Towards the Automated Generation of Focused Proof Systems

TL;DR

This paper presents a method to automatically generate complete focused proof systems from un-focused systems by analyzing permutation relations, demonstrated on linear, intuitionistic, and classical logics.

Contribution

It introduces a graph-based approach to construct focused proof systems automatically from existing un-focused systems, generalizing previous permutation-based proofs.

Findings

Successfully generated focused proof systems for multiple logics

Identified formulas where contraction is admissible

Provided an implementation for permutation graph construction

Abstract

This paper tackles the problem of formulating and proving the completeness of focused-like proof systems in an automated fashion. Focusing is a discipline on proofs which structures them into phases in order to reduce proof search non-determinism. We demonstrate that it is possible to construct a complete focused proof system from a given un-focused proof system if it satisfies some conditions. Our key idea is to generalize the completeness proof based on permutation lemmas given by Miller and Saurin for the focused linear logic proof system. This is done by building a graph from the rule permutation relation of a proof system, called permutation graph. We then show that from the permutation graph of a given proof system, it is possible to construct a complete focused proof system, and additionally infer for which formulas contraction is admissible. An implementation for building the permutation graph of a system is provided. We apply our technique to generate the focused proof systems MALLF, LJF and LKF for linear, intuitionistic and classical logics, respectively.