CERES in Propositional Proof Schemata

TL;DR

This paper investigates the extension of the Ceres cut-elimination algorithm to propositional proof schemata, aiming to establish a complete method for propositional logic by refining resolution refutation schemata.

Contribution

It introduces a modified approach to schematic Ceres for propositional schemata, addressing incompleteness issues and proposing a fixed refutation method for clause sets.

Findings

Naive adaptation of Ceres is incomplete for propositional schemata.

A new method using a generic clause set form improves refutation process.

This work is a step towards a complete cut-elimination algorithm for propositional schemata.

Abstract

Cut-elimination is one of the most famous problems in proof theory, and it was defined and solved for first-order sequent calculus by Gentzen in his celebrated Hauptsatz. Ceres is a different cut-elimination algorithm for first- and higher-order classical logic. Ceres was extended to proof schemata, which are templates for usual first-order proofs, with parameters for natural numbers. However, while Ceres is known to be a complete cut-elimination algorithm for first-order logic, it is not clear whether this holds for first-order schemata too: given in input a proof schema with cuts, does Ceres always produce a schema for a cut-free proof? The difficult step is finding and representing an appropriate refutation schema for the characteristic term schema of a proof schema. In this thesis, we progress in solving this problem by restricting Ceres to propositional schemata, which are templates for propositional proofs. By limiting adequately the expressivity of propositional schemata and proof schemata, we aim at providing a version of schematic Ceres which is a complete cut-elimination algorithm for propositional schemata. We focus on one particular step of Ceres: resolution refutation schemata. First, we prove that by naively adapting Ceres for first-order schemata to our case, we end up with an incomplete algorithm. Then, we modify slightly the concept of resolution refutation schema: to refute a clause set, first we bring it to a generic form, and then we use a fixed refutation of that generic clause set. Our variation of schematic Ceres is the first step towards completeness with respect to propositional schemata.

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See pages 1-last of thesis_final.pdf