mirrors/z3
3
0
Fork 0
mirror of https://github.com/Z3Prover/z3 synced 2025-04-27 02:45:51 +00:00
Code Activity

merged

Browse source 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2012-12-10 14:07:59 -08:00
commit 265bdbe757
76 changed files with 3991 additions and 1422 deletions
Download patch file Download diff file Expand all files Collapse all files

21
src/api/python/z3.py
View file

@ -1126,6 +1126,27 @@ def Var(idx, s):
_z3_assert(is_sort(s), "Z3 sort expected")
return _to_expr_ref(Z3_mk_bound(s.ctx_ref(), idx, s.ast), s.ctx)
def RealVar(idx, ctx=None):
"""
Create a real free variable. Free variables are used to create quantified formulas.
They are also used to create polynomials.
>>> RealVar(0)
Var(0)
"""
return Var(idx, RealSort(ctx))
def RealVarVector(n, ctx=None):
"""
Create a list of Real free variables.
The variables have ids: 0, 1, ..., n-1
>>> x0, x1, x2, x3 = RealVarVector(4)
>>> x2
Var(2)
"""
return [ RealVar(i, ctx) for i in range(n) ]
#########################################
#
# Booleans

564
src/api/python/z3num.py Normal file
View file

@ -0,0 +1,564 @@
############################################
# Copyright (c) 2012 Microsoft Corporation
#
# Z3 Python interface for Z3 numerals
#
# Author: Leonardo de Moura (leonardo)
############################################
from z3 import *
from z3core import *
from z3printer import *
def _to_numeral(num, ctx=None):
if isinstance(num, Numeral):
return num
else:
return Numeral(num, ctx)
class Numeral:
"""
A Z3 numeral can be used to perform computations over arbitrary
precision integers, rationals and real algebraic numbers.
It also automatically converts python numeric values.
>>> Numeral(2)
2
>>> Numeral("3/2") + 1
5/2
>>> Numeral(Sqrt(2))
1.4142135623?
>>> Numeral(Sqrt(2)) + 2
3.4142135623?
>>> Numeral(Sqrt(2)) + Numeral(Sqrt(3))
3.1462643699?
Z3 numerals can be used to perform computations with
values in a Z3 model.
>>> s = Solver()
>>> x = Real('x')
>>> s.add(x*x == 2)
>>> s.add(x > 0)
>>> s.check()
sat
>>> m = s.model()
>>> m[x]
1.4142135623?
>>> m[x] + 1
1.4142135623? + 1
The previous result is a Z3 expression.
>>> (m[x] + 1).sexpr()
'(+ (root-obj (+ (^ x 2) (- 2)) 2) 1.0)'
>>> Numeral(m[x]) + 1
2.4142135623?
>>> Numeral(m[x]).is_pos()
True
>>> Numeral(m[x])**2
2
We can also isolate the roots of polynomials.
>>> x0, x1, x2 = RealVarVector(3)
>>> r0 = isolate_roots(x0**5 - x0 - 1)
>>> r0
[1.1673039782?]
In the following example, we are isolating the roots
of a univariate polynomial (on x1) obtained after substituting
x0 -> r0[0]
>>> r1 = isolate_roots(x1**2 - x0 + 1, [ r0[0] ])
>>> r1
[-0.4090280898?, 0.4090280898?]
Similarly, in the next example we isolate the roots of
a univariate polynomial (on x2) obtained after substituting
x0 -> r0[0] and x1 -> r1[0]
>>> isolate_roots(x1*x2 + x0, [ r0[0], r1[0] ])
[2.8538479564?]
"""
def __init__(self, num, ctx=None):
if isinstance(num, Ast):
self.ast = num
self.ctx = z3._get_ctx(ctx)
elif isinstance(num, RatNumRef) or isinstance(num, AlgebraicNumRef):
self.ast = num.ast
self.ctx = num.ctx
elif isinstance(num, ArithRef):
r = simplify(num)
self.ast = r.ast
self.ctx = r.ctx
else:
v = RealVal(num, ctx)
self.ast = v.ast
self.ctx = v.ctx
Z3_inc_ref(self.ctx_ref(), self.as_ast())
assert Z3_algebraic_is_value(self.ctx_ref(), self.ast)
def __del__(self):
Z3_dec_ref(self.ctx_ref(), self.as_ast())
def is_integer(self):
""" Return True if the numeral is integer.
>>> Numeral(2).is_integer()
True
>>> (Numeral(Sqrt(2)) * Numeral(Sqrt(2))).is_integer()
True
>>> Numeral(Sqrt(2)).is_integer()
False
>>> Numeral("2/3").is_integer()
False
"""
return self.is_rational() and self.denominator() == 1
def is_rational(self):
""" Return True if the numeral is rational.
>>> Numeral(2).is_rational()
True
>>> Numeral("2/3").is_rational()
True
>>> Numeral(Sqrt(2)).is_rational()
False
"""
return Z3_get_ast_kind(self.ctx_ref(), self.as_ast()) == Z3_NUMERAL_AST
def denominator(self):
""" Return the denominator if `self` is rational.
>>> Numeral("2/3").denominator()
3
"""
assert(self.is_rational())
return Numeral(Z3_get_denominator(self.ctx_ref(), self.as_ast()), self.ctx)
def numerator(self):
""" Return the numerator if `self` is rational.
>>> Numeral("2/3").numerator()
2
"""
assert(self.is_rational())
return Numeral(Z3_get_numerator(self.ctx_ref(), self.as_ast()), self.ctx)
def is_irrational(self):
""" Return True if the numeral is irrational.
>>> Numeral(2).is_irrational()
False
>>> Numeral("2/3").is_irrational()
False
>>> Numeral(Sqrt(2)).is_irrational()
True
"""
return not self.is_rational()
def as_long(self):
""" Return a numeral (that is an integer) as a Python long.
>>> (Numeral(10)**20).as_long()
100000000000000000000L
"""
assert(self.is_integer())
return long(Z3_get_numeral_string(self.ctx_ref(), self.as_ast()))
def approx(self, precision=10):
"""Return a numeral that approximates the numeral `self`.
The result `r` is such that |r - self| <= 1/10^precision
If `self` is rational, then the result is `self`.
>>> x = Numeral(2).root(2)
>>> x.approx(20)
6838717160008073720548335/4835703278458516698824704
>>> x.approx(5)
2965821/2097152
>>> Numeral(2).approx(10)
2
"""
return self.upper(precision)
def upper(self, precision=10):
"""Return a upper bound that approximates the numeral `self`.
The result `r` is such that r - self <= 1/10^precision
If `self` is rational, then the result is `self`.
>>> x = Numeral(2).root(2)
>>> x.upper(20)
6838717160008073720548335/4835703278458516698824704
>>> x.upper(5)
2965821/2097152
>>> Numeral(2).upper(10)
2
"""
if self.is_rational():
return self
else:
return Numeral(Z3_get_algebraic_number_upper(self.ctx_ref(), self.as_ast(), precision), self.ctx)
def lower(self, precision=10):
"""Return a lower bound that approximates the numeral `self`.
The result `r` is such that self - r <= 1/10^precision
If `self` is rational, then the result is `self`.
>>> x = Numeral(2).root(2)
>>> x.lower(20)
1709679290002018430137083/1208925819614629174706176
>>> Numeral("2/3").lower(10)
2/3
"""
if self.is_rational():
return self
else:
return Numeral(Z3_get_algebraic_number_lower(self.ctx_ref(), self.as_ast(), precision), self.ctx)
def sign(self):
""" Return the sign of the numeral.
>>> Numeral(2).sign()
1
>>> Numeral(-3).sign()
-1
>>> Numeral(0).sign()
0
"""
return Z3_algebraic_sign(self.ctx_ref(), self.ast)
def is_pos(self):
""" Return True if the numeral is positive.
>>> Numeral(2).is_pos()
True
>>> Numeral(-3).is_pos()
False
>>> Numeral(0).is_pos()
False
"""
return Z3_algebraic_is_pos(self.ctx_ref(), self.ast)
def is_neg(self):
""" Return True if the numeral is negative.
>>> Numeral(2).is_neg()
False
>>> Numeral(-3).is_neg()
True
>>> Numeral(0).is_neg()
False
"""
return Z3_algebraic_is_neg(self.ctx_ref(), self.ast)
def is_zero(self):
""" Return True if the numeral is zero.
>>> Numeral(2).is_zero()
False
>>> Numeral(-3).is_zero()
False
>>> Numeral(0).is_zero()
True
>>> sqrt2 = Numeral(2).root(2)
>>> sqrt2.is_zero()
False
>>> (sqrt2 - sqrt2).is_zero()
True
"""
return Z3_algebraic_is_zero(self.ctx_ref(), self.ast)
def __add__(self, other):
""" Return the numeral `self + other`.
>>> Numeral(2) + 3
5
>>> Numeral(2) + Numeral(4)
6
>>> Numeral("2/3") + 1
5/3
"""
return Numeral(Z3_algebraic_add(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast), self.ctx)
def __radd__(self, other):
""" Return the numeral `other + self`.
>>> 3 + Numeral(2)
5
"""
return Numeral(Z3_algebraic_add(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast), self.ctx)
def __sub__(self, other):
""" Return the numeral `self - other`.
>>> Numeral(2) - 3
-1
"""
return Numeral(Z3_algebraic_sub(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast), self.ctx)
def __rsub__(self, other):
""" Return the numeral `other - self`.
>>> 3 - Numeral(2)
1
"""
return Numeral(Z3_algebraic_sub(self.ctx_ref(), _to_numeral(other, self.ctx).ast, self.ast), self.ctx)
def __mul__(self, other):
""" Return the numeral `self * other`.
>>> Numeral(2) * 3
6
"""
return Numeral(Z3_algebraic_mul(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast), self.ctx)
def __rmul__(self, other):
""" Return the numeral `other * mul`.
>>> 3 * Numeral(2)
6
"""
return Numeral(Z3_algebraic_mul(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast), self.ctx)
def __div__(self, other):
""" Return the numeral `self / other`.
>>> Numeral(2) / 3
2/3
>>> Numeral(2).root(2) / 3
0.4714045207?
>>> Numeral(Sqrt(2)) / Numeral(Sqrt(3))
0.8164965809?
"""
return Numeral(Z3_algebraic_div(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast), self.ctx)
def __rdiv__(self, other):
""" Return the numeral `other / self`.
>>> 3 / Numeral(2)
3/2
>>> 3 / Numeral(2).root(2)
2.1213203435?
"""
return Numeral(Z3_algebraic_div(self.ctx_ref(), _to_numeral(other, self.ctx).ast, self.ast), self.ctx)
def root(self, k):
""" Return the numeral `self^(1/k)`.
>>> sqrt2 = Numeral(2).root(2)
>>> sqrt2
1.4142135623?
>>> sqrt2 * sqrt2
2
>>> sqrt2 * 2 + 1
3.8284271247?
>>> (sqrt2 * 2 + 1).sexpr()
'(root-obj (+ (^ x 2) (* (- 2) x) (- 7)) 2)'
"""
return Numeral(Z3_algebraic_root(self.ctx_ref(), self.ast, k), self.ctx)
def power(self, k):
""" Return the numeral `self^k`.
>>> sqrt3 = Numeral(3).root(2)
>>> sqrt3
1.7320508075?
>>> sqrt3.power(2)
3
"""
return Numeral(Z3_algebraic_power(self.ctx_ref(), self.ast, k), self.ctx)
def __pow__(self, k):
""" Return the numeral `self^k`.
>>> sqrt3 = Numeral(3).root(2)
>>> sqrt3
1.7320508075?
>>> sqrt3**2
3
"""
return self.power(k)
def __lt__(self, other):
""" Return True if `self < other`.
>>> Numeral(Sqrt(2)) < 2
True
>>> Numeral(Sqrt(3)) < Numeral(Sqrt(2))
False
>>> Numeral(Sqrt(2)) < Numeral(Sqrt(2))
False
"""
return Z3_algebraic_lt(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast)
def __rlt__(self, other):
""" Return True if `other < self`.
>>> 2 < Numeral(Sqrt(2))
False
"""
return self > other
def __gt__(self, other):
""" Return True if `self > other`.
>>> Numeral(Sqrt(2)) > 2
False
>>> Numeral(Sqrt(3)) > Numeral(Sqrt(2))
True
>>> Numeral(Sqrt(2)) > Numeral(Sqrt(2))
False
"""
return Z3_algebraic_gt(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast)
def __rgt__(self, other):
""" Return True if `other > self`.
>>> 2 > Numeral(Sqrt(2))
True
"""
return self < other
def __le__(self, other):
""" Return True if `self <= other`.
>>> Numeral(Sqrt(2)) <= 2
True
>>> Numeral(Sqrt(3)) <= Numeral(Sqrt(2))
False
>>> Numeral(Sqrt(2)) <= Numeral(Sqrt(2))
True
"""
return Z3_algebraic_le(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast)
def __rle__(self, other):
""" Return True if `other <= self`.
>>> 2 <= Numeral(Sqrt(2))
False
"""
return self >= other
def __ge__(self, other):
""" Return True if `self >= other`.
>>> Numeral(Sqrt(2)) >= 2
False
>>> Numeral(Sqrt(3)) >= Numeral(Sqrt(2))
True
>>> Numeral(Sqrt(2)) >= Numeral(Sqrt(2))
True
"""
return Z3_algebraic_ge(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast)
def __rge__(self, other):
""" Return True if `other >= self`.
>>> 2 >= Numeral(Sqrt(2))
True
"""
return self <= other
def __eq__(self, other):
""" Return True if `self == other`.
>>> Numeral(Sqrt(2)) == 2
False
>>> Numeral(Sqrt(3)) == Numeral(Sqrt(2))
False
>>> Numeral(Sqrt(2)) == Numeral(Sqrt(2))
True
"""
return Z3_algebraic_eq(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast)
def __ne__(self, other):
""" Return True if `self != other`.
>>> Numeral(Sqrt(2)) != 2
True
>>> Numeral(Sqrt(3)) != Numeral(Sqrt(2))
True
>>> Numeral(Sqrt(2)) != Numeral(Sqrt(2))
False
"""
return Z3_algebraic_neq(self.ctx_ref(), self.ast, _to_numeral(other, self.ctx).ast)
def __str__(self):
if Z3_is_numeral_ast(self.ctx_ref(), self.ast):
return str(RatNumRef(self.ast, self.ctx))
else:
return str(AlgebraicNumRef(self.ast, self.ctx))
def __repr__(self):
return self.__str__()
def sexpr(self):
return Z3_ast_to_string(self.ctx_ref(), self.as_ast())
def as_ast(self):
return self.ast
def ctx_ref(self):
return self.ctx.ref()
def eval_sign_at(p, vs):
"""
Evaluate the sign of the polynomial `p` at `vs`. `p` is a Z3
Expression containing arithmetic operators: +, -, *, ^k where k is
an integer; and free variables x that is_var(x) is True. Moreover,
all variables must be real.
The result is 1 if the polynomial is positive at the given point,
-1 if negative, and 0 if zero.
>>> x0, x1, x2 = RealVarVector(3)
>>> eval_sign_at(x0**2 + x1*x2 + 1, (Numeral(0), Numeral(1), Numeral(2)))
1
>>> eval_sign_at(x0**2 - 2, [ Numeral(Sqrt(2)) ])
0
>>> eval_sign_at((x0 + x1)*(x0 + x2), (Numeral(0), Numeral(Sqrt(2)), Numeral(Sqrt(3))))
1
"""
num = len(vs)
_vs = (Ast * num)()
for i in range(num):
_vs[i] = vs[i].ast
return Z3_algebraic_eval(p.ctx_ref(), p.as_ast(), num, _vs)
def isolate_roots(p, vs=[]):
"""
Given a multivariate polynomial p(x_0, ..., x_{n-1}, x_n), returns the
roots of the univariate polynomial p(vs[0], ..., vs[len(vs)-1], x_n).
Remarks:
* p is a Z3 expression that contains only arithmetic terms and free variables.
* forall i in [0, n) vs is a numeral.
The result is a list of numerals
>>> x0 = RealVar(0)
>>> isolate_roots(x0**5 - x0 - 1)
[1.1673039782?]
>>> x1 = RealVar(1)
>>> isolate_roots(x0**2 - x1**4 - 1, [ Numeral(Sqrt(3)) ])
[-1.1892071150?, 1.1892071150?]
>>> x2 = RealVar(2)
>>> isolate_roots(x2**2 + x0 - x1, [ Numeral(Sqrt(3)), Numeral(Sqrt(2)) ])
[]
"""
num = len(vs)
_vs = (Ast * num)()
for i in range(num):
_vs[i] = vs[i].ast
_roots = AstVector(Z3_algebraic_roots(p.ctx_ref(), p.as_ast(), num, _vs), p.ctx)
return [ Numeral(r) for r in _roots ]
if __name__ == "__main__":
import doctest
if doctest.testmod().failed:
exit(1)

37
src/api/python/z3poly.py Normal file
View file

@ -0,0 +1,37 @@
############################################
# Copyright (c) 2012 Microsoft Corporation
#
# Z3 Python interface for Z3 polynomials
#
# Author: Leonardo de Moura (leonardo)
############################################
from z3 import *
def subresultants(p, q, x):
"""
Return the non-constant subresultants of 'p' and 'q' with respect to the "variable" 'x'.
'p', 'q' and 'x' are Z3 expressions where 'p' and 'q' are arithmetic terms.
Note that, any subterm that cannot be viewed as a polynomial is assumed to be a variable.
Example: f(a) is a considered to be a variable b in the polynomial
f(a)*f(a) + 2*f(a) + 1
>>> x, y = Reals('x y')
>>> subresultants(2*x + y, 3*x - 2*y + 2, x)
[-7*y + 4]
>>> r = subresultants(3*y*x**2 + y**3 + 1, 2*x**3 + y + 3, x)
>>> r[0]
4*y**9 + 12*y**6 + 27*y**5 + 162*y**4 + 255*y**3 + 4
>>> r[1]
-6*y**4 + -6*y
"""
return AstVector(Z3_polynomial_subresultants(p.ctx_ref(), p.as_ast(), q.as_ast(), x.as_ast()), p.ctx)
if __name__ == "__main__":
import doctest
if doctest.testmod().failed:
exit(1)