Why Z3Py does not provide all possible solutions

Question

I ran into a problem where Z3Py does not enumerate all possible solutions for the given Boolean clauses. I was wondering if anyone knows why this is happening.

Here is the code I use for the Z3Py. There are 5 booleans: 1 2 3 4 and 5.

from z3 import *

a,b,c,d,e = Bools('1 2 3 4 5')
solver = Solver()
solver.add(Or(Not(a), Not(b)))
solver.add(Or(Not(b), Not(c)))
solver.add(Or(Not(c), Not(d)))
solver.add(Or(Not(d), Not(e)))

while solver.check() == sat:
    model = solver.model()
    block = []
    for declaration in model:
        constant = declaration()
        block.append(constant != model[declaration])
    solver.append(Or(block))
    solution = []
    for val in model:
        if is_true(model[val]):
            solution.append(str(val()))
        else:
            solution.append('-' + str(val()))
    solution.sort()
    print(solution)

This produces the following models:

['-1', '-2', '-3', '-4', '-5']
['-2', '-3', '-5', '1']
['-2', '-3', '-4', '1', '5']
['-2', '-4', '3', '5']
['-2', '-4', '-5', '3']
['-1', '-3', '-4', '-5', '2']
['-1', '-3', '-4', '2', '5']
['-1', '-2', '-3', '-4', '5']
['-1', '-2', '-3', '-5', '4']
['-1', '-3', '-5', '2', '4']

If I run the same clauses using Pycosat with the following code:

import pycosat
clauses = [(-1, -2), (-2, -3), (-3, -4), (-4, -5)]
for solution in pycosat.itersolve(clauses):
    print(solution)

I get those results:

[-1, -2, -3, -4, -5]
[-1, -2, -3, -4, 5]
[-1, -2, -3, 4, -5]
[-1, -2, 3, -4, -5]
[-1, -2, 3, -4, 5]
[-1, 2, -3, -4, -5]
[-1, 2, -3, -4, 5]
[-1, 2, -3, 4, -5]
[1, -2, -3, 4, -5]
[1, -2, -3, -4, -5]
[1, -2, -3, -4, 5]
[1, -2, 3, -4, -5]
[1, -2, 3, -4, 5]

Hence, Z3Py missing out 3 possible solutions based on the results from Pycosat. And those are:

[1, -2, 3, -4, 5]
[1, -2, -3, 4, -5]
[1, -2, 3, -4, -5]

I know those are the different solvers but based on the same clauses used for both of them, I would expect the results to match. Anyone knows why Z3Py would miss out those solutions in this case? The other thing is that not all the solutions in Z3Py contain all the defined variables.

Answer 1

Note that a model will only contain assignments to variables that do matter in the sat outcome. Any variable that doesn't matter will not get assigned explicitly. To avoid this problem, loop over the variables in your domain and use the parameter model_completion=True to the eval method, like this:

from z3 import *

a,b,c,d,e = Bools('1 2 3 4 5')
solver = Solver()
solver.add(Or(Not(a), Not(b)))
solver.add(Or(Not(b), Not(c)))
solver.add(Or(Not(c), Not(d)))
solver.add(Or(Not(d), Not(e)))

while solver.check() == sat:
    model = solver.model()
    block = []
    solution = []

    for var in [a, b, c, d, e]:
        v = model.eval(var, model_completion=True)
        block.append(var != v)
        solution.append(str(var) if is_true(v) else '-' + str(var))

    solver.add(Or(block))
    solution.sort()
    print(solution)

This prints:

['-1', '-2', '-3', '-4', '-5']
['-2', '-3', '-4', '-5', '1']
['-2', '-3', '-5', '1', '4']
['-1', '-2', '-3', '-5', '4']
['-1', '-2', '-3', '-4', '5']
['-1', '-2', '-4', '3', '5']
['-2', '-4', '-5', '1', '3']
['-2', '-4', '1', '3', '5']
['-2', '-3', '-4', '1', '5']
['-1', '-3', '-4', '2', '5']
['-1', '-3', '-4', '-5', '2']
['-1', '-2', '-4', '-5', '3']
['-1', '-3', '-5', '2', '4']

which I believe is what you're looking for.

Answer 2

The other thing is that not all the solutions in Z3Py contain all the defined variables.

That means that the unmentioned variables do not affect the result. So having ['-2', '-4', '3', '5'] as a solution is equivalent to having both ['1', '-2', '-4', '3', '5'] and ['-1', '-2', '-4', '3', '5'].

If you take that into account when comparing the solutions, the sets of solutions provided by both solvers are equivalent.