Cirquent calculus deepened

TL;DR

This paper introduces cirquent calculus, a circuit-based proof framework that enhances efficiency and expressiveness, and demonstrates its potential to unify classical logic with resource-conscious reasoning.

Contribution

It develops a deep-inference cirquent logic that is resource-aware and shows classical logic as a conservative fragment within this broader framework.

Findings

Cirquent calculus offers exponential improvements in proof complexity.

Classical logic is a conservative fragment of cirquent calculus.

Cirquent calculus surpasses linear logic in expressive power.

Abstract

Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, as opposed to the more traditional approaches that deal with tree-like objects such as formulas or sequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, is just a special, conservative fragment of this more general and, in a sense, more basic logic -- the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof complexities offered by cirquent calculus, and more. Among the main purposes of this paper is to provide an introductory-style starting point for what, as the author wishes to hope, might have a chance to become a new line of research in proof theory -- a proof theory based on circuits instead of formulas.