On Unification Modulo One-Sided Distributivity: Algorithms, Variants and Asymmetry

TL;DR

This paper introduces new polynomial-time algorithms for unification modulo one-sided distributivity, including a subcase and the general case, and also presents the first asymmetric unification algorithm for this theory, addressing previous complexity issues.

Contribution

It provides the first polynomial-time algorithms for unification modulo one-sided distributivity and extends the theory to asymmetric unification.

Findings

A polynomial-time decision algorithm for a subcase of unification.

A polynomial-time decision algorithm for the general unification problem.

The first asymmetric unification algorithm for one-sided distributivity.

Abstract

An algorithm for unification modulo one-sided distributivity is an early result by Tid\'en and Arnborg. More recently this theory has been of interest in cryptographic protocol analysis due to the fact that many cryptographic operators satisfy this property. Unfortunately the algorithm presented in the paper, although correct, has recently been shown not to be polynomial time bounded as claimed. In addition, for some instances, there exist most general unifiers that are exponentially large with respect to the input size. In this paper we first present a new polynomial time algorithm that solves the decision problem for a non-trivial subcase, based on a typed theory, of unification modulo one-sided distributivity. Next we present a new polynomial algorithm that solves the decision problem for unification modulo one-sided distributivity. A construction, employing string compression, is used to achieve the polynomial bound. Lastly, we examine the one-sided distributivity problem in the new asymmetric unification paradigm. We give the first asymmetric unification algorithm for one-sided distributivity.