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/*++
Copyright (c) 2017 Microsoft Corporation
Module Name:
nla_divisions.cpp
Author:
Lev Nachmanson (levnach)
Nikolaj Bjorner (nbjorner)
Description:
Check divisions
--*/
#include "math/lp/nla_core.h"
namespace nla {
void divisions::add_idivision(lpvar q, lpvar x, lpvar y, lpvar r) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar || r == null_lpvar)
return;
m_idivisions.push_back({q, x, y, r});
m_core.trail().push(push_back_vector(m_idivisions));
}
void divisions::add_rdivision(lpvar q, lpvar x, lpvar y, lpvar r) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar || r == null_lpvar)
return;
m_rdivisions.push_back({ q, x, y, r });
m_core.trail().push(push_back_vector(m_rdivisions));
}
void divisions::add_bounded_division(lpvar q, lpvar x, lpvar y, lpvar r) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar || r == null_lpvar)
return;
if (m_core.lra.column_has_term(x) || m_core.lra.column_has_term(y) || m_core.lra.column_has_term(q))
return;
m_bounded_divisions.push_back({ q, x, y, r });
m_core.trail().push(push_back_vector(m_bounded_divisions));
}
void divisions::add_divisibility(lpvar r, lpvar x, lpvar y, lpvar d) {
if (x == null_lpvar || y == null_lpvar || r == null_lpvar || d == null_lpvar)
return;
m_divisibility.push_back({ r, x, y, d });
m_core.trail().push(push_back_vector(m_divisibility));
}
typedef lp::lar_term term;
// y1 >= y2 > 0 & x1 <= x2 => x1/y1 <= x2/y2
// y2 <= y1 < 0 & x1 >= x2 >= 0 => x1/y1 <= x2/y2
// y2 <= y1 < 0 & x1 <= x2 <= 0 => x1/y1 >= x2/y2
void divisions::check() {
core& c = m_core;
if (c.use_nra_model())
return;
auto monotonicity1 = [&](auto x1, auto& x1val, auto y1, auto& y1val, auto& q1, auto& q1val,
auto x2, auto& x2val, auto y2, auto& y2val, auto& q2, auto& q2val) {
if (y1val >= y2val && y2val > 0 && 0 <= x1val && x1val <= x2val && q1val > q2val) {
lemma_builder lemma(c, "y1 >= y2 > 0 & 0 <= x1 <= x2 => x1/y1 <= x2/y2");
lemma |= ineq(term(y1, rational(-1), y2), llc::LT, 0);
lemma |= ineq(y2, llc::LE, 0);
lemma |= ineq(x1, llc::LT, 0);
lemma |= ineq(term(x1, rational(-1), x2), llc::GT, 0);
lemma |= ineq(term(q1, rational(-1), q2), llc::LE, 0);
return true;
}
return false;
};
auto monotonicity2 = [&](auto x1, auto& x1val, auto y1, auto& y1val, auto& q1, auto& q1val,
auto x2, auto& x2val, auto y2, auto& y2val, auto& q2, auto& q2val) {
if (y2val <= y1val && y1val < 0 && x1val >= x2val && x2val >= 0 && q1val > q2val) {
lemma_builder lemma(c, "y2 <= y1 < 0 & x1 >= x2 >= 0 => x1/y1 <= x2/y2");
lemma |= ineq(term(y1, rational(-1), y2), llc::LT, 0);
lemma |= ineq(y1, llc::GE, 0);
lemma |= ineq(term(x1, rational(-1), x2), llc::LT, 0);
lemma |= ineq(x2, llc::LT, 0);
lemma |= ineq(term(q1, rational(-1), q2), llc::LE, 0);
return true;
}
return false;
};
auto monotonicity3 = [&](auto x1, auto& x1val, auto y1, auto& y1val, auto& q1, auto& q1val,
auto x2, auto& x2val, auto y2, auto& y2val, auto& q2, auto& q2val) {
if (y2val <= y1val && y1val < 0 && x1val <= x2val && x2val <= 0 && q1val < q2val) {
lemma_builder lemma(c, "y2 <= y1 < 0 & x1 <= x2 <= 0 => x1/y1 >= x2/y2");
lemma |= ineq(term(y1, rational(-1), y2), llc::LT, 0);
lemma |= ineq(y1, llc::GE, 0);
lemma |= ineq(term(x1, rational(-1), x2), llc::GT, 0);
lemma |= ineq(x2, llc::GT, 0);
lemma |= ineq(term(q1, rational(-1), q2), llc::GE, 0);
return true;
}
return false;
};
auto monotonicity = [&](auto x1, auto& x1val, auto y1, auto& y1val, auto& q1, auto& q1val,
auto x2, auto& x2val, auto y2, auto& y2val, auto& q2, auto& q2val) {
if (monotonicity1(x1, x1val, y1, y1val, q1, q1val, x2, x2val, y2, y2val, q2, q2val))
return true;
if (monotonicity1(x2, x2val, y2, y2val, q2, q2val, x1, x1val, y1, y1val, q1, q1val))
return true;
if (monotonicity2(x1, x1val, y1, y1val, q1, q1val, x2, x2val, y2, y2val, q2, q2val))
return true;
if (monotonicity2(x2, x2val, y2, y2val, q2, q2val, x1, x1val, y1, y1val, q1, q1val))
return true;
if (monotonicity3(x1, x1val, y1, y1val, q1, q1val, x2, x2val, y2, y2val, q2, q2val))
return true;
if (monotonicity3(x2, x2val, y2, y2val, q2, q2val, x1, x1val, y1, y1val, q1, q1val))
return true;
return false;
};
for (auto const & [r, x, y, md] : m_idivisions) {
if (!c.is_relevant(r))
continue;
auto xval = c.val(x);
auto yval = c.val(y);
auto rval = c.val(r);
// idiv semantics
if (!xval.is_int() || !yval.is_int() || yval == 0 || rval == div(xval, yval))
continue;
for (auto const& [q2, x2, y2, md2] : m_idivisions) {
if (q2 == r)
continue;
if (!c.is_relevant(q2))
continue;
auto x2val = c.val(x2);
auto y2val = c.val(y2);
auto q2val = c.val(q2);
if (monotonicity(x, xval, y, yval, r, rval, x2, x2val, y2, y2val, q2, q2val))
return;
}
}
for (auto const& [r, x, y, md] : m_rdivisions) {
if (!c.is_relevant(r))
continue;
auto xval = c.val(x);
auto yval = c.val(y);
auto rval = c.val(r);
// / semantics
if (yval == 0 || rval == xval / yval)
continue;
for (auto const& [q2, x2, y2, md2] : m_rdivisions) {
if (q2 == r)
continue;
if (!c.is_relevant(q2))
continue;
auto x2val = c.val(x2);
auto y2val = c.val(y2);
auto q2val = c.val(q2);
if (monotonicity(x, xval, y, yval, r, rval, x2, x2val, y2, y2val, q2, q2val))
return;
}
}
check_mod_mult();
check_linear_divisibility();
check_mod_congruence();
}
// if p is bounded, q a value, r = eval(p):
// p <= q * div(r, q) + q - 1 => div(p, q) <= div(r, q)
// p >= q * div(r, q) => div(r, q) <= div(p, q)
void divisions::check_bounded_divisions() {
core& c = m_core;
unsigned offset = c.random(), sz = m_bounded_divisions.size();
for (unsigned j = 0; j < sz; ++j) {
unsigned i = (offset + j) % sz;
auto [q, x, y, r] = m_bounded_divisions[i];
if (!c.is_relevant(q))
continue;
auto xv = c.val(x);
auto yv = c.val(y);
auto qv = c.val(q);
if (xv < 0 || !xv.is_int())
continue;
if (yv <= 0 || !yv.is_int())
continue;
if (qv == div(xv, yv))
continue;
rational div_v = div(xv, yv);
// y = yv & x <= yv * div(xv, yv) + yv - 1 => div(x, y) <= div(xv, yv)
// y = yv & x >= y * div(xv, yv) => div(xv, yv) <= div(x, y)
rational mul(1);
rational hi = yv * div_v + yv - 1;
rational lo = yv * div_v;
if (xv > hi) {
lemma_builder lemma(c, "y = yv & x <= yv * div(xv, yv) + yv - 1 => div(p, y) <= div(xv, yv)");
lemma |= ineq(y, llc::NE, yv);
lemma |= ineq(x, llc::GT, hi);
lemma |= ineq(q, llc::LE, div_v);
return;
}
if (xv < lo) {
lemma_builder lemma(c, "y = yv & x >= yv * div(xv, yv) => div(xv, yv) <= div(x, y)");
lemma |= ineq(y, llc::NE, yv);
lemma |= ineq(x, llc::LT, lo);
lemma |= ineq(q, llc::GE, div_v);
return;
}
}
}
// mod(factor, p) = 0 => mod(factor * k, p) = 0
// For each division (q, x, y, r) where x is a monic m = f1 * f2 * ... * fk,
// if some factor fi has mod(fi, p) = 0 (fixed), then mod(x, p) = 0.
void divisions::check_mod_mult() {
core& c = m_core;
unsigned offset = c.random(), sz = m_bounded_divisions.size();
for (unsigned j = 0; j < sz; ++j) {
unsigned i = (offset + j) % sz;
auto [q, x, y, r] = m_bounded_divisions[i];
if (!c.is_relevant(q))
continue;
if (c.var_is_fixed_to_zero(r))
continue;
if (c.val(r).is_zero())
continue;
if (!c.is_monic_var(x))
continue;
auto yv = c.val(y);
if (yv <= 0 || !yv.is_int())
continue;
auto const& m = c.emons()[x];
for (lpvar f : m.vars()) {
for (auto const& [q2, x2, y2, r2] : m_bounded_divisions) {
if (x2 != f)
continue;
if (c.val(y2) != yv)
continue;
if (!c.var_is_fixed_to_zero(r2))
continue;
// mod(factor, p) = 0 => mod(product, p) = 0
lemma_builder lemma(c, "mod(factor, p) = 0 => mod(factor * k, p) = 0");
lemma |= ineq(r2, llc::NE, 0);
lemma |= ineq(r, llc::EQ, 0);
return;
}
}
}
}
// Linear divisibility closure:
// y != 0 & mod(a, y) = 0 & x = c * a (c an integer constant) => mod(x, y) = 0.
// The emitted clause
// y = 0 \/ (y != y2) \/ (x - c*a != 0) \/ (mod(a, y2) != 0) \/ (mod(x, y) = 0)
// is a tautology for every integer c (under the Euclidean semantics of mod),
// so the choice of c/a from the current model can never be unsound. We only
// emit it when the arithmetic literals are false in the current model, which makes
// the clause a real conflict/propagation and guarantees progress.
void divisions::check_linear_divisibility() {
core& c = m_core;
unsigned sz = m_divisibility.size();
for (unsigned i = 0; i < sz; ++i) {
auto const& [rx, x, y, dx] = m_divisibility[i];
if (!c.is_relevant(rx))
continue;
if (c.val(rx).is_zero()) // mod(x, y) already 0 in model: nothing to refute
continue;
auto xval = c.val(x);
auto yval = c.val(y);
if (yval.is_zero())
continue;
if (xval.is_zero())
continue;
for (unsigned j = 0; j < sz; ++j) {
if (i == j)
continue;
auto const& [ra, a, y2, da] = m_divisibility[j];
if (y2 != y && c.val(y2) != yval) // same divisor (by column or value)
continue;
if (!c.is_relevant(ra))
continue;
if (!c.val(ra).is_zero()) // need mod(a, y) = 0 in model
continue;
auto aval = c.val(a);
if (aval.is_zero())
continue;
rational cc = xval / aval;
if (!cc.is_int() || cc.is_zero())
continue;
if (xval != cc * aval) // ensure x = c*a holds exactly in the model
continue;
lemma_builder lemma(c, "y != 0 & y = y2 & mod(a,y2) = 0 & x = c*a => mod(x,y) = 0");
lemma |= ineq(y, llc::EQ, 0); // y = 0 (guard: mod/div uninterpreted when divisor is 0)
if (y2 != y)
lemma |= ineq(term(y, rational(-1), y2), llc::NE, 0); // y != y2 (guard: lemma only applies when divisors are equal)
lemma |= ineq(term(x, -cc, a), llc::NE, 0); // x - c*a != 0
lemma |= ineq(ra, llc::NE, 0); // mod(a, y) != 0
lemma |= ineq(rx, llc::EQ, 0); // mod(x, y) = 0
return;
}
}
}
// Modular congruence over a shared (possibly symbolic) divisor.
//
// For each divisibility fact we have the Euclidean identities (asserted by
// theory_lra::mk_idiv_mod_axioms):
// x = y * div(x,y) + mod(x,y), 0 <= mod(x,y) < |y|.
// For two facts (rx = mod(x,y), dx = div(x,y)) and (rs = mod(s,y), ds = div(s,y))
// sharing divisor y, subtracting the identities gives, for every integer delta,
// div(x,y) - div(s,y) = delta => mod(x,y) - mod(s,y) = (x - s) - delta*y.
// This is a tautology (entailed by the two identities) for any fixed integer
// delta, so choosing delta from the current model can never be unsound. We emit
// the clause
// (div(x,y) - div(s,y) != delta) \/ (mod(x,y) - mod(s,y) - (x - s) + delta*y = 0)
// only when the equality literal is false in the model (delta taken as the model
// value of div(x,y) - div(s,y)), which makes the clause a real propagation and
// guarantees progress. This discharges linear congruences with a symbolic
// modulus (e.g. mod(i + s, n) = i + mod(s, n)) that the nonlinear core does not
// otherwise isolate.
void divisions::check_mod_congruence() {
core& c = m_core;
unsigned sz = m_divisibility.size();
for (unsigned i = 0; i < sz; ++i) {
auto const& [rx, x, y, dx] = m_divisibility[i];
if (!c.is_relevant(rx))
continue;
auto yval = c.val(y);
if (yval.is_zero()) // mod/div uninterpreted when the divisor is 0
continue;
for (unsigned j = i + 1; j < sz; ++j) {
auto const& [rs, s, y2, ds] = m_divisibility[j];
if (!c.is_relevant(rs))
continue;
if (y2 != y && c.val(y2) != yval) // same divisor (by column or value)
continue;
rational delta = c.val(dx) - c.val(ds);
rational lhs = c.val(rx) - c.val(rs);
rational rhs = (c.val(x) - c.val(s)) - delta * yval;
if (lhs == rhs) // residue equation already holds: nothing to propagate
continue;
lemma_builder lemma(c, "y != 0 & y = y2 & div(x,y) - div(s,y) = delta => mod(x,y) - mod(s,y) = (x - s) - delta*y");
lemma |= ineq(y, llc::EQ, 0); // y = 0 (guard: mod/div uninterpreted when divisor is 0)
if (y2 != y)
lemma |= ineq(term(y, rational(-1), y2), llc::NE, 0); // y != y2 (guard: divisors must coincide symbolically)
lemma |= ineq(term(dx, rational(-1), ds), llc::NE, delta); // div(x,y) - div(s,y) != delta
term t;
t.add_monomial(rational::one(), rx);
t.add_monomial(rational(-1), rs);
t.add_monomial(rational(-1), x);
t.add_monomial(rational::one(), s);
t.add_monomial(delta, y);
lemma |= ineq(t, llc::EQ, 0); // mod(x,y) - mod(s,y) - x + s + delta*y = 0
return;
}
}
}
}