Characteristic Clause Sets and Proof Transformations

Proof Theory is the branch of mathematical logic that investigates mathematical reasoning and mathematical proofs. This area emanated from Hilbert’s Program calling for consistency proofs of formal theories. In the 1950s the focus of proof theory began shifting towards applications of formal methods to concrete proofs in order to obtain new mathematical results. The method CERES (cut-elimination by resolution) uses techniques from automated theorem proving for the automation of cut-elimination. The main proof-theoretical tool of this method is the extraction of a characteristic clause set from a proof, a resolution refutation of which serves as the skeleton of a cut-free proof. This thesis is an investigation of the potential of these kind of clause sets for characterizing the mathematical content and structure of a formal proof. We first define a variant of these clause sets, the profile that has several advantages w.r.t. the original characteristic clause sets: It is computationally superior in the sense that it will never generate longer proofs with CERES, but is better in detecting certain redundancies thus allowing even a non-elementary speed-up. Furthermore, it has the nice theoretical property of being invariant under rule permutations which shows that two proofs having the same proof net will also have the same profile. We will isolate a large class of proof transformations and show that they leave the profile invariant. As a basis for this result we will give a detailed analysis of the behavior of the profile under cut-elimination whose result will be particularly natural. We will show that the profile is intimately related to Herbrand-disjunctions. It turns out that the profile has two dual parts corresponding to pruned versions of the two partial Herbrand-disjunctions that can be extracted from a proof with cuts: One being the instances of the end-sequent and one the instances of the cut-formulas. Finally we will perform a case study where two different proofs of a simple mathematical theorem are analyzed by characteristic clause sets in order to demonstrate its potential for applications.