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/*++
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 a.
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! = 1 + ln(a)*y + ln(a)^2*y^2/2 +
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 >= 1, y >= y0 >= 0 => x^y >= x0^y0
- x >= x0 > 0, x0 < 1, 0 <= y <= y0 => x^y >= x0^y0
- 0 < x <= x0, x0 >= 1, 0 <= y <= y0 => x^y <= x0^y0
- 0 < x <= x0 < 1, y >= y0 >= 0 => 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) {
TRACE(nla, tout << r << " == " << x << "^" << y << "\n");
core& c = m_core;
if (x == null_lpvar || y == null_lpvar || r == null_lpvar)
return l_undef;
// if (c.lra.column_has_term(x) || c.lra.column_has_term(y) || c.lra.column_has_term(r))
// return l_undef;
lemmas.reset();
auto x_exp_0 = [&]() {
lemma_builder 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;
};
auto zero_exp_y = [&]() {
lemma_builder 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;
};
auto x_gt_0 = [&]() {
lemma_builder lemma(c, "x > 0 => x^y > 0");
lemma |= ineq(x, llc::LE, rational::zero());
lemma |= ineq(r, llc::GT, rational::zero());
return l_false;
};
auto y_lt_1 = [&]() {
lemma_builder 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;
};
auto y_gt_1 = [&]() {
lemma_builder 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;
};
auto x_ge_3 = [&]() {
lemma_builder lemma(c, "x >= 3, y > 0 => x^y > ln(x)y + 1");
lemma |= ineq(x, llc::LT, rational(3));
lemma |= ineq(y, llc::LE, rational::zero());
lemma |= ineq(lp::lar_term(r, rational::minus_one(), y), llc::GT, rational::one());
return l_false;
};
bool use_rational = !c.use_nra_model();
rational xval, yval, rval;
if (use_rational) {
// the lemmas below are checked against the values used by core::val,
// which ignores the infinitesimal parts, so bail out when the model
// assigns an infinitesimal to one of x, y, r.
if (!c.lra.get_column_value(x).y.is_zero() ||
!c.lra.get_column_value(y).y.is_zero() ||
!c.lra.get_column_value(r).y.is_zero())
return l_undef;
xval = c.val(x);
yval = c.val(y);
rval = c.val(r);
}
else {
auto& am = c.m_nra.am();
if (am.is_rational(c.m_nra.value(x)) &&
am.is_rational(c.m_nra.value(y)) &&
am.is_rational(c.m_nra.value(r))) {
am.to_rational(c.m_nra.value(x), xval);
am.to_rational(c.m_nra.value(y), yval);
am.to_rational(c.m_nra.value(r), rval);
use_rational = true;
}
}
if (use_rational) {
if (xval != 0 && yval == 0 && rval != 1)
return x_exp_0();
else if (xval == 0 && yval != 0 && rval != 0)
return zero_exp_y();
else if (xval > 0 && rval <= 0)
return x_gt_0();
else if (xval > 1 && yval < 0 && rval >= 1)
return y_lt_1();
else if (xval > 1 && yval > 0 && rval <= 1)
return y_gt_1();
else if (xval >= 3 && yval > 0 && rval <= yval + 1)
return x_ge_3();
else if (xval > 0 && yval.is_unsigned()) {
auto r2val = power(xval, yval.get_unsigned());
if (rval == r2val)
return l_true;
if (c.random() % 2 == 0) {
lemma_builder lemma(c, "x == x0, y == y0 => r = x0^y0");
lemma |= ineq(x, llc::NE, xval);
lemma |= ineq(y, llc::NE, yval);
lemma |= ineq(r, llc::EQ, r2val);
return l_false;
}
if (yval > 0 && r2val > rval) {
bool increasing = xval >= 1;
lemma_builder lemma(c, increasing
? "x >= x0 >= 1, y >= y0 > 0 => r >= x0^y0"
: "x >= x0 > 0, x0 < 1, 0 <= y <= y0 => r >= x0^y0");
lemma |= ineq(x, llc::LT, xval);
lemma |= ineq(y, increasing ? llc::LT : llc::GT, yval);
if (!increasing)
lemma |= ineq(y, llc::LT, rational::zero());
lemma |= ineq(r, llc::GE, r2val);
return l_false;
}
if (r2val < rval) {
bool increasing = xval >= 1;
lemma_builder lemma(c, increasing
? "0 < x <= x0, x0 >= 1, 0 <= y <= y0 => r <= x0^y0"
: "0 < x <= x0 < 1, y >= y0 >= 0 => r <= x0^y0");
lemma |= ineq(x, llc::LE, rational::zero());
lemma |= ineq(x, llc::GT, xval);
lemma |= ineq(y, increasing ? llc::GT : llc::LT, yval);
if (increasing)
lemma |= ineq(y, llc::LT, rational::zero());
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);
// r = x^{yn/yd}
// <=>
// r^yd = x^yn
if (ynum.is_unsigned() && yden.is_unsigned()) {
auto ryd = power(rval, yden.get_unsigned());
auto xyn = power(xval, ynum.get_unsigned());
if (ryd == xyn)
return l_true;
}
}
}
if (!use_rational) {
auto& am = c.m_nra.am();
scoped_anum xval(am), yval(am), rval(am);
am.set(xval, c.m_nra.value(x));
am.set(yval, c.m_nra.value(y));
am.set(rval, c.m_nra.value(r));
if (xval != 0 && yval == 0 && rval != 1)
return x_exp_0();
else if (xval == 0 && yval != 0 && rval != 0)
return zero_exp_y();
else if (xval > 0 && rval <= 0)
return x_gt_0();
else if (xval > 1 && yval < 0 && rval >= 1)
return y_lt_1();
else if (xval > 1 && yval > 0 && rval <= 1)
return y_gt_1();
else if (xval >= 3 && yval > 0 && rval <= yval + 1)
return x_ge_3();
else if (xval > 0 && yval > 0 && am.is_rational(yval)) {
rational yr;
am.to_rational(yval, yr);
auto ynum = numerator(yr);
auto yden = denominator(yr);
// r = x^{yn/yd}
// <=>
// r^yd = x^yn
if (ynum.is_unsigned() && yden.is_unsigned()) {
am.set(rval, power(rval, yden.get_unsigned()));
am.set(xval, power(xval, ynum.get_unsigned()));
if (rval == xval)
return l_true;
}
}
}
return l_undef;
}
}