System Description: The Proof Transformation System CERES

The original publication is available at www.springerlink.comInternational audienceCut-elimination is the most prominent form of proof trans- formation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolu- tion) works by extracting a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an ACNF, an LK-proof with only atomic cuts. The system CERES, an implementation of the CERES-method has been used successfully in analyzing nontrivial mathematical proofs (see [4]).In this paper we describe the main features of the CERES system with spe- cial emphasis on the extraction of Herbrand sequents and simplification methods on these sequents. We demonstrate the Herbrand sequent ex- traction and simplification by a mathematical example

Physics and Proof Theory

Axiomatization of Physics (and Science in general) has many drawbacks that are correctly criticized by opposing philosophical views of Science. This paper shows that, by giving formal proofs a more promi- nent role in the formalization, many of the drawbacks can be solved and many of the opposing views are naturally conciliated. Moreover, this ap- proach allows, by means of Proof Theory, to open new conceptual bridges between the disciplines of Physics and Computer Science

Integrating a Global Induction Mechanism into a Sequent Calculus

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.Comment: 16 page

Atomic Cut Introduction by Resolution: Proof Structuring and Compression

The original publication is available at www.springerlink.comInternational audienceThe careful introduction of cut inferences can be used to structure and possibly compress formal sequent calculus proofs. This pa- per presents CIRes, an algorithm for the introduction of atomic cuts based on various modifications and improvements of the CERes method, which was originally devised for efficient cut-elimination. It is also demonstrated that CIRes is capable of compressing proofs, and the amount of compres- sion is shown to be exponential in the length of proofs

Dynamic Logic for an Intermediate Language: Verification, Interaction and Refinement

This thesis is about ensuring that software behaves as it is supposed to behave. More precisely, it is concerned with the deductive verification of the compliance of software implementations with their formal specification. Two successful ideas in program verification are integrated into a new approach: dynamic logic and intermediate verification language. The well-established technique of refinement is used to decompose the difficult task of program verification into two easier tasks

Herbrand sequent extraction

Zsfassung in dt. SpracheNach der Definition einer Verallgemeinerung des Herbrandschen Theorems für Sequente, beschreiben wir drei schon existierende Algorithmen für die Extraktion eines Herbrandsequents aus formalen Beweisen, die im Sequentialkalkül LK geschrieben sind. Darüber hinaus verbessern wir diese drei Algorithmen und entwicklen daraus einen vierten Algorithmus, welcher deren Ideen vereinigt und generalisiert.Dieser Algorithmus wurde dann realisiert und in das CERes (Cut-Elimination by Resolution) Projekt integriert. Die Wichtigkeit der Extraktion von Herbrandsequenten liegt darin, dass Herbrandsequente die Kreativität der Beweise enthält.After defining a generalization of Herbrand's Theorem for sequents, we describe three pre-existing algorithms for the extraction of a Herbrand sequent of the end-sequent of proofs in the Sequent Calculus LK. Furthermore, we improve these three algorithms by designing a new fourth algorithm, which combines and generalizes their essential ideas. The implementation of this new algorithm was realized within the framework of the project CERes (Cut-Elimination by Resolution). The importance of extracting Herbrand sequents from proofs lies on the fact that a Herbrand sequent summarizes the creative content of a proof.7