mirror of
https://github.com/Z3Prover/z3
synced 2025-04-13 12:28:44 +00:00
Add clean_denominators procedure
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura
parent
d60f2db116
commit
1d761ea9a5
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@ -2416,13 +2416,32 @@ namespace realclosure {
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return new (allocator()) rational_value();
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}
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rational_value * mk_rational(mpq & v) {
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/**
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\brief Make a rational and swap its value with v
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*/
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rational_value * mk_rational_and_swap(mpq & v) {
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SASSERT(!qm().is_zero(v));
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rational_value * r = mk_rational();
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::swap(r->m_value, v);
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return r;
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}
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rational_value * mk_rational(mpq const & v) {
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SASSERT(!qm().is_zero(v));
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rational_value * r = mk_rational();
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qm().set(r->m_value, v);
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return r;
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}
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rational_value * mk_rational(mpz const & v) {
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SASSERT(!qm().is_zero(v));
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rational_value * r = mk_rational();
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qm().set(r->m_value, v);
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return r;
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}
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rational_value * mk_rational(mpbq const & v) {
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SASSERT(!bqm().is_zero(v));
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scoped_mpq v_q(qm()); // v as a rational
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::to_mpq(qm(), v, v_q);
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return mk_rational(v_q);
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@ -2921,7 +2940,7 @@ namespace realclosure {
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for (unsigned i = 1; i < sz; i++) {
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mpq i_mpq(i);
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value_ref a_i(*this);
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a_i = mk_rational(i_mpq);
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a_i = mk_rational_and_swap(i_mpq);
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mul(a_i, p[i], a_i);
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r.push_back(a_i);
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}
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@ -3086,30 +3105,42 @@ namespace realclosure {
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return true;
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}
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/**
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\brief r <- p(b)
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*/
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void mk_polynomial_value(unsigned n, value * const * p, value * b, value_ref & r) {
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SASSERT(n > 0);
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if (n == 1 || b == 0) {
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r = p[0];
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}
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else {
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SASSERT(n >= 2);
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// We compute the result using the Horner Sequence
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// ((a_{n-1}*b + a_{n-2})*b + a_{n-3})*b + a_{n-4} ...
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// where a_i's are the coefficients of p.
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mul(p[n - 1], b, r); // r <- a_{n-1} * b
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unsigned i = n - 1;
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while (i > 0) {
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--i;
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if (p[i] != 0)
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add(r, p[i], r); // r <- r + a_i
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if (i > 0)
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mul(r, b, r); // r <- r * b
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}
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}
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}
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/**
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\brief Evaluate the sign of p(b) by computing a value object.
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*/
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int expensive_eval_sign_at(unsigned n, value * const * p, mpbq const & b) {
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SASSERT(n > 0);
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SASSERT(p[n - 1] != 0);
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value_ref bv(*this);
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bv = mk_rational(b);
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// We compute the result using the Horner Sequence
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// ((a_{n-1}*bv + a_{n-2})*bv + a_{n-3})*bv + a_{n-4} ...
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// where a_i's are the coefficients of p.
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value_ref r(*this);
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// r <- a_{n-1} * bv
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mul(p[n - 1], bv, r);
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unsigned i = n - 1;
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while (i > 0) {
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checkpoint();
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--i;
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if (p[i] != 0)
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add(r, p[i], r); // r <- r + a_i
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if (i > 0)
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mul(r, bv, r); // r <- r * bv
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}
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return sign(r);
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value_ref _b(*this);
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_b = mk_rational(b);
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value_ref pb(*this);
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mk_polynomial_value(n, p, _b, pb);
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return sign(pb);
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}
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/**
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@ -4290,7 +4321,7 @@ namespace realclosure {
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if (qm().is_zero(v))
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r = 0;
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else
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r = mk_rational(v);
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r = mk_rational_and_swap(v);
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}
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else {
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INC_DEPTH();
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@ -4319,7 +4350,7 @@ namespace realclosure {
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if (qm().is_zero(v))
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r = 0;
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else
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r = mk_rational(v);
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r = mk_rational_and_swap(v);
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}
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else {
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value_ref neg_b(*this);
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@ -4353,7 +4384,7 @@ namespace realclosure {
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scoped_mpq v(qm());
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qm().set(v, to_mpq(a));
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qm().neg(v);
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r = mk_rational(v);
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r = mk_rational_and_swap(v);
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}
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else {
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neg_rf(to_rational_function(a), r);
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@ -4495,7 +4526,7 @@ namespace realclosure {
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else if (is_nz_rational(a) && is_nz_rational(b)) {
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scoped_mpq v(qm());
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qm().mul(to_mpq(a), to_mpq(b), v);
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r = mk_rational(v);
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r = mk_rational_and_swap(v);
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}
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else {
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INC_DEPTH();
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@ -4530,7 +4561,7 @@ namespace realclosure {
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else if (is_nz_rational(a) && is_nz_rational(b)) {
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scoped_mpq v(qm());
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qm().div(to_mpq(a), to_mpq(b), v);
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r = mk_rational(v);
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r = mk_rational_and_swap(v);
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}
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else {
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value_ref inv_b(*this);
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@ -4561,7 +4592,7 @@ namespace realclosure {
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if (is_nz_rational(a)) {
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scoped_mpq v(qm());
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qm().inv(to_mpq(a), v);
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r = mk_rational(v);
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r = mk_rational_and_swap(v);
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}
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else {
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inv_rf(to_rational_function(a), r);
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@ -4651,6 +4682,280 @@ namespace realclosure {
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return compare(a.m_value, b.m_value);
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}
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// ---------------------------------
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//
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// Structural equality
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//
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// ---------------------------------
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/**
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\brief Values a and b are said to be "structurally" equal if:
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- a and b are 0.
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- a and b are rationals and compare(a, b) == 0
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- a and b are rational function values p_a(x)/q_a(x) and p_b(y)/q_b(y) where x and y are field extensions, and
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* x == y (pointer equality, i.e., they are the same field extension object).
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* Every coefficient of p_a is structurally equal to every coefficient of p_b
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* Every coefficient of q_a is structurally equal to every coefficient of q_b
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Clearly structural equality implies equality, but the reverse is not true.
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*/
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bool struct_eq(value * a, value * b) const {
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if (a == b)
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return true;
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else if (a == 0 || b == 0)
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return false;
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else if (is_nz_rational(a) && is_nz_rational(b))
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return qm().eq(to_mpq(a), to_mpq(b));
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else if (is_nz_rational(a) || is_nz_rational(b))
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return false;
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else {
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SASSERT(is_rational_function(a));
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SASSERT(is_rational_function(b));
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rational_function_value * rf_a = to_rational_function(a);
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rational_function_value * rf_b = to_rational_function(b);
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if (rf_a->ext() != rf_b->ext())
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return false;
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return struct_eq(rf_a->num(), rf_b->num()) && struct_eq(rf_a->den(), rf_b->den());
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}
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}
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/**
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Auxiliary method for
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bool struct_eq(value * a, value * b)
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*/
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bool struct_eq(unsigned sz_a, value * const * p_a, unsigned sz_b, value * const * p_b) const {
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if (sz_a != sz_b)
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return false;
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for (unsigned i = 0; i < sz_a; i++) {
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if (!struct_eq(p_a[i], p_b[i]))
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return false;
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}
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return true;
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}
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/**
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Auxiliary method for
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bool struct_eq(value * a, value * b)
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*/
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bool struct_eq(polynomial const & p_a, polynomial const & p_b) const {
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return struct_eq(p_a.size(), p_a.c_ptr(), p_b.size(), p_b.c_ptr());
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}
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// ---------------------------------
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//
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// Clean denominators
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//
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// ---------------------------------
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/**
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\brief We say 'a' has "clean" denominators if
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- a is 0
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- a is a rational_value that is an integer
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- a is a rational_function_value of the form p_a(x)/1 where the coefficients of p_a also have clean denominators.
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*/
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bool has_clean_denominators(value * a) const {
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if (a == 0)
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return true;
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else if (is_nz_rational(a))
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return qm().is_int(to_mpq(a));
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else {
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rational_function_value * rf_a = to_rational_function(a);
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return is_rational_one(rf_a->den()) && has_clean_denominators(rf_a->num());
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}
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}
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/**
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\brief See comment at has_clean_denominators(value * a)
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*/
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bool has_clean_denominators(polynomial const & p) const {
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unsigned sz = p.size();
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for (unsigned i = 0; i < sz; i++) {
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if (!has_clean_denominators(p[i]))
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return false;
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}
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return true;
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}
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/**
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\brief "Clean" the denominators of 'a'. That is, return p and q s.t.
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a == p/q
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and
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has_clean_denominators(p) and has_clean_denominators(q)
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*/
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void clean_denominators_core(value * a, value_ref & p, value_ref & q) {
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INC_DEPTH();
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TRACE("rcf_clean", tout << "clean_denominators_core [" << m_exec_depth << "]\na: "; display(tout, a, false); tout << "\n";);
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p.reset(); q.reset();
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if (a == 0) {
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p = a;
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q = one();
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}
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else if (is_nz_rational(a)) {
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p = mk_rational(to_mpq(a).numerator());
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q = mk_rational(to_mpq(a).denominator());
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}
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else {
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rational_function_value * rf_a = to_rational_function(a);
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value_ref_buffer p_num(*this), p_den(*this);
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value_ref d_num(*this), d_den(*this);
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clean_denominators_core(rf_a->num(), p_num, d_num);
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clean_denominators_core(rf_a->den(), p_den, d_den);
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value_ref x(*this);
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x = mk_rational_function_value(rf_a->ext());
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mk_polynomial_value(p_num.size(), p_num.c_ptr(), x, p);
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mk_polynomial_value(p_den.size(), p_den.c_ptr(), x, q);
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if (!struct_eq(d_den, d_num)) {
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mul(p, d_den, p);
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mul(q, d_num, q);
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}
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}
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}
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/**
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\brief Clean the denominators of the polynomial p, it returns clean_p and d s.t.
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p = clean_p/d
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and has_clean_denominators(clean_p) && has_clean_denominators(d)
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*/
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void clean_denominators_core(polynomial const & p, value_ref_buffer & norm_p, value_ref & d) {
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value_ref_buffer nums(*this), dens(*this);
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value_ref a_n(*this), a_d(*this);
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bool all_one = true;
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for (unsigned i = 0; i < p.size(); i++) {
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if (p[i]) {
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clean_denominators_core(p[i], a_n, a_d);
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nums.push_back(a_n);
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if (!is_rational_one(a_d))
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all_one = false;
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dens.push_back(a_d);
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}
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else {
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nums.push_back(0);
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dens.push_back(0);
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}
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}
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if (all_one) {
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norm_p = nums;
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d = one();
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}
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else {
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// Compute lcm of the integer elements in dens.
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// This is a little trick to control the coefficient growth.
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// We don't compute lcm of the other elements of dens because it is too expensive.
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scoped_mpq lcm_z(qm());
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bool found_z = false;
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SASSERT(nums.size() == p.size());
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SASSERT(dens.size() == p.size());
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for (unsigned i = 0; i < p.size(); i++) {
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if (!dens[i])
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continue;
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if (is_nz_rational(dens[i])) {
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mpq const & _d = to_mpq(dens[i]);
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SASSERT(qm().is_int(_d));
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if (!found_z) {
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found_z = true;
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qm().set(lcm_z, _d);
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}
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else {
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qm().lcm(lcm_z, _d, lcm_z);
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}
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}
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}
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value_ref lcm(*this);
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if (found_z) {
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lcm = mk_rational(lcm_z);
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}
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else {
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lcm = one();
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}
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// Compute the multipliers for nums.
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// Compute norm_p and d
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//
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// We do NOT use GCD to compute the LCM of the denominators of non-rational values.
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// However, we detect structurally equivalent denominators.
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//
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// Thus a/(b+1) + c/(b+1) is converted into a*c/(b+1) instead of (a*(b+1) + c*(b+1))/(b+1)^2
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norm_p.reset();
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d = lcm;
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value_ref_buffer multipliers(*this);
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value_ref m(*this);
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for (unsigned i = 0; i < p.size(); i++) {
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if (!nums[i]) {
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norm_p.push_back(0);
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}
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else {
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SASSERT(dens[i]);
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bool is_z;
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if (!is_nz_rational(dens[i])) {
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m = lcm;
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is_z = false;
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}
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else {
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scoped_mpq num_z(qm());
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qm().div(lcm_z, to_mpq(dens[i]), num_z);
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SASSERT(qm().is_int(num_z));
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m = mk_rational_and_swap(num_z);
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is_z = true;
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}
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bool found_lt_eq = false;
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for (unsigned j = 0; j < p.size(); j++) {
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TRACE("rcf_clean_bug", tout << "j: " << j << " "; display(tout, m, false); tout << "\n";);
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if (!dens[j])
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continue;
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if (i != j && !is_nz_rational(dens[j])) {
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if (struct_eq(dens[i], dens[j])) {
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if (j < i)
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found_lt_eq = true;
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}
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else {
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mul(m, dens[j], m);
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}
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}
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}
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if (!is_z && !found_lt_eq) {
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mul(dens[i], d, d);
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}
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mul(m, nums[i], m);
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norm_p.push_back(m);
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}
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}
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}
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SASSERT(norm_p.size() == p.size());
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}
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void clean_denominators(value * a, value_ref & p, value_ref & q) {
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if (has_clean_denominators(a)) {
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p = a;
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q = one();
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}
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else {
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clean_denominators_core(a, p, q);
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}
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}
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void clean_denominators(polynomial const & p, value_ref_buffer & norm_p, value_ref & d) {
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if (has_clean_denominators(p)) {
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norm_p.append(p.size(), p.c_ptr());
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d = one();
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}
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else {
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clean_denominators_core(p, norm_p, d);
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}
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}
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void clean_denominators(numeral const & a, numeral & p, numeral & q) {
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value_ref _p(*this), _q(*this);
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clean_denominators(a.m_value, _p, _q);
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set(p, _p);
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set(q, _q);
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}
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// ---------------------------------
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//
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// "Pretty printing"
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//
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// ---------------------------------
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struct collect_algebraic_refs {
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char_vector m_visited; // Set of visited algebraic extensions.
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ptr_vector<algebraic> m_found; // vector/list of visited algebraic extensions.
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@ -4683,11 +4988,20 @@ namespace realclosure {
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}
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};
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static unsigned num_nz_coeffs(polynomial const & p) {
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unsigned r = 0;
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for (unsigned i = 0; i < p.size(); i++) {
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if (p[i])
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r++;
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}
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return r;
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}
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bool use_parenthesis(value * v) const {
|
||||
if (is_zero(v) || is_nz_rational(v))
|
||||
return false;
|
||||
rational_function_value * rf = to_rational_function(v);
|
||||
return rf->num().size() > 1 || !is_rational_one(rf->den());
|
||||
return num_nz_coeffs(rf->num()) > 1 || !is_rational_one(rf->den());
|
||||
}
|
||||
|
||||
template<typename DisplayVar>
|
||||
|
|
@ -5140,6 +5454,10 @@ namespace realclosure {
|
|||
m_imp->display_interval(out, a);
|
||||
}
|
||||
|
||||
void manager::clean_denominators(numeral const & a, numeral & p, numeral & q) {
|
||||
save_interval_ctx ctx(this);
|
||||
m_imp->clean_denominators(a, p, q);
|
||||
}
|
||||
};
|
||||
|
||||
void pp(realclosure::manager::imp * imp, realclosure::polynomial const & p, realclosure::extension * ext) {
|
||||
|
|
@ -5162,6 +5480,10 @@ void pp(realclosure::manager::imp * imp, realclosure::manager::imp::value_ref_bu
|
|||
pp(imp, p[i]);
|
||||
}
|
||||
|
||||
void pp(realclosure::manager::imp * imp, realclosure::manager::imp::value_ref const & v) {
|
||||
pp(imp, v.get());
|
||||
}
|
||||
|
||||
void pp(realclosure::manager::imp * imp, realclosure::mpbqi const & i) {
|
||||
imp->bqim().display(std::cout, i);
|
||||
std::cout << std::endl;
|
||||
|
|
|
|||
|
|
@ -263,6 +263,8 @@ namespace realclosure {
|
|||
|
||||
|
||||
void display_interval(std::ostream & out, numeral const & a) const;
|
||||
|
||||
void clean_denominators(numeral const & a, numeral & p, numeral & q);
|
||||
};
|
||||
|
||||
class value;
|
||||
|
|
|
|||
|
|
@ -135,15 +135,36 @@ static void tst_lin_indep(unsigned m, unsigned n, int _A[], unsigned ex_sz, unsi
|
|||
A.set(i, j, _A[i*n + j]);
|
||||
unsigned_vector r;
|
||||
r.resize(A.n());
|
||||
mm.linear_independent_rows(A, r.c_ptr());
|
||||
scoped_mpz_matrix B(mm);
|
||||
mm.linear_independent_rows(A, r.c_ptr(), B);
|
||||
SASSERT(r.size() == ex_sz);
|
||||
for (unsigned i = 0; i < ex_sz; i++) {
|
||||
SASSERT(r[i] == ex_r[i]);
|
||||
}
|
||||
}
|
||||
|
||||
static void tst_denominators() {
|
||||
unsynch_mpq_manager qm;
|
||||
rcmanager m(qm);
|
||||
scoped_rcnumeral a(m);
|
||||
scoped_rcnumeral t(m);
|
||||
scoped_rcnumeral eps(m);
|
||||
m.mk_pi(a);
|
||||
m.inv(a);
|
||||
m.mk_infinitesimal("eps", eps);
|
||||
t = (a - eps*2) / (a*eps + 1);
|
||||
// t = t + a * 2;
|
||||
scoped_rcnumeral n(m), d(m);
|
||||
std::cout << t << "\n";
|
||||
m.clean_denominators(t, n, d);
|
||||
std::cout << "---->\n" << n << "\n" << d << "\n";
|
||||
}
|
||||
|
||||
void tst_rcf() {
|
||||
enable_trace("rcf_clean");
|
||||
enable_trace("rcf_clean_bug");
|
||||
tst_denominators();
|
||||
return;
|
||||
tst1();
|
||||
tst2();
|
||||
{ int A[] = {0, 1, 1, 1, 0, 1, 1, 1, -1}; int c[] = {10, 4, -4}; int b[] = {-2, 4, 6}; tst_solve(3, A, b, c, true); }
|
||||
|
|
|
|||
Loading…
Reference in a new issue