Z3 returns model not available

问题:

If possible I'd like a second opinion on my code.

The constraints of the problem are:

• a,b,c,d,e,f are non-zero integers
• s1 = [a,b,c] and s2 = [d,e,f] are sets
• The sum s1_i + s2_j for i,j = 0..2 has to be a perfect square

I don't understand why but my code returns model not available. Moreover, when commenting out the following lines:

(assert (and (> sqrtx4 1) (= x4 (* sqrtx4 sqrtx4))))
(assert (and (> sqrtx5 1) (= x5 (* sqrtx5 sqrtx5))))
(assert (and (> sqrtx6 1) (= x6 (* sqrtx6 sqrtx6))))

(assert (and (> sqrtx7 1) (= x7 (* sqrtx7 sqrtx7))))
(assert (and (> sqrtx8 1) (= x8 (* sqrtx8 sqrtx8))))
(assert (and (> sqrtx9 1) (= x9 (* sqrtx9 sqrtx9))))

The values for d, e, f are negative. There is no constraint that requires them to do so. I'm wondering if perhaps there are some hidden constraints that sneaked in and mess up the model.

A valid expected solution would be:

a = 3
b = 168
c = 483
d = 1
e = 193
f = 673

Edit: inserting (assert (= a 3)) and (assert (= b 168)) results in the solver finding the correct values. This only puzzles me further.

Full code:

(declare-fun sqrtx1 () Int)
(declare-fun sqrtx2 () Int)
(declare-fun sqrtx3 () Int)
(declare-fun sqrtx4 () Int)
(declare-fun sqrtx5 () Int)
(declare-fun sqrtx6 () Int)
(declare-fun sqrtx7 () Int)
(declare-fun sqrtx8 () Int)
(declare-fun sqrtx9 () Int)

(declare-fun a     () Int)
(declare-fun b     () Int)
(declare-fun c     () Int)
(declare-fun d     () Int)
(declare-fun e     () Int)
(declare-fun f     () Int)

(declare-fun x1     () Int)
(declare-fun x2     () Int)
(declare-fun x3     () Int)
(declare-fun x4     () Int)
(declare-fun x5     () Int)
(declare-fun x6     () Int)
(declare-fun x7     () Int)
(declare-fun x8     () Int)
(declare-fun x9     () Int)


;all numbers are non-zero integers
(assert (not (= a 0)))
(assert (not (= b 0)))
(assert (not (= c 0)))
(assert (not (= d 0)))
(assert (not (= e 0)))
(assert (not (= f 0)))

;both arrays need to be sets
(assert (not (= a b)))
(assert (not (= a c)))
(assert (not (= b c)))

(assert (not (= d e)))
(assert (not (= d f)))
(assert (not (= e f)))



(assert (and (> sqrtx1 1) (= x1 (* sqrtx1 sqrtx1))))
(assert (and (> sqrtx2 1) (= x2 (* sqrtx2 sqrtx2))))
(assert (and (> sqrtx3 1) (= x3 (* sqrtx3 sqrtx3))))


(assert (and (> sqrtx4 1) (= x4 (* sqrtx4 sqrtx4))))
(assert (and (> sqrtx5 1) (= x5 (* sqrtx5 sqrtx5))))
(assert (and (> sqrtx6 1) (= x6 (* sqrtx6 sqrtx6))))

(assert (and (> sqrtx7 1) (= x7 (* sqrtx7 sqrtx7))))
(assert (and (> sqrtx8 1) (= x8 (* sqrtx8 sqrtx8))))
(assert (and (> sqrtx9 1) (= x9 (* sqrtx9 sqrtx9))))

;all combinations of sums need to be squared
(assert (= (+ a d) x1))
(assert (= (+ a e) x2))
(assert (= (+ a f) x3)) 

(assert (= (+ b d) x4))
(assert (= (+ b e) x5))
(assert (= (+ b f) x6))

(assert (= (+ c d) x7))
(assert (= (+ c e) x8))
(assert (= (+ c f) x9))


(check-sat-using (then simplify solve-eqs smt))
(get-model)
(get-value (a))
(get-value (b))
(get-value (c))
(get-value (d))
(get-value (e))
(get-value (f))

回答1:

Nonlinear integer arithmetic is undecidable. This means that there is no decision procedure that can decide arbitrary non-linear integer constraints to be satisfiable. This is what z3 is telling you when it says "unknown" as the answer your query.

This, of course, does not mean that individual cases cannot be answered. Z3 has certain tactics it applies to solve such formulas, but it is inherently limited in what it can handle. Your problem falls into that category: One that Z3 is just not capable of solving.

Z3 has a dedicated NRA (non-linear real arithmetic) tactic that you can utilize. It essentially treats all variables as reals, solves the problem (nonlinear real arithmetic is decidable and z3 can find all algebraic real solutions), and then checks if the results are actually integer. If not, it tries another solution over the reals. Sometimes this tactic can handle non-linear integer problems, if you happen to hit the right solution. You can trigger it using:

(check-sat-using qfnra)

Unfortunately it doesn't solve your particular problem in the time I allowed it to run. (More than 10 minutes.) It's unlikely it'll ever hit the right solution.

You really don't have many options here. SMT solvers are just not a good fit for nonlinear integer problems. In fact, as I alluded to above, there is no tool that can handle arbitrary nonlinear integer problems due to undecidability; but some tools fare better than others depending on the algorithms they use.

When you tell z3 what a and b are, you are essentially taking away much of the non-linearity, and the rest becomes easy to handle. It is possible that you can find a sequence of tactics to apply that solves your original, but such tricks are very brittle in practice and not easily discovered; as you are essentially introducing heuristics into the search and you don't have much control over how that behaves.

Side note: Your script can be improved slightly. To express that a bunch of numbers are all different, use the distinct predicate:

(assert (distinct (a b c)))
(assert (distinct (d e f)))