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patch for Sturm sequence bug #4961

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This commit is contained in:
Nikolaj Bjorner 2021-01-24 12:58:25 -08:00
parent 2d1684bc2d
commit 7d60d8462d
4 changed files with 124 additions and 114 deletions
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160
src/math/polynomial/algebraic_numbers.cpp
View file

@ -1844,79 +1844,97 @@ namespace algebraic_numbers {
} }
} }
// EXPENSIVE CASE // workaround: Sturm sequences are buggy as exemplified by several open github issues
// Let seq be the Sturm-Tarski sequence for // instead of relying on Sturm check if a simple interval expansion allows to separate
// p_a, p_a' * p_b // a and b.
// Let s_l be the number of sign variations at a_lower. scoped_mpbq la(bqm()), ua(bqm());
// Let s_u be the number of sign variations at a_upper. scoped_mpbq lb(bqm()), ub(bqm());
// By Sturm-Tarski Theorem, we have that unsigned precision = 10;
// V = s_l - s_u = #(p_b(r) > 0) - #(p_b(r) < 0) at roots r of p_a if (get_interval(a, la, ua, precision) &&
// Since there is only one root of p_a in (a_lower, b_lower), get_interval(b, lb, ub, precision)) {
// we are evaluating the sign of p_b at a. IF_VERBOSE(9, verbose_stream() << "sturm 0\n");
// That is V is the sign of p_b at a. if (la > ub)
// return sign_pos;
// We have if (ua < lb)
// V < 0 -> p_b(a) < 0 -> if p_b(b_lower) < 0 then b > a else b < a return sign_neg;
// V == 0 -> p_b(a) == 0 -> a = b }
// V > 0 -> p_b(a) > 0 -> if p_b(b_lower) > 0 then b > a else b < a IF_VERBOSE(9, verbose_stream() << "sturm 1\n");
// Simplifying we have:
// V == 0 --> a = b
// if (V < 0) == (p_b(b_lower) < 0) then b > a else b < a
//
m_compare_sturm++; //
upolynomial::scoped_upolynomial_sequence seq(upm()); // EXPENSIVE CASE
upm().sturm_tarski_seq(cell_a->m_p_sz, cell_a->m_p, cell_b->m_p_sz, cell_b->m_p, seq); // Let seq be the Sturm-Tarski sequence for
unsigned V1 = upm().sign_variations_at(seq, a_lower); // p_a, p_a' * p_b
unsigned V2 = upm().sign_variations_at(seq, a_upper); // Let s_l be the number of sign variations at a_lower.
int V = V1 - V2; // Let s_u be the number of sign variations at a_upper.
TRACE("algebraic", tout << "comparing using sturm\n"; // By Sturm-Tarski Theorem, we have that
display_interval(tout, a) << "\n"; // V = s_l - s_u = #(p_b(r) > 0) - #(p_b(r) < 0) at roots r of p_a
display_interval(tout, b) << "\n"; // Since there is only one root of p_a in (a_lower, b_lower),
tout << "V: " << V << " V1 " << V1 << " V2 " << V2 // we are evaluating the sign of p_b at a.
<< ", sign_lower(a): " << sign_lower(cell_a) // That is V is the sign of p_b at a.
<< ", sign_lower(b): " << sign_lower(cell_b) << "\n"; //
/*upm().display(tout << "sequence: ", seq);*/ // We have
); // V < 0 -> p_b(a) < 0 -> if p_b(b_lower) < 0 then b > a else b < a
if (V == 0) // V == 0 -> p_b(a) == 0 -> a = b
return sign_zero; // V > 0 -> p_b(a) > 0 -> if p_b(b_lower) > 0 then b > a else b < a
if ((V < 0) == (sign_lower(cell_b) < 0)) // Simplifying we have:
return sign_neg; // V == 0 --> a = b
else // if (V < 0) == (p_b(b_lower) < 0) then b > a else b < a
return sign_pos; //
// Here is an unexplored option for comparing numbers.
// m_compare_sturm++;
// The isolating intervals of a and b are still overlapping upolynomial::scoped_upolynomial_sequence seq(upm());
// Then we compute upm().sturm_tarski_seq(cell_a->m_p_sz, cell_a->m_p, cell_b->m_p_sz, cell_b->m_p, seq);
// r(x) = Resultant(x - y1 + y2, p1(y1), p2(y2)) unsigned V1 = upm().sign_variations_at(seq, a_lower);
// where p1(y1) and p2(y2) are the polynomials defining a and b. unsigned V2 = upm().sign_variations_at(seq, a_upper);
// Remarks: int V = V1 - V2;
// 1) The resultant r(x) must not be the zero polynomial, TRACE("algebraic", tout << "comparing using sturm\n";
// since the polynomial x - y1 + y2 does not vanish in any of the roots of p1(y1) and p2(y2) display_interval(tout, a) << "\n";
// display_interval(tout, b) << "\n";
// 2) By resultant calculus, If alpha, beta1, beta2 are roots of x - y1 + y2, p1(y1), p2(y2) tout << "V: " << V << " V1 " << V1 << " V2 " << V2
// then alpha is a root of r(x). << ", sign_lower(a): " << sign_lower(cell_a)
// Thus, we have that a - b is a root of r(x) << ", sign_lower(b): " << sign_lower(cell_b) << "\n";
// /*upm().display(tout << "sequence: ", seq);*/
// 3) If 0 is not a root of r(x), then a != b (by remark 2) );
// if (V == 0)
// 4) Let (l1, u1) and (l2, u2) be the intervals of a and b. return sign_zero;
// Then, a - b must be in (l1 - u2, u1 - l2) if ((V < 0) == (sign_lower(cell_b) < 0))
// return sign_neg;
// 5) Assume a != b, then if we keep refining the isolating intervals for a and b, else
// then eventually, (l1, u1) and (l2, u2) will not overlap. return sign_pos;
// Thus, if 0 is not a root of r(x), we can keep refining until
// the intervals do not overlap. // Here is an unexplored option for comparing numbers.
// //
// 6) If 0 is a root of r(x), we have two possibilities: // The isolating intervals of a and b are still overlapping
// a) Isolate roots of r(x) in the interval (l1 - u2, u1 - l2), // Then we compute
// and then keep refining (l1, u1) and (l2, u2) until they // r(x) = Resultant(x - y1 + y2, p1(y1), p2(y2))
// (l1 - u2, u1 - l2) "convers" only one root. // where p1(y1) and p2(y2) are the polynomials defining a and b.
// // Remarks:
// b) Compute the sturm sequence for r(x), // 1) The resultant r(x) must not be the zero polynomial,
// keep refining the (l1, u1) and (l2, u2) until // since the polynomial x - y1 + y2 does not vanish in any of the roots of p1(y1) and p2(y2)
// (l1 - u2, u1 - l2) contains only one root of r(x) //
// 2) By resultant calculus, If alpha, beta1, beta2 are roots of x - y1 + y2, p1(y1), p2(y2)
// then alpha is a root of r(x).
// Thus, we have that a - b is a root of r(x)
//
// 3) If 0 is not a root of r(x), then a != b (by remark 2)
//
// 4) Let (l1, u1) and (l2, u2) be the intervals of a and b.
// Then, a - b must be in (l1 - u2, u1 - l2)
//
// 5) Assume a != b, then if we keep refining the isolating intervals for a and b,
// then eventually, (l1, u1) and (l2, u2) will not overlap.
// Thus, if 0 is not a root of r(x), we can keep refining until
// the intervals do not overlap.
//
// 6) If 0 is a root of r(x), we have two possibilities:
// a) Isolate roots of r(x) in the interval (l1 - u2, u1 - l2),
// and then keep refining (l1, u1) and (l2, u2) until they
// (l1 - u2, u1 - l2) "convers" only one root.
//
// b) Compute the sturm sequence for r(x),
// keep refining the (l1, u1) and (l2, u2) until
// (l1 - u2, u1 - l2) contains only one root of r(x)
} }
::sign compare(numeral & a, numeral & b) { ::sign compare(numeral & a, numeral & b) {

39
src/nlsat/nlsat_interval_set.cpp
View file

@ -71,20 +71,15 @@ namespace nlsat {
} }
bool check_interval(anum_manager & am, interval const & i) { bool check_interval(anum_manager & am, interval const & i) {
if (i.m_lower_inf) { SASSERT(!i.m_lower_inf || i.m_lower_open);
SASSERT(i.m_lower_open); SASSERT(!i.m_upper_inf || i.m_upper_open);
}
if (i.m_upper_inf) {
SASSERT(i.m_upper_open);
}
if (!i.m_lower_inf && !i.m_upper_inf) { if (!i.m_lower_inf && !i.m_upper_inf) {
auto s = am.compare(i.m_lower, i.m_upper); auto s = am.compare(i.m_lower, i.m_upper);
TRACE("nlsat_interval", tout << "lower: "; am.display_decimal(tout, i.m_lower); tout << ", upper: "; am.display_decimal(tout, i.m_upper); TRACE("nlsat_interval", tout << "lower: "; am.display_decimal(tout, i.m_lower); tout << ", upper: "; am.display_decimal(tout, i.m_upper);
tout << "\ns: " << s << "\n";); tout << "\ns: " << s << "\n";);
SASSERT(s <= 0); SASSERT(s <= 0);
if (::is_zero(s)) { SASSERT(!is_zero(s) || !i.m_lower_open && !i.m_upper_open);
SASSERT(!i.m_lower_open && !i.m_upper_open);
}
} }
return true; return true;
} }
@ -92,12 +87,10 @@ namespace nlsat {
bool check_no_overlap(anum_manager & am, interval const & curr, interval const & next) { bool check_no_overlap(anum_manager & am, interval const & curr, interval const & next) {
SASSERT(!curr.m_upper_inf); SASSERT(!curr.m_upper_inf);
SASSERT(!next.m_lower_inf); SASSERT(!next.m_lower_inf);
int sign = am.compare(curr.m_upper, next.m_lower); sign s = am.compare(curr.m_upper, next.m_lower);
CTRACE("nlsat", sign > 0, display(tout, am, curr); tout << " "; display(tout, am, next); tout << "\n";); CTRACE("nlsat", s > 0, display(tout, am, curr); tout << " "; display(tout, am, next); tout << "\n";);
SASSERT(sign <= 0); SASSERT(s <= 0);
if (sign == 0) { SASSERT(!is_zero(s) || curr.m_upper_open || next.m_lower_open);
SASSERT(curr.m_upper_open || next.m_lower_open);
}
return true; return true;
} }
@ -107,9 +100,7 @@ namespace nlsat {
for (unsigned i = 0; i < sz; i++) { for (unsigned i = 0; i < sz; i++) {
interval const & curr = ints[i]; interval const & curr = ints[i];
SASSERT(check_interval(am, curr)); SASSERT(check_interval(am, curr));
if (i < sz - 1) { SASSERT(i >= sz - 1 || check_no_overlap(am, curr, ints[i+1]));
SASSERT(check_no_overlap(am, curr, ints[i+1]));
}
}); });
return true; return true;
} }
@ -276,10 +267,9 @@ namespace nlsat {
interval_set * interval_set_manager::mk_union(interval_set const * s1, interval_set const * s2) { interval_set * interval_set_manager::mk_union(interval_set const * s1, interval_set const * s2) {
#if 0 #if 0
// issue #2867:
static unsigned s_count = 0; static unsigned s_count = 0;
s_count++; s_count++;
if (s_count == 8442) { if (s_count == 4470) {
enable_trace("nlsat_interval"); enable_trace("nlsat_interval");
enable_trace("algebraic"); enable_trace("algebraic");
} }
@ -416,7 +406,7 @@ namespace nlsat {
else { else {
SASSERT(l1_l2_sign > 0); SASSERT(l1_l2_sign > 0);
if (u1_u2_sign == 0) { if (u1_u2_sign == 0) {
TRACE("nlsat_interval", tout << "l1_l2_sign > 0, u1_u2_sign == 0\n";); TRACE("nlsat_interval", tout << "l2 < l1 <= u1 = u2\n";);
// Case: // Case:
// 1) [ ] // 1) [ ]
// [ ] // [ ]
@ -426,7 +416,7 @@ namespace nlsat {
i2++; i2++;
} }
else if (u1_u2_sign < 0) { else if (u1_u2_sign < 0) {
TRACE("nlsat_interval", tout << "l1_l2_sign > 0, u1_u2_sign > 0\n";); TRACE("nlsat_interval", tout << "l2 < l1 <= u2 < u2\n";);
// Case: // Case:
// 1) [ ] // 1) [ ]
// [ ] // [ ]
@ -436,7 +426,7 @@ namespace nlsat {
else { else {
auto u2_l1_sign = compare_upper_lower(m_am, int2, int1); auto u2_l1_sign = compare_upper_lower(m_am, int2, int1);
if (u2_l1_sign < 0) { if (u2_l1_sign < 0) {
TRACE("nlsat_interval", tout << "l1_l2_sign > 0, u1_u2_sign > 0, u2_l1_sign < 0\n";); TRACE("nlsat_interval", tout << "l2 <= u2 < l1 <= u1\n";);
// Case: // Case:
// 1) [ ] // 1) [ ]
// [ ] // [ ]
@ -457,7 +447,7 @@ namespace nlsat {
i2++; i2++;
} }
else { else {
TRACE("nlsat_interval", tout << "l1_l2_sign > 0, u1_u2_sign > 0, u2_l1_sign > 0\n";); TRACE("nlsat_interval", tout << "l2 < l1 < u2 < u1\n";);
SASSERT(l1_l2_sign > 0); SASSERT(l1_l2_sign > 0);
SASSERT(u1_u2_sign > 0); SASSERT(u1_u2_sign > 0);
SASSERT(u2_l1_sign > 0); SASSERT(u2_l1_sign > 0);
@ -474,6 +464,7 @@ namespace nlsat {
} }
SASSERT(result.size() <= 1 || SASSERT(result.size() <= 1 ||
check_no_overlap(m_am, result[result.size() - 2], result[result.size() - 1])); check_no_overlap(m_am, result[result.size() - 2], result[result.size() - 1]));
} }
SASSERT(!result.empty()); SASSERT(!result.empty());

29
src/util/mpbq.cpp
View file

@ -57,9 +57,8 @@ mpbq_manager::~mpbq_manager() {
} }
void mpbq_manager::reset(mpbq_vector & v) { void mpbq_manager::reset(mpbq_vector & v) {
unsigned sz = v.size(); for (auto & e : v)
for (unsigned i = 0; i < sz; i++) reset(e);
reset(v[i]);
v.reset(); v.reset();
} }
@ -391,23 +390,25 @@ std::string mpbq_manager::to_string(mpbq const & a) {
return buffer.str(); return buffer.str();
} }
void mpbq_manager::display(std::ostream & out, mpbq const & a) { std::ostream& mpbq_manager::display(std::ostream & out, mpbq const & a) {
out << m_manager.to_string(a.m_num); out << m_manager.to_string(a.m_num);
if (a.m_k > 0) if (a.m_k > 0)
out << "/2"; out << "/2";
if (a.m_k > 1) if (a.m_k > 1)
out << "^" << a.m_k; out << "^" << a.m_k;
return out;
} }
void mpbq_manager::display_pp(std::ostream & out, mpbq const & a) { std::ostream& mpbq_manager::display_pp(std::ostream & out, mpbq const & a) {
out << m_manager.to_string(a.m_num); out << m_manager.to_string(a.m_num);
if (a.m_k > 0) if (a.m_k > 0)
out << "/2"; out << "/2";
if (a.m_k > 1) if (a.m_k > 1)
out << "<sup>" << a.m_k << "</sup>"; out << "<sup>" << a.m_k << "</sup>";
return out;
} }
void mpbq_manager::display_smt2(std::ostream & out, mpbq const & a, bool decimal) { std::ostream& mpbq_manager::display_smt2(std::ostream & out, mpbq const & a, bool decimal) {
if (a.m_k == 0) { if (a.m_k == 0) {
m_manager.display_smt2(out, a.m_num, decimal); m_manager.display_smt2(out, a.m_num, decimal);
} }
@ -421,12 +422,12 @@ void mpbq_manager::display_smt2(std::ostream & out, mpbq const & a, bool decimal
if (decimal) out << ".0"; if (decimal) out << ".0";
out << "))"; out << "))";
} }
return out;
} }
void mpbq_manager::display_decimal(std::ostream & out, mpbq const & a, unsigned prec) { std::ostream& mpbq_manager::display_decimal(std::ostream & out, mpbq const & a, unsigned prec) {
if (is_int(a)) { if (is_int(a)) {
out << m_manager.to_string(a.m_num); return out << m_manager.to_string(a.m_num);
return;
} }
mpz two(2); mpz two(2);
mpz ten(10); mpz ten(10);
@ -455,17 +456,16 @@ void mpbq_manager::display_decimal(std::ostream & out, mpbq const & a, unsigned
m_manager.del(n1); m_manager.del(n1);
m_manager.del(v1); m_manager.del(v1);
m_manager.del(two_k); m_manager.del(two_k);
return out;
} }
void mpbq_manager::display_decimal(std::ostream & out, mpbq const & a, mpbq const & b, unsigned prec) { std::ostream& mpbq_manager::display_decimal(std::ostream & out, mpbq const & a, mpbq const & b, unsigned prec) {
mpz two(2); mpz two(2);
mpz ten(10); mpz ten(10);
mpz two_k1, two_k2; mpz two_k1, two_k2;
mpz n1, v1, n2, v2; mpz n1, v1, n2, v2;
if (m_manager.is_neg(a.m_num) != m_manager.is_neg(b.m_num)) { if (m_manager.is_neg(a.m_num) != m_manager.is_neg(b.m_num))
out << "?"; return out << "?";
return;
}
if (m_manager.is_neg(a.m_num)) if (m_manager.is_neg(a.m_num))
out << "-"; out << "-";
m_manager.set(v1, a.m_num); m_manager.set(v1, a.m_num);
@ -512,6 +512,7 @@ void mpbq_manager::display_decimal(std::ostream & out, mpbq const & a, mpbq cons
m_manager.del(v2); m_manager.del(v2);
m_manager.del(two_k1); m_manager.del(two_k1);
m_manager.del(two_k2); m_manager.del(two_k2);
return out;
} }
bool mpbq_manager::to_mpbq(mpq const & q, mpbq & bq) { bool mpbq_manager::to_mpbq(mpq const & q, mpbq & bq) {

10
src/util/mpbq.h
View file

@ -258,17 +258,17 @@ public:
void select_small_core(unsynch_mpq_manager & qm, mpq const & lower, mpq const & upper, mpbq & r); void select_small_core(unsynch_mpq_manager & qm, mpq const & lower, mpq const & upper, mpbq & r);
void display(std::ostream & out, mpbq const & a); std::ostream& display(std::ostream & out, mpbq const & a);
void display_pp(std::ostream & out, mpbq const & a); std::ostream& display_pp(std::ostream & out, mpbq const & a);
void display_decimal(std::ostream & out, mpbq const & a, unsigned prec = 8); std::ostream& display_decimal(std::ostream & out, mpbq const & a, unsigned prec = 8);
/** /**
\brief Display a in decimal while its digits match b digits. \brief Display a in decimal while its digits match b digits.
This function is useful when a and b are representing an interval [a,b] which This function is useful when a and b are representing an interval [a,b] which
contains an algebraic number contains an algebraic number
*/ */
void display_decimal(std::ostream & out, mpbq const & a, mpbq const & b, unsigned prec); std::ostream& display_decimal(std::ostream & out, mpbq const & a, mpbq const & b, unsigned prec);
void display_smt2(std::ostream & out, mpbq const & a, bool decimal); std::ostream& display_smt2(std::ostream & out, mpbq const & a, bool decimal);
/** /**
\brief Approximate n as b/2^k' s.t. k' <= k. \brief Approximate n as b/2^k' s.t. k' <= k.