How can I perform quantifier elimination using the Python API of Z3? Although I checked the tutorial and API, couldn't manage to do it.
I have a formula that has an existential quantifier and I want Z3 to give me a new formula, such that this quantifier is eliminated. I essentially want to do the same thing as this:
How to hide variable with Z3
but with the Python interface. Also my formula is in linear arithmetic.
Thanks!
Addition:
After doing the quantifier elimination I will "add" the quantifier-free formula with another one. So if I make use of the Tactic, is there a way to convert a subgoal (which is the output of the tactic) to an expression in linear arithmetic?
You may use the quantifier elimination tactic for this (see the Tactic.apply docstring):
from z3 import Ints, Tactic, Exists, And
x, t1, t2 = Ints('x t1 t2')
t = Tactic('qe')
print t(Exists(x, And(t1 < x, x < t2)))
Possible solution using Z3Py online:
x, t1, t2 = Reals('x t1 t2')
g = Goal()
g.add(Exists(x, And(t1 < x, x < t2)))
t = Tactic('qe')
print t(g)
Output:
[[¬(0 ≤ t1 + -1·t2)]]
Run this example online here
Possible solution using Redlog of reduce:
Related
Consider these two formulae:
Exists y. Forall x. (y>x), which is unsat.
Forall x. Exists y. (y>x), which is sat.
Note that we cannot find “models” for the sat formula, e.g., using Z3 it outputs Z3Exception: model is not available for the following code:
phi = ForAll([x],Exists([y], lit))
s_def = Solver()
s_def.add(phi)
print(s_def.model())
Also, quantifier elimination does not output an elimination, but a satisfiability result
[[]] (i.e., True):
x, y = Reals('x, y')
t = Tactic("qe")
lit = (y>x)
ae = Goal()
ae.add(ForAll([x],Exists([y], lit)))
ae_qe = t(ae)
print(ae_qe)
I think this happens because the value of y fully depends on x (e.g., if x is 5 then y can be 6). Thus, I have some questions:
Am I right with this interpretation?
What would be the meaning of “a model of a universally quantified formula”?
Do we say a formula accepts quantifier elimination even if it never eliminates the quantifier but "only: evaluate to True or False?
Is there a way to synthetise or construct a model/function that represents a y that holds the constraint (y>x); e.g. f(x)=x+1. In other words, does it make sense that the quantifier elimination of a Forall x. Exists y. Phi(x,y) like the example would be Forall x. Phi(x,f(x))?
You get a model-not-available, because you didn't call check. A model is only available after a call to check. Try this:
from z3 import *
x, y = Ints('x y')
phi = ForAll([x], Exists([y], y > x))
s = Solver()
s.add(phi)
print(s.check())
print(s.model())
This prints:
sat
[]
Now, you're correct that you won't get a meaningful/helpful model when you have a universal-quantifier; because the model will depend on the choice of x in each case. (You only get values for top-level existentials.) This is why the model is printed as the empty-list.
Side Note In certain cases, you can use the skolemization trick (https://en.wikipedia.org/wiki/Skolem_normal_form) to get rid of the nested existential and get a mapping function, but this doesn't always work with SMT solvers as they can only build "finite" skolem functions. (For your example, there isn't such a finite function; i.e., a function that can be written as a case-analysis on the inputs, or an if-then-else chain.)
For your specific questions:
Yes; value of y depends on x and thus cannot be displayed as is. Sometimes skolemization will let you work around this, but SMT solvers can only build finite skolem-functions.
Model of a universally quantified formula is more or less meaningless. It's true for all values of the universally quantified variables. Only top-level existentials are meaningful in a model.
Quantifier elimination did work here; it got rid of the whole formula. Note that QE preserves satisfiability; so it has done its job.
This is the skolem function. As you noted, this skolem function is not finite (can't be written as a chain of finite if-then-elses on concrete values), and thus existing SMT solvers cannot find/print them for you. If you try, you'll see that the solver simply loops:
from z3 import *
x = Int('x')
skolem = Function('skolem', IntSort(), IntSort())
phi = ForAll([x], skolem(x) > x)
s = Solver()
s.add(phi)
print(s)
print(s.check())
print(s.model())
The above never terminates, unfortunately. This sort of problem is simply too complicated for the push-button approach of SMT solving.
In Z3-Py, I am performing quantifier elimination (QE) over the following formulae:
Exists y. Forall x. (x>=2) => ((y>1) /\ (y<=x))
Forall x. Exists y. (x>=2) => ((y>1) /\ (y<=x)),
where both x and y are Integers. I did QE in the following way:
x, y = Ints('x, y')
t = Tactic("qe")
negS0= (x >= 2)
s1 = (y > 1)
s2 = (y <= x)
#EA
ea = Goal()
ea.add(Exists([y],Implies(negS0, (ForAll([x], And(s1,s2))))))
ea_qe = t(ea)
print(ea_qe)
#AE
ae = Goal()
ae.add(ForAll([x],Implies(negS0, (Exists([y], And(s1,s2))))))
ae_qe = t(ae)
print(ae_qe)
Result QE for ae is as expected: [[]] (i.e., True). However, as for ea, QE outputs: [[Not(x, >= 2)]], which is a results that I do not know how to interpret since (1) it has not really performed QE (note the resulting formula still contains x and indeed does not contain y which is the outermost quantified variable) and (2) I do not understand the meaning of the comma in x, >=. I cannot get the model either:
phi = Exists([y],Implies(negS0, (ForAll([x], And(s1,s2)))))
s_def = Solver()
s_def.add(phi)
print(s_def.model())
This results in the error Z3Exception: model is not available.
I think the point is as follows: since (x>=2) is an implication, there are two ways to satisfy the formula; by making the antecedent False or by satisfying the consequent. In the second case, the model would be y=2. But in the first case, the result of QE would be True, thus we cannot get a single model (as it happens with a universal model):
phi = ForAll([x],Implies(negS0, (Exists([y], And(s1,s2)))))
s_def = Solver()
s_def.add(phi)
print(s_def.model())
In any case, I cannot 'philosophically' understand the meaning of a QE of x where x is part of the (quantifier-eliminated) answer.
Any help?
There are two separate issues here, I'll address them separately.
The mysterious comma This is a common gotcha. You declared:
x, y = Ints('x, y')
That is, you gave x the name "x," and y the name "y". Note the comma after the x in the name. This should be
x, y = Ints('x y')
I guess you can see the difference: The name you gave to the variable x is "x," when you do the first; i.e., comma is part of the name. Simply skip the comma on the right hand side, which isn't what you intended anyhow. And the results will start being more meaningful. To be fair, this is a common mistake, and I wish the z3 developers ignored the commas and other punctuation in the string you give; but that's just not the case. They simply break at whitespace.
Quantification
This is another common gotcha. When you write:
ea.add(Exists([y],Implies(negS0, (ForAll([x], And(s1,s2))))))
the x that exists in negS0 is not quantified over by your ForAll, since it's not in the scope. Perhaps you meant:
ea.add(Exists([y],ForAll([x], Implies(negS0, And(s1,s2)))))
It's hard to guess what you were trying to do, but I hope the above makes it clear that the x wasn't quantified. Also, remember that a top-level exist quantifier in a formula is more or less irrelevant. It's equivalent to a top-level declaration for all practical purposes.
Once you make this fix, I think things will become more clear. If not, please ask further clarifying questions. (As a separate question on Stack-overflow; as edits to existing questions only complicate the matters.)
I have a Boolean formula (format: CNF) whose satisfiablity I check using the Z3 SAT solver. I am interested in obtaining partial assignments when the formula is satisfiable. I tried model.partial=true on a simple formula for an OR gate and did not get any partial assignment.
Can you suggest how this can be done? I have no constraints on the assignment other than that it be partial.
Z3's partial model mode are just for models of functions. For propositional formulas there is no assurance that models use a minimal number of assignments. On the contrary, the default mode of SAT solvers is that they find complete assignments.
Suppose you are interested in a minimized number of literals, such that the conjunction of the literals imply the formula. You can use unsat cores to get such subsets. The idea is that you first find a model of your formula F as a conjunction of literals l1, l2, ..., ln. Then given that this is a model of F we have that l1 & l2 & ... & ln & not F is unsatisfiable.
So the idea is to assert "not F" and check satisfiability of "not F" modulo assumptions l1, l2, .., ln. Since the result is unsat, you can query Z3 to retrieve an unsat core among l1, l2, .., ln.
From python what you would do is to create two solver objects:
s1 = Solver()
s2 = Solver()
Then you add F, respectively, Not(F):
s1.add(F)
s2.add(Not(F))
then you find a reduced model for F using both solvers:
is_Sat = s1.check()
if is_Sat != sat:
# do something else, return
m = s1.model()
literals = [sign(m, m[idx]()) for idx in range(len(m)) ]
is_sat = s2.check(literals)
if is_Sat != unsat:
# should not happen
core = s2.unsat_core()
print core
where
def sign(m, c):
val = m.eval(c)
if is_true(val):
return c
else if is_false(val):
return Not(c)
else:
# should not happen for propositional variables.
return BoolVal(True)
There are of course other ways to get reduced set of literals. A partial cheap way is to eagerly evaluate each clause and add literals from the model until each clause is satisfied by at least one literal in the model. In other words, you are looking for a minimal hitting set. You would have to implement this outside of Z3.
I try to find the values for A,B,C,D satisfiying the formula g = And(ForAll([i, j, k], Implies(And(k <= 0, A * i + B * j + C * k + D <= 0), k + i - j <= 0)),Not(And(A==0,B==0,C==0))) using solve(g) . This has many possible solutions, one is A=1,B=-1,C=D=0 but Z3 can't seem to do this (solve(g) does not terminate).
Can Z3 do this kind of nonlinear (but simple) formula ? Perhaps I can specify some QE strategies or something for this ?
Z3 has a quantifier elimination tactic. We can enable it by creating a solver using:
s = Then('qe', 'smt').solver()
This command will create a solver that first applies quantifier elimination and then invokes a SMT solver. However, Z3 has very limited support for quantifier elimination of nonlinear formulas. Your examples is nonlinear because it contains: A * i + B * j + C * k + D <= 0.
So, the qe tactic will not be able to eliminate the quantifier.
It would be great if you could implement better QE support for nonlinear arithmetic.
I have 2 formulas F1 and F2. These two formulas share most variables, except some 'temporary' (or I call them 'free') variables having different names, that are there for some reasons.
Now I want to prove F1 == F2, but prove() method of Z3 always takes into account all the variables. How can I tell prove() to ignore those 'free' variables, and focuses only on a list of variables I really care about?
I mean with all the same input to the list of my variables, if at the output time, F1 and F2 have the same value of all these variables (regardless the values of 'free' variables), then I consider them 'equivalence'
I believe this problem has been solved in other researches before, but I dont know where to look for the information.
Thanks so much.
We can use existential quantifiers to capture 'temporary'/'free' variables.
For example, in the following example, the formulas F and G are not equivalent.
x, y, z, w = Ints('x y z w')
F = And(x >= y, y >= z)
G = And(x > z - 1, w < z)
prove(F == G)
The script will produce the counterexample [z = 0, y = -1, x = 0, w = -1].
If we consider y and w as 'temporary' variables, we may try to prove:
prove(Exists([y], F) == Exists([w], G))
Now, Z3 will return proved. Z3 is essentially showing that for all x and z, there is a y that makes F true if and only if there is a w that makes G true.
Here is the full example.
Remark: when we add quantifiers, we are making the problem much harder for Z3. It may return unknown for problems containing quantifiers.
Apparently, I cannot comment, so I have to add another answer. The process of "disregarding" certain variables is typically called "projection" or "forgetting". I am not familiar with it in contexts going beyond propositional logic, but if direct existential quantification is possible (which Leo described), it is conceptually the simplest way to do it.