What does "quantifier free logic" mean in SMT context? - z3

Question

Even for simplest arithmetic SMT problems the existential quantifier is required to declare symbolic variables. And ∀ quantifier can be turned into ∃ by inverting the constraint. So, I can use both of them in QF_* logics and it works.
I take it, "quantifier free" means something else for such SMT logics, but what exactly?

Answer 1

The claim is that
∀ quantifier can be turned into ∃ by inverting the constraint
AFAIK, the following two relations hold:
∀x.φ(x) <=> ¬∃x.¬φ(x)
¬∀x.φ(x) <=> ∃x.¬φ(x)
Since a quantifier-free SMT formula φ(x) is equisatisfiable to its existential closure ∃x.φ(x), we can use the quantifier-free fragment of an SMT Theory to express a (simple) negated occurrence of universal quantification, and [AFAIK] also a (simple) positive occurrence of universal quantification over trivial formulas (e.g. if [∃x.]φ(x) is unsat then ∀x.¬φ(x)¹).
¹: assuming φ(x) is quantifier-free; As #Levent Erkok points out in his answer, this approach is inconclusive when both φ(x) and ¬φ(x) are satisfiable
However, we cannot, for example, find a model for the following quantified formula using the quantifier-free fragment of SMT:
[∃y.]((∀x.y <= f(x)) and (∃z.y = f(z)))
For the records, this is an encoding of the OMT problem min(y), y=f(x) as a quantified SMT formula. [related paper]
A term t is quantifier-free iff t syntactically contains no quantifiers. A quantifier-free formula φ is equisatisfiable with its existential closure
(∃x1. (∃x2 . . .(∃xn.φ ). . .))
where x1, x2, . . . , xn is any enumeration of free(φ), the free variables in φ.
The set of free variables of a term t, free(t), is defined inductively as:
free(x) = {x} if x is a variable,
free((f t1 t2 . . . tk)) = \cup_{i∈[1,k]} free(ti) for function applications,
free(∀x.φ) = free(φ) \ {x}, and
free(∃x.φ) = free(φ) \ {x}.
[source]

Answer 2

Patrick gave an excellent answer, but here're a few more thoughts. (I'd have put this up as a comment, but StackOverflow thinks it's too long for that!)
Notice that you cannot always play the "negate and check the opposite" trick. This only works because if the negation of a property is unsatisfiable, then the property must be true for all inputs. But it doesn't go the other way around: A property can be satisfiable, and its negation can be satisfiable as well. Simple example: x < 10. This is obviously satisfiable, and so is its negation x >= 10. So, you cannot always get rid of quantifiers by playing this trick. It only works if you want to prove something: Then you can negate it and see if that negation is unsatisfiable. If you're concerned about finding a model to a formula, the method doesn't apply.
You can always skolemize a formula and eliminate all the existential quantifiers by replacing them with uninterpreted functions. What you then end up with is an equisatisfiable formula that has all prefix universals. Clearly, this is not quantifier free, but this is a very common trick that most tools do for you automatically.
Where all this hurts is alternating quantifiers. Regardless of skolemization, if you have alternating quantifiers than your problem is already too difficult to deal with. The wikipedia page on quantifier elimination is rather terse, but it gives a very good introduction: https://en.wikipedia.org/wiki/Quantifier_elimination Bottom line: Not every theory admits quantifier elimination, and even those that do might require exponential algorithms to get rid of them; causing performance issues.