Bounded ACh Unification

Abstract

We consider the problem of the unification modulo an equational theory ACh, which consists of a function symbol $h$ that is homomorphic over an associative-commutative operator $+$. Since the unification modulo ACh theory is undecidable, we define a variant of the problem called \textit{bounded ACh unification}. In this bounded version of ACh unification, we essentially bound the number of times $h$ can be applied to a term recursively, and only allow solutions that satisfy this bound. There is no bound on the number of occurrences of $h$ in a term, and the $+$ symbol can be applied an unlimited number of times. We give inference rules for solving the bounded version of the problem and prove that the rules are sound, complete, and terminating. We have implemented the algorithm in Maude and give experimental results. We argue that this algorithm is useful in cryptographic protocol analysis.