Abstract
Cut-elimination is the most prominent form of proof transformation 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 resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skeleton of a proof with only atomic cuts.
In this paper we present a systematic experiment with the implementation of CERES on a proof of reasonable size and complexity. It turns out that the proof with cuts can be transformed into two mathematically different proofs of the theorem. In particular, the application of positive and negative hyperresolution yield different mathematical arguments. As an unexpected side-effect the derived clauses of the resolution refutation proved particularly interesting as they can be considered as meaningful universal lemmas.
Though the proof under investigation is intuitively simple, the experiment demonstrates that new (and relevant) mathematical information on proofs can be obtained by computational methods. It can be considered as a first step in the development of an experimental culture of computer-aided proof analysis in mathematics.
supported by the Austrian Science Fund (project no. P16264-N05)
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baaz, M., Leitsch, A.: On skolemization and proof complexity. Fundamenta Informaticae 20(4), 353–379 (1994)
Baaz, M., Leitsch, A.: Cut-Elimination and Redundancy-Elimination by Resolution. Journal of Symbolic Computation 29, 149–176 (2000)
Baaz, M., Leitsch, A.: Towards a Clausal Analysis of Cut-Elimination. Journal of Symbolic Computation to appear
Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 405–431 (1934)
Girard, J.Y.: Proof Theory and Logical Complexity. In: Studies in Proof Theory, Bibliopolis, Napoli (1987)
Luckhardt, H.: Herbrand-Analysen zweier Beweise des Satzes von Roth: polynomiale Anzahlschranken. The Journal of Symbolic Logic 54, 234–263 (1989)
Polya, G.: Mathematics and plausible reasoning. Induction and Analogy in Mathematics, vol. I. Princeton University Press, Princeton (1954)
Polya, G.: Mathematics and plausible reasoning. Patterns of Plausible Inference, vol. II. Princeton University Press, Princeton (1954)
Urban, C.: Classical Logic and Computation. Ph.D. Thesis, University of Cambridge Computer Laboratory (2000)
Degtyarev, A., Voronkov, A.: Equality Reasoning in Sequent-Based Calculi. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, ch. 10, pp. 611–706. Elsevier Science, Amsterdam (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H. (2005). Cut-Elimination: Experiments with CERES. In: Baader, F., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2005. Lecture Notes in Computer Science(), vol 3452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32275-7_32
Download citation
DOI: https://doi.org/10.1007/978-3-540-32275-7_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25236-8
Online ISBN: 978-3-540-32275-7
eBook Packages: Computer ScienceComputer Science (R0)