Foundations for an Abstract Proof Theory in the Context of Horn Rules

TL;DR

This paper develops a logic-independent framework for sequent-style proof systems using generalized sequents and inference rule types, enabling analysis, transformation, and complexity assessment of various proof calculi.

Contribution

It introduces g-sequents and a formal framework for analyzing and transforming sequent calculi, unifying diverse proof systems and enabling generic proof transformations.

Findings

Established conditions for rule permutation and simulation.

Developed algorithms for transforming calculi into polynomially equivalent forms.

Analyzed complexity and size relationships of proofs across different calculi.

Abstract

We introduce a novel, logic-independent framework for the study of sequent-style proof systems, which covers a number of proof-theoretic formalisms and concrete proof systems that appear in the literature. In particular, we introduce a generalized form of sequents, dubbed 'g-sequents,' which are taken to be binary graphs of typical, Gentzen-style sequents. We then define a variety of 'inference rule types' as sets of operations that act over such objects, and define 'abstract (sequent) calculi' as pairs consisting of a set of g-sequents together with a finite set of operations. Our approach permits an analysis of how certain inference rule types interact in a general setting, demonstrating under what conditions rules of a specific type can be permuted with or simulated by others, and being applicable to any sequent-style proof system that fits within our framework. We then leverage our permutation and simulation results to establish generic calculus and proof transformation algorithms, which show that every abstract calculus can be effectively transformed into a lattice of polynomially equivalent abstract calculi. We determine the complexity of computing this lattice and compute the relative sizes of proofs and sequents within distinct calculi of a lattice. We recognize that top and bottom elements in lattices correspond to many known deep-inference nested sequent systems and labeled sequent systems (respectively) for logics characterized by Horn properties.