Question

Let’s assume a very simple constraint: solve(x > 0 && x < 5).

Can Z3 (or any other SMT solver, or any other automatic technique)
compute the minimum and maximum values of (integer) variable x that satisfies the given constraints?

In our case, the minimum is 1 and the maximum is 4.

Answer 1

As Leonardo pointed out, this was discussed in detail before: Determine upper/lower bound for variables in an arbitrary propositional formula. Also see: How to optimize a piece of code in Z3? (PI_NON_NESTED_ARITH_WEIGHT related).

To summarize, one can either use a quantified formula, or go iteratively. Unfortunately, these techniques are not equivalent:

• Quantified approach needs no iteration, and can find global min/max in a single call to the solver; at least in theory. However, it does give rise to harder formulas. So, the backend solver can time-out, or simply return “unknown”.

• Iterative approach creates simple formulas for the backend solver to deal with, but it can loop forever if there’s no optimal value; simplest example being trying to find the largest Int value. Quantified version can solve this problem nicely by quickly telling you that there is no such value, while the iterative version would go on indefinitely. This can be a problem if you don’t know ahead of time that your constraints do have an optimal solution. (Needless to say, the “sufficient” iteration count is typically hard to guess, and might depend on random factors, like the seed used by the solver.)

Also keep in mind that if there is a custom optimization algorithm for the problem domain at hand, it’s unlikely that a general purpose SMT solver can outperform it.