The Unification Type of an Equational Theory May Depend on the Instantiation Preorder: From Results for Single Theories to Results for Classes of Theories

TL;DR

This paper investigates how the unification type of equational theories depends on the instantiation preorder, providing new results for classes of theories and showing that certain properties prevent the unification type from being zero.

Contribution

It extends previous results by determining the unrestricted unification type for more theories and establishing general conditions that prevent unification type zero.

Findings

ACUI is infinitary w.r.t. the unrestricted instantiation preorder.

Unification type can improve from zero to infinitary when switching preorders.

Certain classes of theories are Noetherian and cannot have unification type zero.

Abstract

The unification type of an equational theory is defined using a preorder on substitutions, called the instantiation preorder, whose scope is either restricted to the variables occurring in the unification problem, or unrestricted such that all variables are considered. It has been known for more than three decades that the unification type of an equational theory may vary, depending on which instantiation preorder is used. More precisely, it was shown in 1991 that the theory ACUI of an associative, commutative, and idempotent binary function symbol with a unit is unitary w.r.t. the restricted instantiation preorder, but not unitary w.r.t. the unrestricted one. In 2016 this result was strengthened by showing that the unrestricted type of this theory also cannot be finitary. In the conference version of this article, we considerably improved on this result by proving that ACUI is infinitary w.r.t. the unrestricted instantiation preorder, thus precluding type zero. We also showed that, w.r.t. this preorder, the unification type of ACU (where idempotency is removed from the axioms) and of AC (where additionally the unit is removed) is infinitary, though it is respectively unitary and finitary in the restricted case. In the other direction, we proved (using the example of unification in the description logic EL) that the unification type may actually improve from type zero to infinitary when switching from the restricted instantiation preorder to the unrestricted one. In the present article, we not only determine the unrestricted unification type of considerably more equational theories, but we also prove general results for whole classes of theories. In particular, we show that theories that are regular and finite, regular and locally finite, or regular, monoidal, and satisfy an additional condition are Noetherian, and thus cannot have unrestricted unification type zero.