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|
############################################
# Copyright (c) 2012 Microsoft Corporation
#
# Z3 Python interface
#
# Authors: Leonardo de Moura (leonardo)
# ThanhVu (Vu) Nguyen <tnguyen@cs.unm.edu>
############################################
"""
Usage:
import common_z3 as CM_Z3
"""
from .z3 import *
def vset(seq, idfun=None, as_list=True):
# This functions preserves the order of arguments while removing duplicates.
# This function is from https://code.google.com/p/common-python-vu/source/browse/vu_common.py
# (Thanhu's personal code). It has been copied here to avoid a dependency on vu_common.py.
"""
order preserving
>>> vset([[11,2],1, [10,['9',1]],2, 1, [11,2],[3,3],[10,99],1,[10,['9',1]]],idfun=repr)
[[11, 2], 1, [10, ['9', 1]], 2, [3, 3], [10, 99]]
"""
def _uniq_normal(seq):
d_ = {}
for s in seq:
if s not in d_:
d_[s] = None
yield s
def _uniq_idfun(seq,idfun):
d_ = {}
for s in seq:
h_ = idfun(s)
if h_ not in d_:
d_[h_] = None
yield s
if idfun is None:
res = _uniq_normal(seq)
else:
res = _uniq_idfun(seq,idfun)
return list(res) if as_list else res
def get_z3_version(as_str=False):
major = ctypes.c_uint(0)
minor = ctypes.c_uint(0)
build = ctypes.c_uint(0)
rev = ctypes.c_uint(0)
Z3_get_version(major,minor,build,rev)
rs = map(int,(major.value,minor.value,build.value,rev.value))
if as_str:
return "{}.{}.{}.{}".format(*rs)
else:
return rs
def ehash(v):
"""
Returns a 'stronger' hash value than the default hash() method.
The result from hash() is not enough to distinguish between 2
z3 expressions in some cases.
Note: the following doctests will fail with Python 2.x as the
default formatting doesn't match that of 3.x.
>>> x1 = Bool('x'); x2 = Bool('x'); x3 = Int('x')
>>> print(x1.hash(),x2.hash(),x3.hash()) #BAD: all same hash values
783810685 783810685 783810685
>>> print(ehash(x1), ehash(x2), ehash(x3))
x_783810685_1 x_783810685_1 x_783810685_2
"""
if __debug__:
assert is_expr(v)
return "{}_{}_{}".format(str(v),v.hash(),v.sort_kind())
"""
In Z3, variables are called *uninterpreted* consts and
variables are *interpreted* consts.
"""
def is_expr_var(v):
"""
EXAMPLES:
>>> is_expr_var(Int('7'))
True
>>> is_expr_var(IntVal('7'))
False
>>> is_expr_var(Bool('y'))
True
>>> is_expr_var(Int('x') + 7 == Int('y'))
False
>>> LOnOff, (On,Off) = EnumSort("LOnOff",['On','Off'])
>>> Block,Reset,SafetyInjection=Consts("Block Reset SafetyInjection",LOnOff)
>>> is_expr_var(LOnOff)
False
>>> is_expr_var(On)
False
>>> is_expr_var(Block)
True
>>> is_expr_var(SafetyInjection)
True
"""
return is_const(v) and v.decl().kind()==Z3_OP_UNINTERPRETED
def is_expr_val(v):
"""
EXAMPLES:
>>> is_expr_val(Int('7'))
False
>>> is_expr_val(IntVal('7'))
True
>>> is_expr_val(Bool('y'))
False
>>> is_expr_val(Int('x') + 7 == Int('y'))
False
>>> LOnOff, (On,Off) = EnumSort("LOnOff",['On','Off'])
>>> Block,Reset,SafetyInjection=Consts("Block Reset SafetyInjection",LOnOff)
>>> is_expr_val(LOnOff)
False
>>> is_expr_val(On)
True
>>> is_expr_val(Block)
False
>>> is_expr_val(SafetyInjection)
False
"""
return is_const(v) and v.decl().kind()!=Z3_OP_UNINTERPRETED
def get_vars(f,rs=[]):
"""
>>> x,y = Ints('x y')
>>> a,b = Bools('a b')
>>> get_vars(Implies(And(x+y==0,x*2==10),Or(a,Implies(a,b==False))))
[x, y, a, b]
"""
if __debug__:
assert is_expr(f)
if is_const(f):
if is_expr_val(f):
return rs
else: #variable
return vset(rs + [f],str)
else:
for f_ in f.children():
rs = get_vars(f_,rs)
return vset(rs,str)
def mk_var(name,vsort):
if vsort.kind() == Z3_INT_SORT:
v = Int(name)
elif vsort.kind() == Z3_REAL_SORT:
v = Real(name)
elif vsort.kind() == Z3_BOOL_SORT:
v = Bool(name)
elif vsort.kind() == Z3_DATATYPE_SORT:
v = Const(name,vsort)
else:
assert False, 'Cannot handle this sort (s: %sid: %d)'\
%(vsort,vsort.kind())
return v
def prove(claim,assume=None,verbose=0):
"""
>>> r,m = prove(BoolVal(True),verbose=0); r,model_str(m,as_str=False)
(True, None)
#infinite counter example when proving contradiction
>>> r,m = prove(BoolVal(False)); r,model_str(m,as_str=False)
(False, [])
>>> x,y,z=Bools('x y z')
>>> r,m = prove(And(x,Not(x))); r,model_str(m,as_str=True)
(False, '[]')
>>> r,m = prove(True,assume=And(x,Not(x)),verbose=0)
Traceback (most recent call last):
...
AssertionError: Assumption is alway False!
>>> r,m = prove(Implies(x,x),assume=y,verbose=2); r,model_str(m,as_str=False)
assume:
y
claim:
Implies(x, x)
to_prove:
Implies(y, Implies(x, x))
(True, None)
>>> r,m = prove(And(x,True),assume=y,verbose=0); r,model_str(m,as_str=False)
(False, [(x, False), (y, True)])
>>> r,m = prove(And(x,y),assume=y,verbose=0)
>>> print(r)
False
>>> print(model_str(m,as_str=True))
x = False
y = True
>>> a,b = Ints('a b')
>>> r,m = prove(a**b == b**a,assume=None,verbose=0)
E: cannot solve !
>>> r is None and m is None
True
"""
if __debug__:
assert not assume or is_expr(assume)
to_prove = claim
if assume:
if __debug__:
is_proved,_ = prove(Not(assume))
def _f():
emsg = "Assumption is alway False!"
if verbose >= 2:
emsg = "{}\n{}".format(assume,emsg)
return emsg
assert is_proved==False, _f()
to_prove = Implies(assume,to_prove)
if verbose >= 2:
print('assume: ')
print(assume)
print('claim: ')
print(claim)
print('to_prove: ')
print(to_prove)
f = Not(to_prove)
models = get_models(f,k=1)
if models is None: #unknown
print('E: cannot solve !')
return None, None
elif models == False: #unsat
return True,None
else: #sat
if __debug__:
assert isinstance(models,list)
if models:
return False, models[0] #the first counterexample
else:
return False, [] #infinite counterexample,models
def get_models(f,k):
"""
Returns the first k models satisfiying f.
If f is not satisfiable, returns False.
If f cannot be solved, returns None
If f is satisfiable, returns the first k models
Note that if f is a tautology, e.g.\ True, then the result is []
Based on http://stackoverflow.com/questions/11867611/z3py-checking-all-solutions-for-equation
EXAMPLES:
>>> x, y = Ints('x y')
>>> len(get_models(And(0<=x,x <= 4),k=11))
5
>>> get_models(And(0<=x**y,x <= 1),k=2) is None
True
>>> get_models(And(0<=x,x <= -1),k=2)
False
>>> len(get_models(x+y==7,5))
5
>>> len(get_models(And(x<=5,x>=1),7))
5
>>> get_models(And(x<=0,x>=5),7)
False
>>> x = Bool('x')
>>> get_models(And(x,Not(x)),k=1)
False
>>> get_models(Implies(x,x),k=1)
[]
>>> get_models(BoolVal(True),k=1)
[]
"""
if __debug__:
assert is_expr(f)
assert k>=1
s = Solver()
s.add(f)
models = []
i = 0
while s.check() == sat and i < k:
i = i + 1
m = s.model()
if not m: #if m == []
break
models.append(m)
#create new constraint to block the current model
block = Not(And([v() == m[v] for v in m]))
s.add(block)
if s.check() == unknown:
return None
elif s.check() == unsat and i==0:
return False
else:
return models
def is_tautology(claim,verbose=0):
"""
>>> is_tautology(Implies(Bool('x'),Bool('x')))
True
>>> is_tautology(Implies(Bool('x'),Bool('y')))
False
>>> is_tautology(BoolVal(True))
True
>>> is_tautology(BoolVal(False))
False
"""
return prove(claim=claim,assume=None,verbose=verbose)[0]
def is_contradiction(claim,verbose=0):
"""
>>> x,y=Bools('x y')
>>> is_contradiction(BoolVal(False))
True
>>> is_contradiction(BoolVal(True))
False
>>> is_contradiction(x)
False
>>> is_contradiction(Implies(x,y))
False
>>> is_contradiction(Implies(x,x))
False
>>> is_contradiction(And(x,Not(x)))
True
"""
return prove(claim=Not(claim),assume=None,verbose=verbose)[0]
def exact_one_model(f):
"""
return True if f has exactly 1 model, False otherwise.
EXAMPLES:
>>> x, y = Ints('x y')
>>> exact_one_model(And(0<=x**y,x <= 0))
False
>>> exact_one_model(And(0<=x,x <= 0))
True
>>> exact_one_model(And(0<=x,x <= 1))
False
>>> exact_one_model(And(0<=x,x <= -1))
False
"""
models = get_models(f,k=2)
if isinstance(models,list):
return len(models)==1
else:
return False
def myBinOp(op,*L):
"""
>>> myAnd(*[Bool('x'),Bool('y')])
And(x, y)
>>> myAnd(*[Bool('x'),None])
x
>>> myAnd(*[Bool('x')])
x
>>> myAnd(*[])
>>> myAnd(Bool('x'),Bool('y'))
And(x, y)
>>> myAnd(*[Bool('x'),Bool('y')])
And(x, y)
>>> myAnd([Bool('x'),Bool('y')])
And(x, y)
>>> myAnd((Bool('x'),Bool('y')))
And(x, y)
>>> myAnd(*[Bool('x'),Bool('y'),True])
Traceback (most recent call last):
...
AssertionError
"""
if __debug__:
assert op == Z3_OP_OR or op == Z3_OP_AND or op == Z3_OP_IMPLIES
if len(L)==1 and (isinstance(L[0],list) or isinstance(L[0],tuple)):
L = L[0]
if __debug__:
assert all(not isinstance(l,bool) for l in L)
L = [l for l in L if is_expr(l)]
if L:
if len(L)==1:
return L[0]
else:
if op == Z3_OP_OR:
return Or(L)
elif op == Z3_OP_AND:
return And(L)
else: #IMPLIES
return Implies(L[0],L[1])
else:
return None
def myAnd(*L): return myBinOp(Z3_OP_AND,*L)
def myOr(*L): return myBinOp(Z3_OP_OR,*L)
def myImplies(a,b):return myBinOp(Z3_OP_IMPLIES,[a,b])
Iff = lambda f: And(Implies(f[0],f[1]),Implies(f[1],f[0]))
def model_str(m,as_str=True):
"""
Returned a 'sorted' model (so that it's easier to see)
The model is sorted by its key,
e.g. if the model is y = 3 , x = 10, then the result is
x = 10, y = 3
EXAMPLES:
see doctest exampels from function prove()
"""
if __debug__:
assert m is None or m == [] or isinstance(m,ModelRef)
if m :
vs = [(v,m[v]) for v in m]
vs = sorted(vs,key=lambda a,_: str(a))
if as_str:
return '\n'.join(['{} = {}'.format(k,v) for (k,v) in vs])
else:
return vs
else:
return str(m) if as_str else m
|