Deciding dependence in logic and algebra

TL;DR

This paper generalizes the concept of dependence in logic and algebra, providing methods to decide dependence in various algebraic structures using constructive proofs and interpolation techniques.

Contribution

It introduces a universal algebraic framework for dependence, linking it to existing notions and establishing decidability results for multiple algebraic varieties.

Findings

Decidability of dependence in lattices

Deciding dependence in MV-algebras and semigroups

Constructive proofs for uniform interpolation

Abstract

We introduce a universal algebraic generalization of de Jongh's notion of dependence for formulas of intuitionistic propositional logic, relating it to a notion of dependence defined by Marczewski for elements of an algebraic structure. Following ideas of de Jongh and Chagrova, we show how constructive proofs of (weak forms of) uniform interpolation can be used to decide dependence for varieties of abelian l-groups, MV-algebras, semigroups, and modal algebras. We also consider minimal provability results for dependence, obtaining in particular a complete description and decidability of dependence for the variety of lattices.