Are there functions defining set operations , e.g set, intersection, union, members etc , over Z3 expressions ? Also, are there functions to check if a formula is a cnf or dnf ?
If not I can try to implement them in the z3utils file.
We can use Python sets to encode sets of expressions. The only problem is that the operator __eq__ for Z3Py expressions will build a new expression instead of comparing whether to expressions are equal or not. To fix that, we can use a wrapper that invokes the correct compares Z3 expressions. Here is a sample (available online at rise4fun).
class AstRefKey:
def __init__(self, n):
self.n = n
def __hash__(self):
return self.n.hash()
def __eq__(self, other):
return self.n.eq(other.n)
def __repr__(self):
return str(self.n)
def askey(n):
assert isinstance(n, AstRef)
return AstRefKey(n)
x = Int('x')
s = set()
s.add(askey(x+1))
s.add(askey(x))
print s
print askey(x + 1) in s
s2 = set()
s2.add(askey(x+2))
s2.add(askey(x))
print s2
print s.union(s2)
The only inconvenience is that we have to keep using askey. We can avoid this inconvenience by defining a class ASTSet that wraps a Python set object an invokes askey for us.
Regarding, dnf and cnf recognizers. This functionality is not exposed in the external APIs.
Related
Per this answer the Z3 set sort is implemented using arrays, which makes sense given the SetAdd and SetDel methods available in the API. It is also claimed here that if the array modification functions are never used, it's wasteful overhead to use arrays instead of uninterpreted functions. Given that, if my only uses of a set are to apply constraints with IsMember (either on individual values or as part of a quantification), is it a better idea to use an uninterpreted function mapping from the underlying element sort to booleans? So:
from z3 import *
s = Solver()
string_set = SetSort(StringSort())
x = String('x')
s.add(IsMember(x, string_set))
becomes
from z3 import *
s = Solver()
string_set = Function('string_set', StringSort(), BoolSort())
x = String('x')
s.add(string_set(x))
Are there any drawbacks to this approach? Alternative representations with even less overhead?
Those are really your only options, as long as you want to restrict yourself to the standard interface. In the past, I also had luck with representing sets (and in general relations) outside of the solver, keeping the processing completely outside. Here's what I mean:
from z3 import *
def FSet_Empty():
return lambda x: False
def FSet_Insert(val, s):
return lambda x: If(x == val, True, s(val))
def FSet_Delete(val, s):
return lambda x: If(x == val, False, s(val))
def FSet_Member(val, s):
return s(val)
x, y, z = Ints('x y z')
myset = FSet_Insert(x, FSet_Insert(y, FSet_Insert(z, FSet_Empty())))
s = Solver()
s.add(FSet_Member(2, myset))
print(s.check())
print(s.model())
Note how we model sets by unary relations, i.e., functions from values to booleans. You can generalize this to arbitrary relations and the ideas carry over. This prints:
sat
[x = 2, z = 4, y = 3]
You can easily add union (essentially Or), intersection (essentially And), and complement (essentially Not) operations. Doing cardinality is harder, especially in the presence of complement, but that's true for all the other approaches too.
As is usual with these sorts of modeling questions, there's no single approach that will work best across all problems. They'll all have their strengths and weaknesses. I'd recommend creating a single API, and implementing it using all three of these ideas, and benchmarking your problem domain to see what works the best; keeping in mind if you start working on a different problem the answer might be different. Please report your findings!
Using Z3Py, I tried to build a program which Z3 would decide means that the sort Human is empty.
from z3 import *
from z3_helper import Z3Helper
Human = DeclareSort("Human")
is_mortal = Function("is_mortal", Human, BoolSort())
h = Const('h', Human)
s = Solver()
s.add([
ForAll([h], And(is_mortal(h), Not(is_mortal(h))))
])
print s.check()
s.model()
But instead of returning a model where Human is empty, it returns unsat. Why is this?
If I remove the "all men are mortal" axiom, it returns an empty set as the model.
Is the problem that the existence of const h means that the existence of at least one Human is required?
SMT-LIB and Z3 take the view that simply typed first-order logic assumes that all sorts are non-empty. See also http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-draft-3.pdf, section 5.1 onwards.
For solving a SAT-problem I decided to use the Z3-solver from Microsoft and Python 3. The aim is to take a long model (up to 500,000 features) and find all possible solutions. To find them I want to add the first solution S1 to the initial equation and exclude S1 it and so forth. I will do it using a while-loop.
Solving a SAT-problem is important for me, because I wanna analyse feature models.
But I'm facing a problem with adding sth to the initial equation. I will share a minimal example:
# Import statements
import sys
sys.path.insert(0,'/.../z3/bin')
from z3 import * # https://github.com/Z3Prover/z3/wiki
def main():
'''
Solves after transformation a given boolean equation by using the Z3-Solver from Microsoft.
'''
fd = dict()
fd['_r'] = Bool('_r')
fd['_r_1'] = Bool('_r_1')
equation = '''And(fd.get("_r"),Or(Not(fd.get("_r")),fd.get("_r_1")))'''
# Solve the equation
s = Solver()
exec('s.add(' + equation + ')')
s.check()
print(s.model())
###################################
# Successfull until here.
###################################
s.add(Or(fd['_r'] != bool(s.model()[fd.get('_r')])))
# s.add(Or(fd['_r'] != False))
s.check()
print(s.model())
if __name__=='__main__':
main()
The first coded line after # Successfull... throws an z3types.Z3Exception: model is not available error. So i tried the line above adding simply false to the model. That works just fine.
I'm stucked here. I believe the error is easy to solve, but I don't see the solution. Does one of you? Thanks!
Models become available only after s.check() returns 'sat'.
The model maps Boolean propositions to {True, False} and
generally maps constants and functions to fixed values.
The requirement is that the model provides an interpretation
that satisfies the formula that is added to the solver 's'.
We don't know whether the solver state is satisfiable before
we have called 's.check()'.
Suppose you want to say:
s.add(Or(fd['_r'] != bool(s.model()[fd.get('_r')])))
meaning that in the model that satisfies the constraint should
have the property that if '_r' is true under the model, then
fd['_r'] != True, and if '_r' is false under the model, then
fd['_r'] != False. This is equivalent to saying
fd['_r'] != '_r'. So it isn't really necessary to access the value
of '_r' in whatever model may evaluate '_r' in order to say something
about the evaluation of it.
As per my knowledge, since z3 doesn't recognize transcendental functions its throwing me an error while conversion using following code.
def convertor(f, status="unknown", name="benchmark", logic=""):
v = (Ast * 0)()
if isinstance(f, Solver):
a = f.assertions()
if len(a) == 0:
f = BoolVal(True)
else:
f = And(*a)
return Z3_benchmark_to_smtlib_string(f.ctx_ref(), name, logic, status, "", 0, v, f.as_ast())
pi, EI, kA , kB, N = Reals('pi EI kA kB N')
s= Solver()
s.add(pi == 3.1415926525)
s.add(EI == 175.2481)
s.add(kA>= 0)
s.add(kA<= 100)
s.add(kB>= 0)
s.add(kB<= 100)
s.add(N>= 100)
s.add(N<= 200)
please change the path of the input file "beamfinv3.bch", which can be found at: link
continue_read=False
input_file = open('/home/mani/downloads/new_z3/beamfinv3.bch', 'r')
for line in input_file:
if line.strip()=="Constraints":
continue_read=True
continue
if line.strip()=="end":
continue_read=False
if continue_read==True:
parts = line.split(';')
if (parts[0]!="end"):
#print parts[0]
s.add(eval(parts[0]))
input_file.close()
file=open('cyber.smt2','w')
result=convertor(s, logic="None")
file.write (result)
error:
File "<string>", line 1, in <module>
NameError: name 'sin' is not defined
Any way out? or help?
Thanks.
The core of this problem is that eval tries to execute a Python script, i.e., all functions that occur within parts[0] must have a corresponding Python function of the same name, which is not the case for the trigonometric functions (the are neither in the Python API nor the C API, the former being based on the latter). For now you could try to add those functions yourself, perhaps with an implementation based on parse_smt2_string, or perhaps by replacing the Python strings with SMT2 strings altogether.
Z3 can represent expressions containing trigonometric functions, but it will refuse to do so when the logic is set to something; see arith_decl_plugin. I don't know Python well enough, but it might have to be None instead of "".
While Z3 can represent these expressions, it's probably not very good at solving them. See comments on the limitations in Can Z3 handle sinusoidal and exponential functions, Z3 supports for nonlinear arithmetics, and Z3 Performance with Non-Linear Arithmetic.
I am trying to solve a problem, for example I have a 4 point and each two point has a cost between them. Now I want to find a sequence of nodes which total cost would be less than a bound. I have written a code but it seems not working. The main problem is I have define a python function and trying to call it with in a constraint.
Here is my code: I have a function def getVal(n1,n2): where n1, n2 are Int Sort. The line Nodes = [ Int("n_%s" % (i)) for i in range(totalNodeNumber) ] defines 4 points as Int sort and when I am adding a constraint s.add(getVal(Nodes[0], Nodes[1]) + getVal(Nodes[1], Nodes[2]) < 100) then it calls getVal function immediately. But I want that, when Z3 will decide a value for Nodes[0], Nodes[1], Nodes[2], Nodes[3] then the function should be called for getting the cost between to points.
from z3 import *
import random
totalNodeNumber = 4
Nodes = [ Int("n_%s" % (i)) for i in range(totalNodeNumber) ]
def getVal(n1,n2):
# I need n1 and n2 values those assigned by Z3
cost = random.randint(1,20)
print cost
return IntVal(cost)
s = Solver()
#constraint: Each Nodes value should be distinct
nodes_index_distinct_constraint = Distinct(Nodes)
s.add(nodes_index_distinct_constraint)
#constraint: Each Nodes value should be between 0 and totalNodeNumber
def get_node_index_value_constraint(i):
return And(Nodes[i] >= 0, Nodes[i] < totalNodeNumber)
nodes_index_constraint = [ get_node_index_value_constraint(i) for i in range(totalNodeNumber)]
s.add(nodes_index_constraint)
#constraint: Problem with this constraint
# Here is the problem it's just called python getVal function twice without assiging Nodes[0],Nodes[1],Nodes[2] values
# But I want to implement that - Z3 will call python function during his decission making of variables
s.add(getVal(Nodes[0], Nodes[1]) + getVal(Nodes[1], Nodes[2]) + getVal(Nodes[2], Nodes[3]) < 100)
if s.check() == sat:
print "SAT"
print "Model: "
m = s.model()
nodeIndex = [ m.evaluate(Nodes[i]) for i in range(totalNodeNumber) ]
print nodeIndex
else:
print "UNSAT"
print "No solution found !!"
If this is not a right way to solve the problem then could you please tell me what would be other alternative way to solve it. Can I encode this kind of problem to find optimal sequence of way points using Z3 solver?
I don't understand what problem you need to solve. Definitely, the way getVal is formulated does not make sense. It does not use the arguments n1, n2. If you want to examine values produced by a model, then you do this after Z3 returns from a call to check().
I don't think you can use a python function in your SMT logic. What you could alternatively is define getVal as a Function like this
getVal = Function('getVal',IntSort(),IntSort(),IntSort())
And constraint the edge weights as
s.add(And(getVal(0,1)==1,getVal(1,2)==2,getVal(0,2)==3))
The first two input parameters of getVal represent the node ids and the last integer represents the weight.