Bounded Second-Order Unification Is NP-Complete

Bounded Second-Order Unification is the problem of deciding, for a given second-order equation t ? = u and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F , the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable. We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time.