Enumerating Independent Linear Inferences

TL;DR

This paper introduces an efficient method for enumerating independent linear inferences in Boolean algebra, discovering minimal inferences up to 9 variables, and providing counterexamples to existing conjectures.

Contribution

It develops a graphical approach that surpasses previous methods in finding minimal independent linear inferences, including those that challenge prior conjectures.

Findings

Found four minimal 8-variable independent inferences, proving no smaller ones exist.

Identified ten minimal 9-variable inferences independent of previous ones.

Discovered inferences that contradict a conjecture of Das and Strassburger.

Abstract

A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find four `minimal' 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. Two of these new inferences derive some previously found independent linear inferences. The other two (which are dual) exhibit structure seemingly beyond the scope of previous approaches we are aware of; in particular, their existence contradicts a conjecture of Das and Strassburger. We were also able to identify 10 minimal 9-variable linear inferences independent of all the aforementioned inferences, comprising 5 dual pairs, and present applications of our implementation to recent `graph logics'.