A Decidable Case of the Semi-Unification Problem

Semi-unification is a common generalization of unification and matching. The semiunification problem is to decide solvability of finite sets of equations s = t and inequations s̃ ≤i t̃ between first-order terms, with different inequality relations ≤i, i ∈ I. A solution consists of a substitution T0 and residual substitutions Ti, i ∈ I, such that , respectively, T0(s) = T0(t) and Ti(T0(s̃)) = T0(t̃). The semi-unification problem has recently been shown to be undecidable [9]. We present a new subclass of decidable semi-unification problems, properly containing those over monadic languages. In our ‘quasi-monadic’ problems, function symbols may be of arity > 1, but only terms with at most one free variable are admitted.