CERES in Propositional Proof Schemata

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 firstand higher-order classical logic. It is based on the notion of a characteristic set of clauses which is extracted from a proof in sequent calculus and is always unsatisfiable. A resolution refutation of this clause set is used as a skeleton for a proof with only atomic cuts. This is obtained by replacing clauses from the resolution refutation with the corresponding proof projection derived from the original proof. Ceres was extended to proof schemata, which are templates for usual first-order proofs, with parameters for natural numbers. Every instantiation of the parameters to concrete numbers yields a new first-order proof. We could apply existing algorithms for cutelimination to each proof in this infinite sequence, obtaining an infinite sequence of cut-free proofs. The goal of Ceres for first-order schemata is instead to give a uniform, schematic description of this sequence of cut-free proofs. To this aim, every concept in Ceres was made schematic: there are characteristic clause set schemata, resolution refutation schemata, projection schemata, etc. 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 cut-free proofs? 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.