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add module for handling axioms for powers

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This commit is contained in:
Nikolaj Bjorner 2023-01-25 13:34:13 -08:00
parent 9e2ec9d018
commit e41dd91893
7 changed files with 343 additions and 201 deletions
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1
src/math/lp/CMakeLists.txt
View file

@ -38,6 +38,7 @@ z3_add_component(lp
nla_intervals.cpp
nla_monotone_lemmas.cpp
nla_order_lemmas.cpp
nla_powers.cpp
nla_solver.cpp
nla_tangent_lemmas.cpp
nra_solver.cpp

8
src/math/lp/nla_core.cpp
View file

@ -29,6 +29,7 @@ core::core(lp::lar_solver& s, reslimit & lim) :
m_basics(this),
m_order(this),
m_monotone(this),
m_powers(*this),
m_intervals(this, lim),
m_monomial_bounds(this),
m_horner(this),
@ -120,9 +121,8 @@ bool core::canonize_sign(const monic& m) const {
bool core::canonize_sign(const factorization& f) const {
bool r = false;
for (const factor & a : f) {
for (const factor & a : f)
r ^= canonize_sign(a);
}
return r;
}
@ -1477,6 +1477,10 @@ void core::check_weighted(unsigned sz, std::pair<unsigned, std::function<void(vo
}
}
lbool core::check_power(lpvar r, lpvar x, lpvar y, vector<lemma>& l_vec) {
m_lemma_vec = &l_vec;
return m_powers.check(r, x, y, l_vec);
}
lbool core::check(vector<lemma>& l_vec) {
lp_settings().stats().m_nla_calls++;

106
src/math/lp/nla_core.h
View file

@ -19,6 +19,7 @@
#include "math/lp/nla_order_lemmas.h"
#include "math/lp/nla_monotone_lemmas.h"
#include "math/lp/nla_grobner.h"
#include "math/lp/nla_powers.h"
#include "math/lp/emonics.h"
#include "math/lp/nla_settings.h"
#include "math/lp/nex.h"
@ -42,104 +43,6 @@ bool try_insert(const A& elem, B& collection) {
return true;
}
typedef lp::constraint_index lpci;
typedef lp::lconstraint_kind llc;
typedef lp::constraint_index lpci;
typedef lp::explanation expl_set;
typedef lp::var_index lpvar;
const lpvar null_lpvar = UINT_MAX;
inline int rat_sign(const rational& r) { return r.is_pos()? 1 : ( r.is_neg()? -1 : 0); }
inline rational rrat_sign(const rational& r) { return rational(rat_sign(r)); }
inline bool is_set(unsigned j) { return j != null_lpvar; }
inline bool is_even(unsigned k) { return (k & 1) == 0; }
class ineq {
lp::lconstraint_kind m_cmp;
lp::lar_term m_term;
rational m_rs;
public:
ineq(lp::lconstraint_kind cmp, const lp::lar_term& term, const rational& rs) : m_cmp(cmp), m_term(term), m_rs(rs) {}
ineq(const lp::lar_term& term, lp::lconstraint_kind cmp, int i) : m_cmp(cmp), m_term(term), m_rs(rational(i)) {}
ineq(const lp::lar_term& term, lp::lconstraint_kind cmp, const rational& rs) : m_cmp(cmp), m_term(term), m_rs(rs) {}
ineq(lpvar v, lp::lconstraint_kind cmp, int i): m_cmp(cmp), m_term(v), m_rs(rational(i)) {}
ineq(lpvar v, lp::lconstraint_kind cmp, rational const& r): m_cmp(cmp), m_term(v), m_rs(r) {}
bool operator==(const ineq& a) const {
return m_cmp == a.m_cmp && m_term == a.m_term && m_rs == a.m_rs;
}
const lp::lar_term& term() const { return m_term; };
lp::lconstraint_kind cmp() const { return m_cmp; };
const rational& rs() const { return m_rs; };
};
class lemma {
vector<ineq> m_ineqs;
lp::explanation m_expl;
public:
void push_back(const ineq& i) { m_ineqs.push_back(i);}
size_t size() const { return m_ineqs.size() + m_expl.size(); }
const vector<ineq>& ineqs() const { return m_ineqs; }
vector<ineq>& ineqs() { return m_ineqs; }
lp::explanation& expl() { return m_expl; }
const lp::explanation& expl() const { return m_expl; }
bool is_conflict() const { return m_ineqs.empty() && !m_expl.empty(); }
};
class core;
//
// lemmas are created in a scope.
// when the destructor of new_lemma is invoked
// all constraints are assumed added to the lemma
// correctness of the lemma can be checked at this point.
//
class new_lemma {
char const* name;
core& c;
lemma& current() const;
public:
new_lemma(core& c, char const* name);
~new_lemma();
lemma& operator()() { return current(); }
std::ostream& display(std::ostream& out) const;
new_lemma& operator&=(lp::explanation const& e);
new_lemma& operator&=(const monic& m);
new_lemma& operator&=(const factor& f);
new_lemma& operator&=(const factorization& f);
new_lemma& operator&=(lpvar j);
new_lemma& operator|=(ineq const& i);
new_lemma& explain_fixed(lpvar j);
new_lemma& explain_equiv(lpvar u, lpvar v);
new_lemma& explain_var_separated_from_zero(lpvar j);
new_lemma& explain_existing_lower_bound(lpvar j);
new_lemma& explain_existing_upper_bound(lpvar j);
lp::explanation& expl() { return current().expl(); }
unsigned num_ineqs() const { return current().ineqs().size(); }
};
inline std::ostream& operator<<(std::ostream& out, new_lemma const& l) {
return l.display(out);
}
struct pp_fac {
core const& c;
factor const& f;
pp_fac(core const& c, factor const& f): c(c), f(f) {}
};
struct pp_var {
core const& c;
lpvar v;
pp_var(core const& c, lpvar v): c(c), v(v) {}
};
struct pp_factorization {
core const& c;
factorization const& f;
pp_factorization(core const& c, factorization const& f): c(c), f(f) {}
};
class core {
friend struct common;
@ -149,6 +52,7 @@ class core {
friend struct basics;
friend struct tangents;
friend class monotone;
friend class powers;
friend struct nla_settings;
friend class intervals;
friend class horner;
@ -183,6 +87,7 @@ class core {
basics m_basics;
order m_order;
monotone m_monotone;
powers m_powers;
intervals m_intervals;
monomial_bounds m_monomial_bounds;
nla_settings m_nla_settings;
@ -463,9 +368,8 @@ public:
bool conflict_found() const;
lbool check(vector<lemma>& l_vec);
void set_lemma_vec(vector<lemma>& l_vec) { m_lemma_vec = &l_vec; }
lbool check(vector<lemma>& l_vec);
lbool check_power(lpvar r, lpvar x, lpvar y, vector<lemma>& l_vec);
bool no_lemmas_hold() const;

181
src/math/lp/nla_powers.cpp Normal file
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@ -0,0 +1,181 @@
/*++
Copyright (c) 2017 Microsoft Corporation
Module Name:
nla_powers.cpp
Author:
Lev Nachmanson (levnach)
Nikolaj Bjorner (nbjorner)
Description:
Refines bounds on powers.
Reference: TOCL-2018, Cimatti et al.
Special cases:
1. Exponentiation. x is fixed numeral.
TOCL18 axioms:
a^y > 0 (if a > 0)
y = 0 <=> a^y = 1 (if a != 0)
y < 0 <=> a^y < 1 (if a > 1)
y > 0 <=> a^y > 1 (if a > 1)
y != 0 <=> a^y > y + 1 (if a >= 2)
y1 < y2 <=> a^y1 < a^y2 (**)
Other special case:
y = 1 <= a^y = a
TOCL18 approach: Polynomial abstractions
Taylor: a^y = sum_i ln(a)*y^i/i!
Truncation: P(n, a) = sum_{i=0}^n ln(a)*y^i/i!
y = 0: handled by axiom a^y = 1
y < 0: P(2n-1, y) <= a^y <= P(2n, y), n > 0 because Taylor contribution is negative at odd powers.
y > 0: P(n, y) <= a^y <= P(n, y)*(1 - y^{n+1}/(n+1)!)
2. Powers. y is fixed positive integer.
3. Other
General case:
For now the solver integrates just weak monotonicity lemmas:
- x >= x0 > 0, y >= y0 => x^y >= x0^y0
- 0 < x <= x0, y <= y0 => x^y <= x0^y0
TODO:
- Comprehensive integration for truncation polynomial approximation.
- TOCL18 approach includes refinement loop based on precision epsilon.
- accept solvability if r is within a small range of x^y, when x^y is not rational.
- integrate algebraic numbers, or even extension fields (for 'e').
- integrate monotonicy axioms (**) by tracking exponents across instances.
anum isn't initialized unless nra_solver is invoked.
there is no proviso for using algebraic numbers outside of the nra solver.
so either we have a rational refinement version _and_ an algebraic numeral refinement
loop or we introduce algebraic numerals outside of the nra_solver
scoped_anum xval(am()), yval(am()), rval(am());
am().set(xval, am_value(x));
am().set(yval, am_value(y));
am().set(rval, am_value(r));
--*/
#include "math/lp/nla_core.h"
namespace nla {
lbool powers::check(lpvar r, lpvar x, lpvar y, vector<lemma>& lemmas) {
if (x == null_lpvar || y == null_lpvar || r == null_lpvar)
return l_undef;
core& c = m_core;
auto xval = c.val(x);
auto yval = c.val(y);
auto rval = c.val(r);
lemmas.reset();
if (xval != 0 && yval == 0 && rval != 1) {
new_lemma lemma(c, "x != 0 => x^0 = 1");
lemma |= ineq(x, llc::EQ, rational::zero());
lemma |= ineq(y, llc::NE, rational::zero());
lemma |= ineq(r, llc::EQ, rational::one());
return l_false;
}
if (xval == 0 && yval > 0 && rval != 0) {
new_lemma lemma(c, "y != 0 => 0^y = 0");
lemma |= ineq(x, llc::NE, rational::zero());
lemma |= ineq(y, llc::EQ, rational::zero());
lemma |= ineq(r, llc::EQ, rational::zero());
return l_false;
}
if (xval > 0 && rval < 0 && rval <= 0) {
new_lemma lemma(c, "x > 0 => x^y > 0");
lemma |= ineq(x, llc::LE, rational::zero());
lemma |= ineq(r, llc::GT, rational::zero());
return l_false;
}
if (xval > 1 && rval >= 1 && yval < 0) {
new_lemma lemma(c, "x > 1, y < 0 => x^y < 1");
lemma |= ineq(x, llc::LE, rational::one());
lemma |= ineq(y, llc::GE, rational::zero());
lemma |= ineq(r, llc::LT, rational::one());
return l_false;
}
if (xval > 1 && rval <= 1 && yval > 0) {
new_lemma lemma(c, "x > 1, y > 0 => x^y > 1");
lemma |= ineq(x, llc::LE, rational::one());
lemma |= ineq(y, llc::LE, rational::zero());
lemma |= ineq(r, llc::GT, rational::one());
return l_false;
}
if (xval >= 2 && yval != 0 & rval <= yval + 1) {
new_lemma lemma(c, "x >= 2, y != 0 => x^y > y + 1");
lemma |= ineq(x, llc::LT, rational(2));
lemma |= ineq(y, llc::EQ, rational::zero());
lemma |= ineq(lp::lar_term(r, rational::minus_one(), y), llc::GT, rational::one());
return l_false;
}
if (xval > 0 && yval.is_unsigned()) {
auto r2val = power(xval, yval.get_unsigned());
if (rval == r2val)
return l_true;
if (xval > 0 && r2val < rval) {
SASSERT(yval > 0);
new_lemma lemma(c, "x >= x0 > 0, y >= y0 > 0 => r >= x0^y0");
lemma |= ineq(x, llc::LT, xval);
lemma |= ineq(y, llc::LT, yval);
lemma |= ineq(r, llc::GE, r2val);
return l_false;
}
if (xval > 0 && r2val < rval) {
new_lemma lemma(c, "x >= x0 > 0, y <= y0 => r <= x0^y0");
lemma |= ineq(x, llc::LT, xval);
lemma |= ineq(y, llc::GT, yval);
lemma |= ineq(r, llc::LE, r2val);
return l_false;
}
}
if (xval > 0 && yval > 0 && !yval.is_int()) {
auto ynum = numerator(yval);
auto yden = denominator(yval);
if (!ynum.is_unsigned())
return l_undef;
if (!yden.is_unsigned())
return l_undef;
// r = x^{yn/yd}
// <=>
// r^yd = x^yn
auto ryd = power(rval, yden.get_unsigned());
auto xyn = power(xval, ynum.get_unsigned());
if (ryd == xyn)
return l_true;
}
return l_undef;
}
}

29
src/math/lp/nla_powers.h Normal file
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@ -0,0 +1,29 @@
/*++
Copyright (c) 2017 Microsoft Corporation
Module Name:
nla_powers.h
Author:
Lev Nachmanson (levnach)
Nikolaj Bjorner (nbjorner)
Description:
Refines bounds on powers.
--*/
#include "math/lp/nla_types.h"
namespace nla {
class core;
class powers {
core& m_core;
public:
powers(core& c):m_core(c) {}
lbool check(lpvar r, lpvar x, lpvar y, vector<lemma>&);
};
}

99
src/math/lp/nla_solver.cpp
View file

@ -77,104 +77,7 @@ namespace nla {
// ensure r = x^y, add abstraction/refinement lemmas
lbool solver::check_power(lpvar r, lpvar x, lpvar y, vector<lemma>& lemmas) {
if (x == null_lpvar || y == null_lpvar || r == null_lpvar)
return l_undef;
if (use_nra_model())
return l_undef;
auto xval = m_core->val(x);
auto yval = m_core->val(y);
auto rval = m_core->val(r);
core& c = get_core();
c.set_lemma_vec(lemmas);
lemmas.reset();
// x >= x0 > 0, y >= y0 > 0 => r >= x0^y0
// x >= x0 > 0, y <= y0 => r <= x0^y0
// x != 0, y = 0 => r = 1
// x = 0, y != 0 => r = 0
//
// for x fixed, it is exponentiation
// => use tangent lemmas and error tolerance.
if (xval > 0 && yval.is_unsigned()) {
auto r2val = power(xval, yval.get_unsigned());
if (rval == r2val)
return l_true;
if (xval != 0 && yval == 0) {
new_lemma lemma(c, "x != 0 => x^0 = 1");
lemma |= ineq(x, llc::EQ, rational::zero());
lemma |= ineq(y, llc::NE, rational::zero());
lemma |= ineq(r, llc::EQ, rational::one());
return l_false;
}
if (xval == 0 && yval > 0) {
new_lemma lemma(c, "y != 0 => 0^y = 0");
lemma |= ineq(x, llc::NE, rational::zero());
lemma |= ineq(y, llc::EQ, rational::zero());
lemma |= ineq(r, llc::EQ, rational::zero());
return l_false;
}
if (xval > 0 && r2val < rval) {
SASSERT(yval > 0);
new_lemma lemma(c, "x >= x0 > 0, y >= y0 > 0 => r >= x0^y0");
lemma |= ineq(x, llc::LT, xval);
lemma |= ineq(y, llc::LT, yval);
lemma |= ineq(r, llc::GE, r2val);
return l_false;
}
if (xval > 0 && r2val < rval) {
new_lemma lemma(c, "x >= x0 > 0, y <= y0 => r <= x0^y0");
lemma |= ineq(x, llc::LT, xval);
lemma |= ineq(y, llc::GT, yval);
lemma |= ineq(r, llc::LE, r2val);
return l_false;
}
}
if (xval > 0 && yval > 0 && !yval.is_int()) {
auto ynum = numerator(yval);
auto yden = denominator(yval);
if (!ynum.is_unsigned())
return l_undef;
if (!yden.is_unsigned())
return l_undef;
// r = x^{yn/yd}
// <=>
// r^yd = x^yn
auto ryd = power(rval, yden.get_unsigned());
auto xyn = power(xval, ynum.get_unsigned());
if (ryd == xyn)
return l_true;
#if 0
// try some root approximation instead?
if (ryd > xyn) {
// todo
}
if (ryd < xyn) {
// todo
}
#endif
}
return l_undef;
// anum isn't initialized unless nra_solver is invoked.
// there is no proviso for using algebraic numbers outside of the nra solver.
// so either we have a rational refinement version _and_ an algebraic numeral refinement
// loop or we introduce algebraic numerals outside of the nra_solver
#if 0
scoped_anum xval(am()), yval(am()), rval(am());
am().set(xval, am_value(x));
am().set(yval, am_value(y));
am().set(rval, am_value(r));
#endif
return m_core->check_power(r, x, y, lemmas);
}
}

120
src/math/lp/nla_types.h Normal file
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@ -0,0 +1,120 @@
/*++
Copyright (c) 2017 Microsoft Corporation
Module Name:
nla_types.h
Author:
Lev Nachmanson (levnach)
Nikolaj Bjorner (nbjorner)
Description:
Types used for nla solver.
--*/
#pragma once
namespace nla {
typedef lp::constraint_index lpci;
typedef lp::lconstraint_kind llc;
typedef lp::constraint_index lpci;
typedef lp::explanation expl_set;
typedef lp::var_index lpvar;
const lpvar null_lpvar = UINT_MAX;
inline int rat_sign(const rational& r) { return r.is_pos()? 1 : ( r.is_neg()? -1 : 0); }
inline rational rrat_sign(const rational& r) { return rational(rat_sign(r)); }
inline bool is_set(unsigned j) { return j != null_lpvar; }
inline bool is_even(unsigned k) { return (k & 1) == 0; }
class ineq {
lp::lconstraint_kind m_cmp;
lp::lar_term m_term;
rational m_rs;
public:
ineq(lp::lconstraint_kind cmp, const lp::lar_term& term, const rational& rs) : m_cmp(cmp), m_term(term), m_rs(rs) {}
ineq(const lp::lar_term& term, lp::lconstraint_kind cmp, int i) : m_cmp(cmp), m_term(term), m_rs(rational(i)) {}
ineq(const lp::lar_term& term, lp::lconstraint_kind cmp, const rational& rs) : m_cmp(cmp), m_term(term), m_rs(rs) {}
ineq(lpvar v, lp::lconstraint_kind cmp, int i): m_cmp(cmp), m_term(v), m_rs(rational(i)) {}
ineq(lpvar v, lp::lconstraint_kind cmp, rational const& r): m_cmp(cmp), m_term(v), m_rs(r) {}
bool operator==(const ineq& a) const {
return m_cmp == a.m_cmp && m_term == a.m_term && m_rs == a.m_rs;
}
const lp::lar_term& term() const { return m_term; };
lp::lconstraint_kind cmp() const { return m_cmp; };
const rational& rs() const { return m_rs; };
};
class lemma {
vector<ineq> m_ineqs;
lp::explanation m_expl;
public:
void push_back(const ineq& i) { m_ineqs.push_back(i);}
size_t size() const { return m_ineqs.size() + m_expl.size(); }
const vector<ineq>& ineqs() const { return m_ineqs; }
vector<ineq>& ineqs() { return m_ineqs; }
lp::explanation& expl() { return m_expl; }
const lp::explanation& expl() const { return m_expl; }
bool is_conflict() const { return m_ineqs.empty() && !m_expl.empty(); }
};
class core;
//
// lemmas are created in a scope.
// when the destructor of new_lemma is invoked
// all constraints are assumed added to the lemma
// correctness of the lemma can be checked at this point.
//
class new_lemma {
char const* name;
core& c;
lemma& current() const;
public:
new_lemma(core& c, char const* name);
~new_lemma();
lemma& operator()() { return current(); }
std::ostream& display(std::ostream& out) const;
new_lemma& operator&=(lp::explanation const& e);
new_lemma& operator&=(const monic& m);
new_lemma& operator&=(const factor& f);
new_lemma& operator&=(const factorization& f);
new_lemma& operator&=(lpvar j);
new_lemma& operator|=(ineq const& i);
new_lemma& explain_fixed(lpvar j);
new_lemma& explain_equiv(lpvar u, lpvar v);
new_lemma& explain_var_separated_from_zero(lpvar j);
new_lemma& explain_existing_lower_bound(lpvar j);
new_lemma& explain_existing_upper_bound(lpvar j);
lp::explanation& expl() { return current().expl(); }
unsigned num_ineqs() const { return current().ineqs().size(); }
};
inline std::ostream& operator<<(std::ostream& out, new_lemma const& l) {
return l.display(out);
}
struct pp_fac {
core const& c;
factor const& f;
pp_fac(core const& c, factor const& f): c(c), f(f) {}
};
struct pp_var {
core const& c;
lpvar v;
pp_var(core const& c, lpvar v): c(c), v(v) {}
};
struct pp_factorization {
core const& c;
factorization const& f;
pp_factorization(core const& c, factorization const& f): c(c), f(f) {}
};
}