Exact Unification

TL;DR

This paper introduces a new hierarchy of 'exact' unification types based on set inclusion, leading to smaller unifier sets and simplifying the unification process in certain algebraic classes.

Contribution

It defines a novel exact unification preordering, demonstrating its advantages and algebraic interpretation, especially for classes with nullary unification type.

Findings

Exact unification often yields smaller unifier sets than standard methods.

Classes like distributive lattices and MV-algebras have unitary or finitary exact types.

Algebraic interpretation of exact unification enhances understanding of unification in various classes.

Abstract

A new hierarchy of "exact" unification types is introduced, motivated by the study of admissibility for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always have a smaller or equal cardinality than those provided by the standard instantiation preordering, and in significant cases a dramatic reduction may be observed. In particular, the classes of distributive lattices, idempotent semigroups, and MV-algebras, which all have nullary unification type, have unitary or finitary exact type. These results are obtained via an algebraic interpretation of exact unification, inspired by Ghilardi's algebraic approach to equational unification.