The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite set G of ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.
