Set Unification

TL;DR

This paper provides a comprehensive, formalized, and unified framework for set unification problems, introducing new algorithms and classifying various problem types to advance applications in logic and computer science.

Contribution

It offers a uniform formalization of set unification, classifies problem types, and introduces new algorithms for ACI1 and (Ab)(Cl) unification.

Findings

New goal-driven algorithm for ACI1 unification

Simpler algorithm for (Ab)(Cl) unification

Unified classification of set unification problems

Abstract

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.