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z3/src/test/polysat.cpp
Jakob Rath 4a3fe8ab82 fix
2022-07-21 13:00:36 +02:00

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#include "math/polysat/log.h"
#include "math/polysat/solver.h"
#include "ast/ast.h"
#include "parsers/smt2/smt2parser.h"
#include "util/util.h"
#include <vector>
namespace {
using namespace dd;
void permute_args(unsigned k, pdd& a, pdd& b, pdd& c) {
SASSERT(k < 6);
unsigned i = k % 3;
unsigned j = k % 2;
if (i == 1)
std::swap(a, b);
else if (i == 2)
std::swap(a, c);
if (j == 1)
std::swap(b, c);
}
void permute_args(unsigned n, pdd& a, pdd& b, pdd& c, pdd& d) {
SASSERT(n < 24);
switch (n % 4) {
case 1:
std::swap(a, b);
break;
case 2:
std::swap(a, c);
break;
case 3:
std::swap(a, d);
break;
default:
break;
}
switch (n % 3) {
case 1:
std::swap(b, c);
break;
case 2:
std::swap(b, d);
break;
default:
break;
}
switch (n % 2) {
case 1:
std::swap(c, d);
break;
default:
break;
}
}
}
namespace polysat {
// test resolve, factoring routines
// auxiliary
struct solver_scope {
reslimit lim;
};
class scoped_solver : public solver_scope, public solver {
std::string m_name;
lbool m_last_result = l_undef;
public:
scoped_solver(std::string name): solver(lim), m_name(name) {
LOG("\n\n\n" << std::string(78, '#') << "\n\nSTART: " << m_name << "\n");
set_max_conflicts(10);
}
void set_max_conflicts(unsigned c) {
params_ref p;
p.set_uint("max_conflicts", c);
updt_params(p);
}
void check() {
m_last_result = check_sat();
LOG(m_name << ": " << m_last_result << "\n");
statistics st;
collect_statistics(st);
LOG(st << "\n" << *this << "\n");
std::cout << st << "\n";
}
void expect_unsat() {
if (m_last_result != l_false && m_last_result != l_undef) {
LOG_H1("FAIL: " << m_name << ": expected UNSAT, got " << m_last_result << "!");
VERIFY(false);
}
}
void expect_sat(std::vector<std::pair<dd::pdd, unsigned>> const& expected_assignment = {}) {
if (m_last_result == l_true) {
for (auto const& p : expected_assignment) {
auto const& v_pdd = p.first;
auto const expected_value = p.second;
SASSERT(v_pdd.is_monomial() && !v_pdd.is_val());
auto const v = v_pdd.var();
if (get_value(v) != expected_value) {
LOG_H1("FAIL: " << m_name << ": expected assignment v" << v << " := " << expected_value << ", got value " << get_value(v) << "!");
VERIFY(false);
}
}
}
else if (m_last_result == l_false) {
LOG_H1("FAIL: " << m_name << ": expected SAT, got " << m_last_result << "!");
VERIFY(false);
}
}
};
class test_polysat {
public:
/**
* Testing the solver's internal state.
*/
/// Creates two separate conflicts (from narrowing) before solving loop is started.
static void test_add_conflicts() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(3));
auto b = s.var(s.add_var(3));
s.add_eq(a + 1);
s.add_eq(a + 2);
s.add_eq(b + 1);
s.add_eq(b + 2);
s.check();
s.expect_unsat();
}
/// Has constraints which must be inserted into other watchlist to discover UNSAT
static void test_wlist() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(3));
auto b = s.var(s.add_var(3));
auto c = s.var(s.add_var(3));
auto d = s.var(s.add_var(3));
s.add_eq(d + c + b + a + 1);
s.add_eq(d + c + b + a);
s.add_eq(d + c + b);
s.add_eq(d + c);
s.add_eq(d);
s.check();
s.expect_unsat();
}
/// Has a constraint in cjust[a] where a does not occur.
static void test_cjust() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(3));
auto b = s.var(s.add_var(3));
auto c = s.var(s.add_var(3));
// 1. Decide a = 0.
s.add_eq(a*a + b + 7); // 2. Propagate b = 1
s.add_eq(b*b + c*c*c*(b+7) + c + 5); // 3. Propagate c = 2
s.add_eq(b*b + c*c); // 4. Conflict
// Resolution fails because second constraint has c*c*c
// => cjust[a] += b*b + c*c
s.check();
s.expect_unsat();
}
/**
* most basic linear equation solving.
* they should be solvable.
* they also illustrate some limitations of basic solver even if it solves them.
* Example
* the value to a + 1 = 0 is fixed at 3, there should be no search.
*/
static void test_l1() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(2));
s.add_eq(a + 1);
s.check();
s.expect_sat({{a, 3}});
}
static void test_l2() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(2));
auto b = s.var(s.add_var(2));
s.add_eq(2*a + b + 1);
s.add_eq(2*b + a);
s.check();
s.expect_sat({{a, 2}, {b, 3}});
}
static void test_l3() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(2));
auto b = s.var(s.add_var(2));
s.add_eq(3*b + a + 2);
s.check();
s.expect_sat();
}
static void test_l4() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(3));
s.add_eq(4*a + 2);
s.check();
s.expect_unsat();
}
static void test_l5() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(3));
auto b = s.var(s.add_var(3));
s.add_diseq(b);
s.add_eq(a + 2*b + 4);
s.add_eq(a + 4*b + 4);
s.check();
s.expect_sat({{a, 4}, {b, 4}});
}
/**
* This one is unsat because a*a*(a*a - 1)
* is 0 for all values of a.
*/
static void test_p1() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(2));
auto p = a*a*(a*a - 1) + 1;
s.add_eq(p);
s.check();
s.expect_unsat();
}
/**
* has solutions a = 2 and a = 3
*/
static void test_p2() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(2));
auto p = a*(a-1) + 2;
s.add_eq(p);
s.check();
s.expect_sat();
}
/**
* unsat
* - learns 3*x + 1 == 0 by polynomial resolution
* - this forces x == 5, which means the first constraint is unsatisfiable by parity.
*/
static void test_p3() {
scoped_solver s(__func__);
auto x = s.var(s.add_var(4));
auto y = s.var(s.add_var(4));
auto z = s.var(s.add_var(4));
s.add_eq(x*x*y + 3*y + 7);
s.add_eq(2*y + z + 8);
s.add_eq(3*x + 4*y*z + 2*z*z + 1);
s.check();
s.expect_unsat();
}
// Unique solution: u = 5
static void test_ineq_basic1() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(4));
s.add_ule(u, 5);
s.add_ule(5, u);
s.check();
s.expect_sat({{u, 5}});
}
// Unsatisfiable
static void test_ineq_basic2() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(4));
s.add_ult(u, 5);
s.add_ule(5, u);
s.check();
s.expect_unsat();
}
// Solutions with u = v = w
static void test_ineq_basic3() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(4));
auto v = s.var(s.add_var(4));
auto w = s.var(s.add_var(4));
s.add_ule(u, v);
s.add_ule(v, w);
s.add_ule(w, u);
s.check();
s.expect_sat();
SASSERT_EQ(s.get_value(u.var()), s.get_value(v.var()));
SASSERT_EQ(s.get_value(u.var()), s.get_value(w.var()));
}
// Unsatisfiable
static void test_ineq_basic4() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(4));
auto v = s.var(s.add_var(4));
auto w = s.var(s.add_var(4));
s.add_ule(u, v);
s.add_ult(v, w);
s.add_ule(w, u);
s.check();
s.expect_unsat();
}
// Satisfiable
// Without forbidden intervals, we just try values for u until it works
static void test_ineq_basic5() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(4));
auto v = s.var(s.add_var(4));
s.add_ule(12, u + v);
s.add_ule(v, 2);
s.check();
s.expect_sat(); // e.g., u = 12, v = 0
}
// Like test_ineq_basic5 but the other forbidden interval will be the longest
static void test_ineq_basic6() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(4));
auto v = s.var(s.add_var(4));
s.add_ule(14, u + v);
s.add_ule(v, 2);
s.check();
s.expect_sat();
}
/**
* Check unsat of:
* u = v*q + r
* r < u
* v*q > u
*/
static void test_ineq1() {
scoped_solver s(__func__);
auto u = s.var(s.add_var(5));
auto v = s.var(s.add_var(5));
auto q = s.var(s.add_var(5));
auto r = s.var(s.add_var(5));
s.add_eq(u - (v*q) - r);
s.add_ult(r, u);
s.add_ult(u, v*q);
s.check();
s.expect_unsat();
}
/**
* Check unsat of:
* n*q1 = a - b
* n*q2 + r2 = c*a - c*b
* n > r2 > 0
*/
static void test_ineq2() {
scoped_solver s(__func__);
auto n = s.var(s.add_var(5));
auto q1 = s.var(s.add_var(5));
auto a = s.var(s.add_var(5));
auto b = s.var(s.add_var(5));
auto c = s.var(s.add_var(5));
auto q2 = s.var(s.add_var(5));
auto r2 = s.var(s.add_var(5));
s.add_eq(n*q1 - a + b);
s.add_eq(n*q2 + r2 - c*a + c*b);
s.add_ult(r2, n);
s.add_diseq(r2);
s.check();
s.expect_unsat();
}
/**
* Monotonicity example from certora
*
* We do overflow checks by doubling the base bitwidth here.
*/
static void test_monot(unsigned base_bw = 5) {
scoped_solver s(std::string{__func__} + "(" + std::to_string(base_bw) + ")");
auto max_int_const = rational::power_of_two(base_bw) - 1;
unsigned bw = 2 * base_bw;
auto max_int = s.var(s.add_var(bw));
s.add_eq(max_int + (-max_int_const));
auto tb1 = s.var(s.add_var(bw));
s.add_ule(tb1, max_int);
auto tb2 = s.var(s.add_var(bw));
s.add_ule(tb2, max_int);
auto a = s.var(s.add_var(bw));
s.add_ule(a, max_int);
auto v = s.var(s.add_var(bw));
s.add_ule(v, max_int);
auto base1 = s.var(s.add_var(bw));
s.add_ule(base1, max_int);
auto base2 = s.var(s.add_var(bw));
s.add_ule(base2, max_int);
auto elastic1 = s.var(s.add_var(bw));
s.add_ule(elastic1, max_int);
auto elastic2 = s.var(s.add_var(bw));
s.add_ule(elastic2, max_int);
auto err = s.var(s.add_var(bw));
s.add_ule(err, max_int);
auto rem1 = s.var(s.add_var(bw));
auto quot2 = s.var(s.add_var(bw));
s.add_ule(quot2, max_int);
auto rem2 = s.var(s.add_var(bw));
auto rem3 = s.var(s.add_var(bw));
auto quot4 = s.var(s.add_var(bw));
s.add_ule(quot4, max_int);
auto rem4 = s.var(s.add_var(bw));
s.add_diseq(elastic1);
// division: tb1 = (v * base1) / elastic1;
s.add_eq((tb1 * elastic1) + rem1 - (v * base1));
s.add_ult(rem1, elastic1);
s.add_ule((tb1 * elastic1), max_int);
// division: quot2 = (a * base1) / elastic1
s.add_eq((quot2 * elastic1) + rem2 - (a * base1));
s.add_ult(rem2, elastic1);
s.add_ule((quot2 * elastic1), max_int);
s.add_eq(base1 + quot2 - base2);
s.add_eq(elastic1 + a - elastic2);
// division: tb2 = ((v * base2) / elastic2);
s.add_eq((tb2 * elastic2) + rem3 - (v * base2));
s.add_ult(rem3, elastic2);
s.add_ule((tb2 * elastic2), max_int);
// division: quot4 = v / elastic2;
s.add_eq((quot4 * elastic2) + rem4 - v);
s.add_ult(rem4, elastic2);
s.add_ule((quot4 * elastic2), max_int);
s.add_eq(quot4 + 1 - err);
s.push();
s.add_ult(tb1, tb2);
s.check();
s.expect_unsat();
s.pop();
s.push();
s.add_ult(tb2 + err, tb1);
s.check();
s.expect_unsat();
s.pop();
}
/*
* Mul-then-div in fixed point arithmetic is (roughly) neutral.
*
* I.e. we prove "(((a * b) / sf) * sf) / b" to be equal to a, up to some error margin.
*
* sf is the scaling factor (we could leave this unconstrained, but non-zero, to make the benchmark a bit harder)
* em is the error margin
*
* We do overflow checks by doubling the base bitwidth here.
*/
static void test_fixed_point_arith_mul_div_inverse() {
scoped_solver s(__func__);
auto baseBw = 5;
auto max_int_const = 31; // (2^5 - 1) -- change this when you change baseBw
auto bw = 2 * baseBw;
auto max_int = s.var(s.add_var(bw));
s.add_eq(max_int - max_int_const);
// "input" variables
auto a = s.var(s.add_var(bw));
s.add_ule(a, max_int);
auto b = s.var(s.add_var(bw));
s.add_ule(b, max_int);
s.add_ult(0, b); // b > 0
// scaling factor (setting it, somewhat arbitrarily, to max_int/3)
auto sf = s.var(s.add_var(bw));
s.add_eq(sf - (max_int_const/3));
// (a * b) / sf = quot1 <=> quot1 * sf + rem1 - (a * b) = 0
auto quot1 = s.var(s.add_var(bw));
auto rem1 = s.var(s.add_var(bw));
s.add_eq((quot1 * sf) + rem1 - (a * b));
s.add_ult(rem1, sf);
s.add_ule(quot1 * sf, max_int);
// (((a * b) / sf) * sf) / b <=> quot2 * b + rem2 - (((a * b) / sf) * sf) = 0
auto quot2 = s.var(s.add_var(bw));
auto rem2 = s.var(s.add_var(bw));
s.add_eq((quot2 * b) + rem2 - (quot1 * sf));
s.add_ult(rem2, b);
s.add_ule(quot2 * b, max_int);
// sf / b = quot3 <=> quot3 * b + rem3 = sf
auto quot3 = s.var(s.add_var(bw));
auto rem3 = s.var(s.add_var(bw));
s.add_eq((quot3 * b) + rem3 - sf);
s.add_ult(rem3, b);
s.add_ule(quot3 * b, max_int);
// em = sf / b + 1
auto em = s.var(s.add_var(bw));
s.add_eq(quot3 + 1 - em);
// we prove quot3 <= a and quot3 + em >= a
s.push();
s.add_ult(a, quot3);
s.check();
s.expect_unsat();
s.pop();
// s.push();
// s.add_ult(quot3 + em, a);
// s.check();
// s.expect_unsat();
// s.pop();
}
/*
* Div-then-mul in fixed point arithmetic is (roughly) neutral.
*
* I.e. we prove "(b * ((a * sf) / b)) / sf" to be equal to a, up to some error margin.
*
* sf is the scaling factor (we could leave this unconstrained, but non-zero, to make the benchmark a bit harder)
* em is the error margin
*
* We do overflow checks by doubling the base bitwidth here.
*/
static void test_fixed_point_arith_div_mul_inverse(unsigned base_bw = 5) {
scoped_solver s(__func__);
auto max_int_const = rational::power_of_two(base_bw) - 1;
auto bw = 2 * base_bw;
auto max_int = s.var(s.add_var(bw));
s.add_eq(max_int - max_int_const);
// "input" variables
auto a = s.var(s.add_var(bw));
s.add_ule(a, max_int);
auto b = s.var(s.add_var(bw));
s.add_ule(b, max_int);
s.add_ult(0, b); // b > 0
// scaling factor (setting it, somewhat arbitrarily, to max_int/3)
auto sf = s.var(s.add_var(bw));
s.add_eq(sf - floor(max_int_const/3));
// (a * sf) / b = quot1 <=> quot1 * b + rem1 - (a * sf) = 0
auto quot1 = s.var(s.add_var(bw));
auto rem1 = s.var(s.add_var(bw));
s.add_eq((quot1 * b) + rem1 - (a * sf));
s.add_ult(rem1, b);
s.add_ule(quot1 * b, max_int);
// (b * ((a * sf) / b)) / sf = quot2 <=> quot2 * sf + rem2 - (b * ((a * sf) / b)) = 0
auto quot2 = s.var(s.add_var(bw));
auto rem2 = s.var(s.add_var(bw));
s.add_eq((quot2 * sf) + rem2 - (b * quot1));
s.add_ult(rem2, sf);
s.add_ule(quot2 * sf, max_int);
// b / sf = quot3 <=> quot3 * sf + rem3 - b = 0
auto quot3 = s.var(s.add_var(bw));
auto rem3 = s.var(s.add_var(bw));
s.add_eq((quot3 * sf) + rem3 - b);
s.add_ult(rem3, sf);
s.add_ule(quot3 * sf, max_int);
// em = b / sf + 1
auto em = s.var(s.add_var(bw));
s.add_eq(quot3 + 1 - em);
// we prove quot3 <= a and quot3 + em >= a
s.push();
s.add_ult(quot3 + em, a);
s.check();
// s.expect_unsat();
s.pop();
s.push();
s.add_ult(a, quot3);
s.check();
// s.expect_unsat();
s.pop();
//exit(0);
}
/** Monotonicity under bounds,
* puzzle extracted from https://github.com/NikolajBjorner/polysat/blob/main/puzzles/bv.smt2
*
* x, y, z \in [0..2^64[
* x, y, z < 2^32
* y <= x
* x*z < 2^32
* y*z >= 2^32
*/
static void test_monot_bounds(unsigned base_bw = 32) {
scoped_solver s(std::string{__func__} + "(" + std::to_string(base_bw) + ")");
unsigned bw = 2 * base_bw;
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
auto x = s.var(s.add_var(bw));
auto bound = rational::power_of_two(base_bw);
#if 1
s.add_ult(x, bound);
s.add_ult(y, bound);
// s.add_ult(z, bound); // not required
#else
s.add_ule(x, bound - 1);
s.add_ule(y, bound - 1);
// s.add_ule(z, bound - 1); // not required
#endif
unsigned a = 13;
s.add_ule(z, y);
s.add_ult(x*y, a);
s.add_ule(a, x*z);
s.check();
s.expect_unsat();
}
/** Monotonicity under bounds, simplified even more.
*
* x, y, z \in [0..2^64[
* x, y, z < 2^32
* z <= y
* y*x < z*x
*/
static void test_monot_bounds_simple(unsigned base_bw = 32) {
scoped_solver s(__func__);
unsigned bw = 2 * base_bw;
/*
auto z = s.var(s.add_var(bw));
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
*/
auto y = s.var(s.add_var(bw));
auto x = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
auto bound = rational::power_of_two(base_bw);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_ult(z, bound);
s.add_ule(z, y);
s.add_ult(y*x, z*x);
s.check();
s.expect_unsat();
}
/*
* Transcribed from https://github.com/NikolajBjorner/polysat/blob/main/puzzles/bv.smt2
*
* We do overflow checks by doubling the base bitwidth here.
*/
static void test_monot_bounds_full(unsigned base_bw = 5) {
scoped_solver s(__func__);
auto const max_int_const = rational::power_of_two(base_bw) - 1;
auto const bw = 2 * base_bw;
auto const max_int = s.var(s.add_var(bw));
s.add_eq(max_int - max_int_const);
auto const first = s.var(s.add_var(bw));
s.add_ule(first, max_int);
auto const second = s.var(s.add_var(bw));
s.add_ule(second, max_int);
auto const idx = s.var(s.add_var(bw));
s.add_ule(idx, max_int);
auto const q = s.var(s.add_var(bw));
s.add_ule(q, max_int);
auto const r = s.var(s.add_var(bw));
s.add_ule(r, max_int);
// q = max_int / idx <=> q * idx + r - max_int = 0
s.add_eq((q * idx) + r - max_int);
s.add_ult(r, idx);
s.add_ule(q * idx, max_int);
/* last assertion:
(not
(=> (bvugt second first)
(=>
(=> (not (= idx #x00000000))
(bvule (bvsub second first) q))
(bvumul_noovfl (bvsub second first) idx))))
transforming negated boolean skeleton:
(not (=> a (=> (or b c) d))) <=> (and a (not d) (or b c))
*/
// (bvugt second first)
s.add_ult(first, second);
// (not (bvumul_noovfl (bvsub second first) idx))
s.add_ult(max_int, (second - first) * idx);
// s.add_ule((second - first) * idx, max_int);
// resolving disjunction via push/pop
// first disjunct: (= idx #x00000000)
s.push();
s.add_eq(idx);
s.check();
s.expect_unsat();
s.pop();
// second disjunct: (bvule (bvsub second first) q)
s.push();
s.add_ule(second - first, q);
s.check();
s.expect_unsat();
s.pop();
}
static void test_var_minimize(unsigned bw = 32) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
s.add_eq(x);
s.add_eq(4 * y + 8 * z + x + 2); // should only depend on assignment to x
s.check();
s.expect_unsat();
}
/**
* x*x <= z
* (x+1)*(x+1) <= z
* y == x+1
* ¬(y*y <= z)
*
* The original version had signed comparisons but that doesn't matter for the UNSAT result.
* UNSAT can be seen easily by substituting the equality.
*
* Possible ways to solve:
* - Integrate AC congruence closure
* See: Deepak Kapur. A Modular Associative Commutative (AC) Congruence Closure Algorithm, FSCD 2021. https://doi.org/10.4230/LIPIcs.FSCD.2021.15
* - Propagate equalities as substitutions
* x=t /\ p(x) ==> p(t)
* Ackermann-like reduction
* (index, watch lists over boolean literals)
* - Augment explain:
* conflict: y=x+1 /\ y^2 > z
* explain could then derive (x+1)^2 > z
*/
static void test_subst(unsigned bw = 32) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
s.add_ule(x * x, z); // optional
s.add_ule((x + 1) * (x + 1), z);
s.add_eq(x + 1 - y);
s.add_ult(z, y*y);
s.check();
s.expect_unsat();
}
static void test_subst_signed(unsigned bw = 32) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
s.add_sle(x * x, z); // optional
s.add_sle((x + 1) * (x + 1), z);
s.add_eq(x + 1 - y);
s.add_slt(z, y*y);
s.check();
s.expect_unsat();
}
// xy < xz and !Omega(x*y) => y < z
static void test_ineq_axiom1(unsigned bw = 32, std::optional<unsigned> perm = std::nullopt) {
if (perm) {
scoped_solver s(std::string(__func__) + " perm=" + std::to_string(*perm));
auto const bound = rational::power_of_two(bw/2);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
permute_args(*perm, x, y, z);
s.add_ult(x * y, x * z);
s.add_ule(z, y);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.check();
s.expect_unsat();
}
else {
for (unsigned i = 0; i < 6; ++i) {
test_ineq_axiom1(bw, i);
}
}
}
static void test_ineq_non_axiom1(unsigned bw = 32) {
auto const bound = rational::power_of_two(bw - 1);
for (unsigned i = 0; i < 6; ++i) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
permute_args(i, x, y, z);
s.add_ult(x * y, x * z);
s.add_ule(z, y);
//s.add_ult(x, bound);
//s.add_ult(y, bound);
s.check();
s.expect_sat();
}
}
// xy <= xz & !Omega(x*y) => y <= z or x = 0
static void test_ineq_axiom2(unsigned bw = 32) {
auto const bound = rational::power_of_two(bw/2);
for (unsigned i = 0; i < 6; ++i) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto z = s.var(s.add_var(bw));
permute_args(i, x, y, z);
s.add_ult(x * y, x * z);
s.add_ult(z, y);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_diseq(x);
s.check();
s.expect_unsat();
}
}
// xy < b & a <= x & !Omega(x*y) => a*y < b
static void test_ineq_axiom3(unsigned bw = 32) {
auto const bound = rational::power_of_two(bw/2);
for (unsigned i = 0; i < 24; ++i) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto a = s.var(s.add_var(bw));
auto b = s.var(s.add_var(bw));
permute_args(i, x, y, a, b);
s.add_ult(x * y, b);
s.add_ule(a, x);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_ule(b, a * y);
s.check();
s.expect_unsat();
}
}
// x*y <= b & a <= x & !Omega(x*y) => a*y <= b
static void test_ineq_axiom4(unsigned bw = 32) {
auto const bound = rational::power_of_two(bw/2);
for (unsigned i = 0; i < 24; ++i) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto a = s.var(s.add_var(bw));
auto b = s.var(s.add_var(bw));
permute_args(i, x, y, a, b);
s.add_ule(x * y, b);
s.add_ule(a, x);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_ult(b, a * y);
s.check();
s.expect_unsat();
}
}
// x*y <= b & a <= x & !Omega(x*y) => a*y <= b
static void test_ineq_non_axiom4(unsigned bw, unsigned i) {
auto const bound = rational::power_of_two(bw - 1);
scoped_solver s(__func__);
LOG("permutation: " << i);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto a = s.var(s.add_var(bw));
auto b = s.var(s.add_var(bw));
permute_args(i, x, y, a, b);
s.add_ule(x * y, b);
s.add_ule(a, x);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_ult(b, a * y);
s.check();
s.expect_sat();
}
static void test_ineq_non_axiom4(unsigned bw = 32) {
for (unsigned i = 0; i < 24; ++i)
test_ineq_non_axiom4(bw, i);
}
// a < xy & x <= b & !Omega(x*y) => a < b*y
static void test_ineq_axiom5(unsigned bw = 32) {
auto const bound = rational::power_of_two(bw/2);
for (unsigned i = 0; i < 24; ++i) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto a = s.var(s.add_var(bw));
auto b = s.var(s.add_var(bw));
permute_args(i, x, y, a, b);
s.add_ult(a, x * y);
s.add_ule(x, b);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_ule(b * y, a);
s.check();
s.expect_unsat();
}
}
// a <= xy & x <= b & !Omega(x*y) => a <= b*y
static void test_ineq_axiom6(unsigned bw = 32) {
auto const bound = rational::power_of_two(bw/2);
for (unsigned i = 0; i < 24; ++i) {
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
auto y = s.var(s.add_var(bw));
auto a = s.var(s.add_var(bw));
auto b = s.var(s.add_var(bw));
permute_args(i, x, y, a, b);
s.add_ule(a, x * y);
s.add_ule(x, b);
s.add_ult(x, bound);
s.add_ult(y, bound);
s.add_ult(b * y, a);
s.check();
s.expect_unsat();
}
}
static void test_quot_rem_incomplete() {
unsigned bw = 4;
scoped_solver s(__func__);
s.set_max_conflicts(5);
auto quot = s.var(s.add_var(bw));
auto rem = s.var(s.add_var(bw));
auto a = s.value(rational(2), bw);
auto b = s.value(rational(5), bw);
// Incomplete axiomatization of quotient/remainder.
// quot_rem(2, 5) should have single solution (0, 2),
// but with the usual axioms we also get (3, 3).
s.add_eq(b * quot + rem - a);
s.add_umul_noovfl(b, quot);
s.add_ult(rem, b);
// To force a solution that's different from the correct one.
s.add_diseq(quot - 0);
s.check();
s.expect_sat({{quot, 3}, {rem, 3}});
}
static void test_quot_rem_fixed() {
unsigned bw = 4;
scoped_solver s(__func__);
s.set_max_conflicts(5);
auto a = s.value(rational(2), bw);
auto b = s.value(rational(5), bw);
auto [quot, rem] = s.quot_rem(a, b);
s.add_diseq(quot - 0); // to force a solution that's different from the correct one
s.check();
s.expect_unsat();
}
static void test_quot_rem(unsigned bw = 32) {
scoped_solver s(__func__);
s.set_max_conflicts(5);
auto a = s.var(s.add_var(bw));
auto quot = s.var(s.add_var(bw));
auto rem = s.var(s.add_var(bw));
auto x = a * 123;
auto y = 123;
// quot = udiv(a*123, 123)
s.add_eq(quot * y + rem - x);
s.add_diseq(a - quot);
s.add_umul_noovfl(quot, y);
s.add_ult(rem, x);
s.check();
s.expect_sat();
}
static void test_quot_rem2(unsigned bw = 32) {
scoped_solver s(__func__);
s.set_max_conflicts(5);
auto q = s.var(s.add_var(bw));
auto r = s.var(s.add_var(bw));
auto idx = s.var(s.add_var(bw));
auto second = s.var(s.add_var(bw));
auto first = s.var(s.add_var(bw));
s.add_eq(q*idx + r, UINT_MAX);
s.add_ult(r, idx);
s.add_umul_noovfl(q, idx);
s.add_ult(first, second);
s.add_diseq(idx, 0);
s.add_ule(second - first, q);
s.add_umul_noovfl(second - first, idx);
s.check();
}
static void test_band(unsigned bw = 32) {
{
scoped_solver s(__func__);
auto p = s.var(s.add_var(bw));
auto q = s.var(s.add_var(bw));
s.add_diseq(p - s.band(p, q));
s.add_diseq(p - q);
s.check();
s.expect_sat();
}
{
scoped_solver s(__func__);
auto p = s.var(s.add_var(bw));
auto q = s.var(s.add_var(bw));
s.add_ult(p, s.band(p, q));
s.check();
s.expect_unsat();
}
{
scoped_solver s(__func__);
auto p = s.var(s.add_var(bw));
auto q = s.var(s.add_var(bw));
s.add_ult(q, s.band(p, q));
s.check();
s.expect_unsat();
}
{
scoped_solver s(__func__);
auto p = s.var(s.add_var(bw));
auto q = s.var(s.add_var(bw));
s.add_ule(p, s.band(p, q));
s.check();
s.expect_sat();
}
{
scoped_solver s(__func__);
auto p = s.var(s.add_var(bw));
auto q = s.var(s.add_var(bw));
s.add_ule(p, s.band(p, q));
s.add_diseq(p - s.band(p, q));
s.check();
s.expect_unsat();
}
}
static void test_fi_zero() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(256));
auto b = s.var(s.add_var(256));
auto c = s.var(s.add_var(256));
s.add_eq(a, 0);
s.add_eq(c, 0); // add c to prevent simplification by leading coefficient
s.add_eq(4*a - 123456789*b + c);
s.check();
s.expect_sat({{a, 0}, {b, 0}});
}
static void test_fi_nonzero() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(5));
auto b = s.var(s.add_var(5));
s.add_ult(b*b*b, 7*a + 6);
s.check();
}
static void test_fi_nonmax() {
scoped_solver s(__func__);
auto a = s.var(s.add_var(5));
auto b = s.var(s.add_var(5));
s.add_ult(a + 8, b*b*b);
s.check();
}
static void test_fi_disequal_mild() {
{
// small version
scoped_solver s(__func__);
auto a = s.var(s.add_var(6));
auto b = s.var(s.add_var(6));
// v > -3*v
s.add_eq(a - 3);
s.add_ult(-a*b, b);
s.check();
}
{
// large version
scoped_solver s(__func__);
auto a = s.var(s.add_var(256));
auto b = s.var(s.add_var(256));
// v > -100*v
s.add_eq(a - 100);
s.add_ult(-a*b, b);
s.check();
}
}
// Goal: we probably mix up polysat variables and PDD variables at several points; try to uncover such cases
// NOTE: actually, add_var seems to keep them in sync, so this is not an issue at the moment (but we should still test it later)
// static void test_mixed_vars() {
// scoped_solver s(__func__);
// auto a = s.var(s.add_var(2));
// auto b = s.var(s.add_var(4));
// auto c = s.var(s.add_var(2));
// s.add_eq(a + 2*c + 4);
// s.add_eq(3*b + 4);
// s.check();
// // Expected result:
// }
}; // class test_polysat
// Here we deal with linear constraints of the form
//
// a1*x + b1 <= a2*x + b2 (mod m = 2^bw)
//
// and their negation.
class test_fi {
static bool is_violated(rational const& a1, rational const& b1, rational const& a2, rational const& b2,
rational const& val, bool negated, rational const& m) {
rational const lhs = (a1*val + b1) % m;
rational const rhs = (a2*val + b2) % m;
if (negated)
return lhs <= rhs;
else
return lhs > rhs;
}
// Returns true if the input is valid and the test did useful work
static bool check_one(rational const& a1, rational const& b1, rational const& a2, rational const& b2, rational const& val, bool negated, unsigned bw) {
rational const m = rational::power_of_two(bw);
if (a1.is_zero() && a2.is_zero())
return false;
if (!is_violated(a1, b1, a2, b2, val, negated, m))
return false;
scoped_solver s(__func__);
auto x = s.var(s.add_var(bw));
signed_constraint c = s.ule(a1*x + b1, a2*x + b2);
if (negated)
c.negate();
viable& v = s.m_viable;
v.intersect(x.var(), c);
// Trigger forbidden interval refinement
v.is_viable(x.var(), val);
auto* e = v.m_units[x.var()];
if (!e) {
std::cout << "test_fi: no interval for a1=" << a1 << " b1=" << b1 << " a2=" << a2 << " b2=" << b2 << " val=" << val << " neg=" << negated << std::endl;
// VERIFY(false);
return false;
}
VERIFY(e);
auto* first = e;
SASSERT(e->next() == e); // the result is expected to be a single interval (although for this check it doesn't really matter if there's more...)
do {
rational const& lo = e->interval.lo_val();
rational const& hi = e->interval.hi_val();
for (rational x = lo; x != hi; x = (x + 1) % m) {
// LOG("lo=" << lo << " hi=" << hi << " x=" << x);
if (!is_violated(a1, b1, a2, b2, val, negated, m)) {
std::cout << "test_fi: unsound for a1=" << a1 << " b1=" << b1 << " a2=" << a2 << " b2=" << b2 << " val=" << val << " neg=" << negated << std::endl;
VERIFY(false);
}
}
e = e->next();
}
while (e != first);
return true;
}
public:
static void exhaustive(unsigned bw = 0) {
if (bw == 0) {
exhaustive(1);
exhaustive(2);
exhaustive(3);
exhaustive(4);
exhaustive(5);
}
else {
std::cout << "test_fi::exhaustive for bw=" << bw << std::endl;
rational const m = rational::power_of_two(bw);
for (rational p(1); p < m; ++p) {
for (rational r(1); r < m; ++r) {
// TODO: remove this condition to test the cases other than disequal_lin! (also start p,q from 0)
if (p == r)
continue;
for (rational q(0); q < m; ++q)
for (rational s(0); s < m; ++s)
for (rational v(0); v < m; ++v)
for (bool negated : {true, false})
check_one(p, q, r, s, v, negated, bw);
}
}
}
}
static void randomized(unsigned num_rounds = 100000, unsigned bw = 16) {
std::cout << "test_fi::randomized for bw=" << bw << " (" << num_rounds << " rounds)" << std::endl;
rational const m = rational::power_of_two(bw);
VERIFY(bw <= 32 && "random_gen generates 32-bit numbers");
random_gen rng;
unsigned round = num_rounds;
while (round) {
// rational a1 = (rational(rng()) % (m - 1)) + 1;
// rational a2 = (rational(rng()) % (m - 1)) + 1;
rational a1 = rational(rng()) % m;
rational a2 = rational(rng()) % m;
if (a1.is_zero() || a2.is_zero() || a1 == a2)
continue;
rational b1 = rational(rng()) % m;
rational b2 = rational(rng()) % m;
rational val = rational(rng()) % m;
bool useful =
check_one(a1, b1, a2, b2, val, true, bw)
|| check_one(a1, b1, a2, b2, val, false, bw);
if (useful)
round--;
}
}
}; // class test_fi
// convert assertions into internal solver state
// support small grammar of formulas.
pdd to_pdd(ast_manager& m, solver& s, obj_map<expr, pdd*>& expr2pdd, expr* e) {
pdd* r = nullptr;
if (expr2pdd.find(e, r))
return *r;
bv_util bv(m);
rational n;
unsigned sz = bv.get_bv_size(e);
expr* a, *b;
if (bv.is_bv_add(e, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
r = alloc(pdd, pa + pb);
}
else if (bv.is_bv_sub(e, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
r = alloc(pdd, pa - pb);
}
else if (bv.is_bv_mul(e, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
r = alloc(pdd, pa * pb);
}
else if (bv.is_bv_udiv(e, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
auto qr = s.quot_rem(pa, pb);
r = alloc(pdd, std::get<0>(qr));
}
else if (bv.is_bv_urem(e, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
auto qr = s.quot_rem(pa, pb);
r = alloc(pdd, std::get<1>(qr));
}
else if (bv.is_bv_lshr(e, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
r = alloc(pdd, s.lshr(pa, pb));
}
else if (bv.is_bv_and(e) && to_app(e)->get_num_args() == 2) {
a = to_app(e)->get_arg(0);
b = to_app(e)->get_arg(1);
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
r = alloc(pdd, s.band(pa, pb));
}
else if (bv.is_bv_neg(e)) {
a = to_app(e)->get_arg(0);
auto pa = to_pdd(m, s, expr2pdd, a);
r = alloc(pdd, -pa);
}
else if (bv.is_numeral(e, n, sz))
r = alloc(pdd, s.value(n, sz));
else if (is_uninterp(e))
r = alloc(pdd, s.var(s.add_var(sz)));
else {
std::cout << "UNKNOWN " << mk_pp(e, m) << "\n";
NOT_IMPLEMENTED_YET();
r = alloc(pdd, s.var(s.add_var(sz)));
}
expr2pdd.insert(e, r);
return *r;
}
void internalize(ast_manager& m, solver& s, ptr_vector<expr>& fmls) {
bv_util bv(m);
obj_map<expr, pdd*> expr2pdd;
for (expr* fm : fmls) {
bool is_not = m.is_not(fm, fm);
expr* a, *b;
if (m.is_eq(fm, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
if (is_not)
s.add_diseq(pa - pb);
else
s.add_eq(pa - pb);
}
else if (bv.is_ult(fm, a, b) || bv.is_ugt(fm, b, a)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
if (is_not)
s.add_ule(pb, pa);
else
s.add_ult(pa, pb);
}
else if (bv.is_ule(fm, a, b) || bv.is_uge(fm, b, a)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
if (is_not)
s.add_ult(pb, pa);
else
s.add_ule(pa, pb);
}
else if (bv.is_slt(fm, a, b) || bv.is_sgt(fm, b, a)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
if (is_not)
s.add_sle(pb, pa);
else
s.add_slt(pa, pb);
}
else if (bv.is_sle(fm, a, b) || bv.is_sge(fm, b, a)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
if (is_not)
s.add_slt(pb, pa);
else
s.add_sle(pa, pb);
}
else if (bv.is_bv_umul_no_ovfl(fm, a, b)) {
auto pa = to_pdd(m, s, expr2pdd, a);
auto pb = to_pdd(m, s, expr2pdd, b);
if (is_not)
s.add_umul_ovfl(pa, pb);
else
s.add_umul_noovfl(pa, pb);
}
else {
std::cout << "SKIP: " << mk_pp(fm, m) << "\n";
}
}
for (auto const& [k,v] : expr2pdd)
dealloc(v);
}
} // namespace polysat
void tst_polysat() {
using namespace polysat;
test_polysat::test_fi_zero();
test_polysat::test_fi_nonzero();
test_polysat::test_fi_nonmax();
test_polysat::test_fi_disequal_mild();
#if 0
// looks like a fishy conflict lemma?
test_polysat::test_monot_bounds();
return;
test_polysat::test_quot_rem_incomplete();
test_polysat::test_quot_rem_fixed();
//return;
test_polysat::test_band();
return;
test_polysat::test_quot_rem();
return;
test_polysat::test_ineq_axiom1();
test_polysat::test_ineq_axiom2();
test_polysat::test_ineq_axiom3();
test_polysat::test_ineq_axiom4();
test_polysat::test_ineq_axiom5();
test_polysat::test_ineq_axiom6();
return;
#endif
// test_polysat::test_monot_bounds(8);
test_polysat::test_add_conflicts();
test_polysat::test_wlist();
test_polysat::test_l1();
test_polysat::test_l2();
test_polysat::test_l3();
test_polysat::test_l4();
test_polysat::test_l5();
test_polysat::test_p1();
test_polysat::test_p2();
test_polysat::test_p3();
test_polysat::test_ineq_basic1();
test_polysat::test_ineq_basic2();
test_polysat::test_ineq_basic3();
test_polysat::test_ineq_basic4();
test_polysat::test_ineq_basic5();
test_polysat::test_ineq_basic6();
test_polysat::test_cjust();
test_polysat::test_subst();
test_polysat::test_var_minimize();
test_polysat::test_ineq1();
test_polysat::test_ineq2();
test_polysat::test_monot();
test_polysat::test_monot_bounds(2);
return;
test_polysat::test_ineq_axiom1();
test_polysat::test_ineq_axiom2();
test_polysat::test_ineq_axiom3();
test_polysat::test_ineq_axiom4();
test_polysat::test_ineq_axiom5();
test_polysat::test_ineq_axiom6();
test_fi::exhaustive();
test_fi::randomized();
return;
#if 0
test_polysat::test_ineq_non_axiom4(32, 5);
#endif
// inefficient conflicts:
// Takes time: test_polysat::test_monot_bounds_full();
test_polysat::test_monot_bounds_simple(8);
test_polysat::test_fixed_point_arith_div_mul_inverse();
}
#include "ast/bv_decl_plugin.h"
#include <signal.h>
polysat::scoped_solver* g_solver = nullptr;
static void display_statistics() {
if (g_solver) {
statistics st;
g_solver->collect_statistics(st);
std::cout << st << "\n";
}
}
static void STD_CALL on_ctrl_c(int) {
signal (SIGINT, SIG_DFL);
display_statistics();
raise(SIGINT);
}
void tst_polysat_argv(char** argv, int argc, int& i) {
// set up SMT2 parser to extract assertions
// assume they are simple bit-vector equations (and inequations)
// convert to solver state.
signal(SIGINT, on_ctrl_c);
if (argc < 3) {
std::cerr << "Usage: " << argv[0] << " FILE\n";
return;
}
std::cout << "processing " << argv[2] << "\n";
std::ifstream is(argv[2]);
if (is.bad() || is.fail()) {
std::cout << "failed to open " << argv[2] << "\n";
return;
}
cmd_context ctx(false);
ast_manager& m = ctx.m();
ctx.set_ignore_check(true);
VERIFY(parse_smt2_commands(ctx, is));
ptr_vector<expr> fmls = ctx.assertions();
polysat::scoped_solver s("polysat");
s.set_max_conflicts(1000);
g_solver = &s;
polysat::internalize(m, s, fmls);
std::cout << "checking\n";
s.check();
}
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