Admissibility in Finitely Generated Quasivarieties

TL;DR

This paper presents algorithms for efficiently checking admissibility of quasiequations in finitely generated quasivarieties by generating minimal finite algebra sets, improving computational feasibility over naive methods.

Contribution

The paper introduces algorithms that generate minimal algebra sets for admissibility checks in finitely generated quasivarieties, enabling practical decision procedures.

Findings

Algorithms successfully identify minimal algebra sets for various quasivarieties.

Structural and almost structural completeness can be effectively verified.

Method adapts to admissibility of rules in finite-valued logics.

Abstract

Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in $\Q$ is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.