e)) {
unsigned id = it->m_value;
func_decl* f = to_app(e)->get_decl();
expr_ref val(a.mk_numeral(mbo.get_value(id), false), m);
mdl.register_decl(f, val);
}
else {
TRACE("qe", tout << "omitting model update for non-uninterpreted constant " << mk_pp(e, m) << "\n";);
}
}
expr_ref val(a.mk_numeral(value.get_rational(), false), m);
expr_ref tval(m);
eval(t, tval);
// update the predicate 'bound' which forces larger values when 'strict' is true.
// strict: bound := valuue < t
// !strict: bound := value <= t
if (!value.is_finite()) {
ge = a.mk_ge(t, tval);
gt = m.mk_false();
}
else if (value.get_infinitesimal().is_neg()) {
ge = a.mk_ge(t, tval);
gt = a.mk_ge(t, val);
}
else {
ge = a.mk_ge(t, val);
gt = a.mk_gt(t, val);
}
validate_model(mdl, fmls0);
return value;
}
bool validate_model(model& mdl, expr_ref_vector const& fmls) {
bool valid = true;
for (unsigned i = 0; i < fmls.size(); ++i) {
expr_ref val(m);
VERIFY(mdl.eval(fmls[i], val));
if (!m.is_true(val)) {
valid = false;
TRACE("qe", tout << mk_pp(fmls[i], m) << " := " << val << "\n";);
}
}
return valid;
}
void extract_coefficients(opt::model_based_opt& mbo, model_evaluator& eval, obj_map<expr, rational> const& ts, obj_map<expr, unsigned>& tids, vars& coeffs) {
coeffs.reset();
eval.set_model_completion(true);
obj_map<expr, rational>::iterator it = ts.begin(), end = ts.end();
for (; it != end; ++it) {
unsigned id;
expr* v = it->m_key;
if (!tids.find(v, id)) {
rational r;
expr_ref val(m);
eval(v, val);
if (a.is_numeral(val, r)) {
id = mbo.add_var(r, a.is_int(v));
}
else {
TRACE("qe", tout << "extraction of coefficients cancelled\n";);
return;
}
tids.insert(v, id);
m_trail.push_back(v);
}
CTRACE("qe", it->m_value.is_zero(), tout << mk_pp(v, m) << " has coefficeint 0\n";);
if (!it->m_value.is_zero()) {
coeffs.push_back(var(id, it->m_value));
}
}
}
};
arith_project_plugin::arith_project_plugin(ast_manager& m) {
m_imp = alloc(imp, m);
}
arith_project_plugin::~arith_project_plugin() {
dealloc(m_imp);
}
bool arith_project_plugin::operator()(model& model, app* var, app_ref_vector& vars, expr_ref_vector& lits) {
return (*m_imp)(model, var, vars, lits);
}
void arith_project_plugin::operator()(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
(*m_imp)(model, vars, lits);
}
bool arith_project_plugin::solve(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
return m_imp->solve(model, vars, lits);
}
family_id arith_project_plugin::get_family_id() {
return m_imp->a.get_family_id();
}
opt::inf_eps arith_project_plugin::maximize(expr_ref_vector const& fmls, model& mdl, app* t, expr_ref& ge, expr_ref& gt) {
return m_imp->maximize(fmls, mdl, t, ge, gt);
}
bool arith_project(model& model, app* var, expr_ref_vector& lits) {
ast_manager& m = lits.get_manager();
arith_project_plugin ap(m);
app_ref_vector vars(m);
return ap(model, var, vars, lits);
}
}