On linear rewriting systems for Boolean logic and some applications to proof theory

TL;DR

This paper investigates linear rewriting systems for Boolean logic, showing that derivations are polynomially bounded and highlighting complexity limitations in designing sound, complete systems.

Contribution

It establishes polynomial bounds on derivation lengths and demonstrates the non-existence of polynomial-time sound, complete linear inference systems unless P=NP, connecting multiple theoretical areas.

Findings

Derivations in linear systems are polynomially bounded.

No polynomial-time sound and complete linear inference system exists unless P=NP.

Introduces a new theoretical framework linking rewriting, Boolean functions, and graph theory.

Abstract

Linear rules have played an increasing role in structural proof theory in recent years. It has been observed that the set of all sound linear inference rules in Boolean logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left- and right-)linear rewrite rule. In this paper we study properties of systems consisting only of linear inferences. Our main result is that the length of any 'nontrivial' derivation in such a system is bound by a polynomial. As a consequence there is no polynomial-time decidable sound and complete system of linear inferences, unless coNP=NP. We draw tools and concepts from term rewriting, Boolean function theory and graph theory in order to access some required intermediate results. At the same time we make several connections between these areas that, to our knowledge, have not yet been presented and constitute a rich theoretical framework for reasoning about linear TRSs for Boolean logic.