Boolean unification with predicates

In this article, we deal with the following problem which we call Boolean unification with predicates: For a given formula F[X ] in first-order logic with equality containing an n-ary predicate variable X , is there a quantifier-free formula G[x1,...,xn] such that the formula F[G] is valid in first-order logic with equality? We obtain the following results. Boolean unification with predicates for quantifier-free F is 2 -complete. In addition, there exists an EXPTIME algorithm which for an input formula F[X ], given as above, constructs a formula G such that F[G] being valid in first-order logic with equality, if such a formula exists. For F of the form ∀yF ′[X ,y] with F ′ quantifier-free, we prove that Boolean unification with predicates is already undecidable. The same holds for F of the form ∃yF ′[X ,y] for F ′ quantifier-free. Instances of Boolean unification with predicates naturally occur in the context of automated theorem proving. Our results are relevant for cut-introduction and the automated search for induction invariants.