First-order quantified separators

Quantified first-order formulas, often with quantifier alternations, are increasingly used in the verification of complex systems. While automated theorem provers for first-order logic are becoming more robust, invariant inference tools that handle quantifiers are currently restricted to purely universal formulas. We define and analyze first-order quantified separators and their application to inferring quantified invariants with alternations. A separator for a given set of positively and negatively labeled structures is a formula that is true on positive structures and false on negative structures. We investigate the problem of finding a separator from the class of formulas in prenex normal form with a bounded number of quantifiers and show this problem is NP-complete by reduction to and from SAT. We also give a practical separation algorithm, which we use to demonstrate the first invariant inference procedure able to infer invariants with quantifier alternations.

• Theory of computation → Complexity theory and logic; • Software and its engineering → Formal methods.