Unification Modulo ACU I Plus Homomorphisms/Distributivity

E-unification problems are central in automated deduction. In this paper, we consider theories that are extensions of the well-known ACI or ACUI , obtained by adding finitely many homomorphism symbols, or a symbol ‘∗’ that distributes over the ACUIsymbol denoted ‘+’. We first show that when we adjoin a set of commuting homomorphisms to ACUI , unification is undecidable. We then consider the ACUIDl-unification problem, i.e., unification modulo ACUI plus left-distributivity of a given ‘∗’ w.r.t. ‘+’, and prove its NEXPTIME-decidability. When we assume the symbol ‘∗’ to be 2-sided distributive w.r.t. ‘+’, we get the theory ACUID, for which the unification problem remains decidable. But if equations of associativity-commutativity, or just of associativity, on ‘∗’ are added on to ACUID, then the unification problem becomes undecidable.