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reorganize polysat functionality to use abstract solver interface

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make dependency be self-contained
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Nikolaj Bjorner 2023-12-09 09:38:18 -08:00
parent 837e111d93
commit 81c6f00c99
23 changed files with 381 additions and 123 deletions
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src/sat/smt/polysat_ule.cpp
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/*++
Copyright (c) 2021 Microsoft Corporation
Module Name:
polysat unsigned <= constraints
Author:
Nikolaj Bjorner (nbjorner) 2021-03-19
Jakob Rath 2021-04-06
Notes:
Canonical representation of equation p == 0 is the constraint p <= 0.
The alternatives p < 1, -1 <= q, q > -2 are eliminated.
Rewrite rules to simplify expressions.
In the following let k, k1, k2 be values.
- k1 <= k2 ==> 0 <= 0 if k1 <= k2
- k1 <= k2 ==> 1 <= 0 if k1 > k2
- 0 <= p ==> 0 <= 0
- p <= 0 ==> 1 <= 0 if p is never zero due to parity
- p <= -1 ==> 0 <= 0
- k <= p ==> p - k <= - k - 1
- k*2^n*p <= 0 ==> 2^n*p <= 0 if k is odd, leading coeffient is always a power of 2.
Note: the rules will rewrite alternative formulations of equations:
- -1 <= p ==> p + 1 <= 0
- 1 <= p ==> p - 1 <= -2
Rewrite rules on signed constraints:
- p > -2 ==> p + 1 <= 0
- p <= -2 ==> p + 1 > 0
At this point, all equations are in canonical form.
TODO: clause simplifications:
- p + k <= p ==> p + k <= k & p != 0 for k != 0
- p*q = 0 ==> p = 0 or q = 0 applies to any factoring
- 2*p <= 2*q ==> (p >= 2^n-1 & q < 2^n-1) or (p >= 2^n-1 = q >= 2^n-1 & p <= q)
==> (p >= 2^n-1 => q < 2^n-1 or p <= q) &
(p < 2^n-1 => p <= q) &
(p < 2^n-1 => q < 2^n-1)
- 3*p <= 3*q ==> ?
Note:
case p <= p + k is already covered because we test (lhs - rhs).is_val()
It can be seen as an instance of lemma 5.2 of Supratik and John.
The following forms are equivalent:
p <= q
p <= p - q - 1
q - p <= q
q - p <= -p - 1
-q - 1 <= -p - 1
-q - 1 <= p - q - 1
Useful lemmas:
p <= q && q+1 != 0 ==> p+1 <= q+1
p <= q && p != 0 ==> -q <= -p
--*/
#include "sat/smt/polysat_constraints.h"
#include "sat/smt/polysat_ule.h"
#define LOG(_msg_) verbose_stream() << _msg_ << "\n"
namespace polysat {
// Simplify lhs <= rhs.
//
// NOTE: the result should not depend on the initial value of is_positive;
// the purpose of is_positive is to allow flipping the sign as part of a rewrite rule.
static void simplify_impl(bool& is_positive, pdd& lhs, pdd& rhs) {
SASSERT_EQ(lhs.power_of_2(), rhs.power_of_2());
unsigned const N = lhs.power_of_2();
// 0 <= p --> 0 <= 0
if (lhs.is_zero()) {
rhs = 0;
return;
}
// p <= -1 --> 0 <= 0
if (rhs.is_max()) {
lhs = 0, rhs = 0;
return;
}
// p <= p --> 0 <= 0
if (lhs == rhs) {
lhs = 0, rhs = 0;
return;
}
// Evaluate constants
if (lhs.is_val() && rhs.is_val()) {
if (lhs.val() <= rhs.val())
lhs = 0, rhs = 0;
else
lhs = 0, rhs = 0, is_positive = !is_positive;
return;
}
// Try to reduce the number of variables on one side using one of these rules:
//
// p <= q --> p <= p - q - 1
// p <= q --> q - p <= q
//
// Possible alternative to 1:
// p <= q --> q - p <= -p - 1
// Possible alternative to 2:
// p <= q --> -q-1 <= p - q - 1
//
// Example:
//
// x <= x + y --> x <= - y - 1
// x + y <= x --> -y <= x
if (!lhs.is_val() && !rhs.is_val()) {
unsigned const lhs_vars = lhs.free_vars().size();
unsigned const rhs_vars = rhs.free_vars().size();
unsigned const diff_vars = (lhs - rhs).free_vars().size();
if (diff_vars < lhs_vars || diff_vars < rhs_vars) {
LOG("reduce number of varables");
// verbose_stream() << "IN: " << ule_pp(to_lbool(is_positive), lhs, rhs) << "\n";
if (lhs_vars <= rhs_vars)
rhs = lhs - rhs - 1;
else
lhs = rhs - lhs;
// verbose_stream() << "OUT: " << ule_pp(to_lbool(is_positive), lhs, rhs) << "\n";
}
}
// -p + k <= k --> p <= k
if (rhs.is_val() && !rhs.is_zero() && lhs.offset() == rhs.val()) {
LOG("-p + k <= k --> p <= k");
lhs = rhs - lhs;
}
// k <= p + k --> p <= -k-1
if (lhs.is_val() && !lhs.is_zero() && lhs.val() == rhs.offset()) {
LOG("k <= p + k --> p <= -k-1");
pdd k = lhs;
lhs = rhs - lhs;
rhs = -k - 1;
}
// k <= -p --> p-1 <= -k-1
if (lhs.is_val() && rhs.leading_coefficient().get_bit(N - 1) && !rhs.offset().is_zero()) {
LOG("k <= -p --> p-1 <= -k-1");
pdd k = lhs;
lhs = -(rhs + 1);
rhs = -k - 1;
}
// -p <= k --> -k-1 <= p-1
// if (rhs.is_val() && lhs.leading_coefficient() > rational::power_of_two(N - 1) && !lhs.offset().is_zero()) {
if (rhs.is_val() && lhs.leading_coefficient().get_bit(N - 1) && !lhs.offset().is_zero()) {
LOG("-p <= k --> -k-1 <= p-1");
pdd k = rhs;
rhs = -(lhs + 1);
lhs = -k - 1;
}
// NOTE: do not use pdd operations in conditions when comparing pdd values.
// e.g.: "lhs.offset() == (rhs + 1).val()" is problematic with the following evaluation:
// 1. return reference into pdd_manager::m_values from lhs.offset()
// 2. compute rhs+1, which may reallocate pdd_manager::m_values
// 3. now the reference returned from lhs.offset() may be invalid
pdd const rhs_plus_one = rhs + 1;
// p - k <= -k - 1 --> k <= p
// TODO: potential bug here: first call offset(), then rhs+1 has to reallocate pdd_manager::m_values, then the reference to offset is broken.
if (rhs.is_val() && !rhs.is_zero() && lhs.offset() == rhs_plus_one.val()) {
LOG("p - k <= -k - 1 --> k <= p");
pdd k = -(rhs + 1);
rhs = lhs + k;
lhs = k;
}
pdd const lhs_minus_one = lhs - 1;
// k <= 2^(N-1)*p + q + k-1 --> k <= 2^(N-1)*p - q
if (lhs.is_val() && rhs.leading_coefficient() == rational::power_of_two(N-1) && rhs.offset() == lhs_minus_one.val()) {
LOG("k <= 2^(N-1)*p + q + k-1 --> k <= 2^(N-1)*p - q");
rhs = (lhs - 1) - rhs;
}
// -1 <= p --> p + 1 <= 0
if (lhs.is_max()) {
lhs = rhs + 1;
rhs = 0;
}
// 1 <= p --> p > 0
if (lhs.is_one()) {
lhs = rhs;
rhs = 0;
is_positive = !is_positive;
}
// p > -2 --> p + 1 <= 0
// p <= -2 --> p + 1 > 0
if (rhs.is_val() && !rhs.is_zero() && (rhs + 2).is_zero()) {
// Note: rhs.is_zero() iff rhs.manager().power_of_2() == 1 (the rewrite is not wrong for M=2, but useless)
lhs = lhs + 1;
rhs = 0;
is_positive = !is_positive;
}
// 2p + 1 <= 0 --> 0 < 0
if (rhs.is_zero() && lhs.is_never_zero()) {
lhs = 0;
is_positive = !is_positive;
return;
}
// a*p + q <= 0 --> p + a^-1*q <= 0 for a odd
if (rhs.is_zero() && !lhs.leading_coefficient().is_power_of_two()) {
rational lc = lhs.leading_coefficient();
rational x, y;
gcd(lc, lhs.manager().two_to_N(), x, y);
if (x.is_neg())
x = mod(x, lhs.manager().two_to_N());
lhs *= x;
SASSERT(lhs.leading_coefficient().is_power_of_two());
}
} // simplify_impl
}
namespace polysat {
ule_constraint::ule_constraint(pdd const& l, pdd const& r) :
m_lhs(l), m_rhs(r) {
SASSERT_EQ(m_lhs.power_of_2(), m_rhs.power_of_2());
vars().append(m_lhs.free_vars());
for (auto v : m_rhs.free_vars())
if (!vars().contains(v))
vars().push_back(v);
}
void ule_constraint::simplify(bool& is_positive, pdd& lhs, pdd& rhs) {
SASSERT_EQ(lhs.power_of_2(), rhs.power_of_2());
#ifndef NDEBUG
bool const old_is_positive = is_positive;
pdd const old_lhs = lhs;
pdd const old_rhs = rhs;
#endif
simplify_impl(is_positive, lhs, rhs);
#ifndef NDEBUG
if (old_is_positive != is_positive || old_lhs != lhs || old_rhs != rhs) {
ule_pp const old_ule(to_lbool(old_is_positive), old_lhs, old_rhs);
ule_pp const new_ule(to_lbool(is_positive), lhs, rhs);
// always-false and always-true constraints should be rewritten to 0 != 0 and 0 == 0, respectively.
if (is_always_false(old_is_positive, old_lhs, old_rhs)) {
SASSERT(!is_positive);
SASSERT(lhs.is_zero());
SASSERT(rhs.is_zero());
}
if (is_always_true(old_is_positive, old_lhs, old_rhs)) {
SASSERT(is_positive);
SASSERT(lhs.is_zero());
SASSERT(rhs.is_zero());
}
}
SASSERT(is_simplified(lhs, rhs)); // rewriting should be idempotent
#endif
}
bool ule_constraint::is_simplified(pdd const& lhs0, pdd const& rhs0) {
bool const pos0 = true;
bool pos1 = pos0;
pdd lhs1 = lhs0;
pdd rhs1 = rhs0;
simplify_impl(pos1, lhs1, rhs1);
bool const is_simplified = (pos1 == pos0 && lhs1 == lhs0 && rhs1 == rhs0);
DEBUG_CODE({
// check that simplification doesn't depend on initial sign
bool pos2 = !pos0;
pdd lhs2 = lhs0;
pdd rhs2 = rhs0;
simplify_impl(pos2, lhs2, rhs2);
SASSERT_EQ(pos2, !pos1);
SASSERT_EQ(lhs2, lhs1);
SASSERT_EQ(rhs2, rhs1);
});
return is_simplified;
}
std::ostream& ule_constraint::display(std::ostream& out, lbool status, pdd const& lhs, pdd const& rhs) {
out << lhs;
if (rhs.is_zero() && status == l_true) out << " == ";
else if (rhs.is_zero() && status == l_false) out << " != ";
else if (status == l_true) out << " <= ";
else if (status == l_false) out << " > ";
else out << " <=/> ";
return out << rhs;
}
std::ostream& ule_constraint::display(std::ostream& out, lbool status) const {
return display(out, status, m_lhs, m_rhs);
}
std::ostream& ule_constraint::display(std::ostream& out) const {
return display(out, l_true, m_lhs, m_rhs);
}
// Evaluate lhs <= rhs
lbool ule_constraint::eval(pdd const& lhs, pdd const& rhs) {
// NOTE: don't assume simplifications here because we also call this on partially substituted constraints
if (lhs.is_zero())
return l_true; // 0 <= p
if (lhs == rhs)
return l_true; // p <= p
if (rhs.is_max())
return l_true; // p <= -1
if (rhs.is_zero() && lhs.is_never_zero())
return l_false; // p <= 0 implies p == 0
if (lhs.is_one() && rhs.is_never_zero())
return l_true; // 1 <= p implies p != 0
if (lhs.is_val() && rhs.is_val())
return to_lbool(lhs.val() <= rhs.val());
return l_undef;
}
lbool ule_constraint::eval() const {
return eval(lhs(), rhs());
}
lbool ule_constraint::eval(assignment const& a) const {
return eval(a.apply_to(lhs()), a.apply_to(rhs()));
}
}