On Asymmetric Unification and the Combination Problem in Disjoint Theories

Abstract

Abstract. Asymmetric unification is a new paradigm for unification modulo theories that introduces irreducibility constraints on one side of a unification problem. It has important applications in symbolic cryptographic protocol analysis, for which it is often necessary to put irreducibility constraints on portions of a state. However many facets of asymmetric unification that are of particular interest, including its behavior under combinations of disjoint theories, remain poorly understood. In this paper we give a new formulation of the method for unification in the combination of disjoint equational theories developed by Baader and Schulz that both gives additional insights into the disjoint combination problem in general, and furthermore allows us to extend the method to asymmetric unification, giving the first unification method for asymmetric unification in the combination of disjoint theories.

On Asymmetric Unification for the Theory of XOR with a Homomorphism

“…Proof. The result follows directly from the proof contained in [7]. There it is shown that Algorithm 3 is both sound and complete.…”

“…The problem of asymmetric unification in the combination of disjoint theories was studied in [7] where an algorithm is developed for the problem. However, the algorithm of [7] does not immediately apply to the two methods developed in this paper. This is due to the nature of the two automata based approaches.…”

“…Let ∆ be the combination, ∆ = ∆ 1 ∪ ∆ 2 , of the two theories having signature Σ 1 ∪ Σ 2 . The algorithm of [7] solves the asymmetric ∆ -unification problem. It assumes that there exists a finitary complete asymmetric ∆ i -unification algorithm with linear constant restrictions, A i .…”

“…For the automaton it is not always possible to check solutions, however, it is possible to build constraints into the automaton that enforce these conditions. Algorithm 3 is a modification of the algorithm from [7] with the following properties:…”

“…There it is shown that Algorithm 3 is both sound and complete. The only modification is that in [7] the combination algorithm checks the ∆ 1 solutions for the properties of being injective and theory preserving, while in Algorithm 3 it is assumed that the algorithm A 1 itself will enforce these restrictions.…”