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 $\boldsymbol G\boldsymbol\lbrack{\boldsymbol x}_\mathbf1\boldsymbol,\boldsymbol\dots\boldsymbol,{\boldsymbol x}_\mathbf n\boldsymbol\rbrack$ such that the formula $\boldsymbol F\boldsymbol\lbrack\boldsymbol G\boldsymbol\rbrack$ is valid in first-order logic with equality? We obtain the following results. Boolean unification with predicates for quantifier-free $F$ is $\Pi^P_2$-complete. In addition, there exists an EXPTIME algorithm which for an input formula $\boldsymbol F\boldsymbol\lbrack\boldsymbol X\boldsymbol\rbrack$, given as above, constructs a formula $\boldsymbol G$ such that $\boldsymbol F\boldsymbol\lbrack\boldsymbol G\boldsymbol\rbrack$ being valid in first-order logic with equality, if such a formula exists. For $F$ of the form $\exists\bar yF'[X,\bar y]$ with $f'$ quantifier-free, we prove that Boolean unification with predicates is already undecidable. The same holds for $F$ of the form $\forall\bar yF'[X,\bar y]$ 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.