Abstract
Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentzen’s sequent calculus LK for classical first-order logic. The extracted Herbrand sequent summarizes the creative information of the formal proof, which lies in the instantiations chosen for the quantifiers, and can be used to facilitate its analysis by humans. Furthermore, we also demonstrate the usage of the algorithm in the analysis of a proof of the equivalence of two different definitions for the mathematical concept of lattice, obtained with the proof transformation system CERES.
Supported by the Austrian Science Fund (project P19875) and by the Programme Alban (project E05M054053BR).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: Ceres: An Analysis of Fürstenberg’s Proof of the Infinity of Primes. Theoretical Computer Science (to appear)
Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: Cut-Elimination: Experiments with CERES. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 481–495. Springer, Heidelberg (2005)
Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: Proof Transformation by CERES. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006. LNCS (LNAI), vol. 4108, pp. 82–93. Springer, Heidelberg (2006)
Baaz, M., Leitsch, A.: On skolemization and proof complexity. Fundamenta Matematicae 20, 353–379 (1994)
Baaz, M., Leitsch, A.: Cut normal forms and proof complexity. Annals of Pure and Applied Logic 97, 127–177 (1999)
Baaz, M., Leitsch, A.: Cut-elimination and Redundancy-elimination by Resolution. Journal of Symbolic Computation 29(2), 149–176 (2000)
Buss, S.R.: On Herbrand’s theorem. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, p. 195. Springer, Heidelberg (1995)
Gentzen, G.: Untersuchungen über das logische Schließen. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland Publishing Company, Amsterdam (1969)
Gerhardy, P., Kohlenbach, U.: Extracting herbrand disjunctions by functional interpretation. Archive for Mathematical Logic 44(5) (2005)
Herbrand, J.: Recherches sur la théorie de la démonstration. Travaux de la Societé des Sciences et des Lettres de Varsovie, Class III, Sciences Mathématiques et Physiques 33 (1930)
Hetzl, S.: Characteristic Clause Sets and Proof Transformations. PhD thesis, Vienna University of Technology (2007)
Hetzl, S., Leitsch, A.: Proof transformations and structural invariance. In: Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., Marra, V. (eds.) ManyVal 2006. LNCS (LNAI), vol. 4460, pp. 201–230. Springer, Heidelberg (2007)
Luckhardt, H.: Herbrand-Analysen zweier Beweise des Satzes von Roth: polynomiale Anzahlschranken. Journal of Symbolic Logic 54, 234–263 (1989)
Statman, R.: Lower bounds on Herbrand’s theorem. Proceedings of the American Mathematical Society 75, 104–107 (1979)
Sutcliffe, G., Suttner, C.: The State of CASC. AI Communications 19(1), 35–48 (2006)
Paleo, B.W.: Herbrand sequent extraction. Master’s thesis, Technische Universitaet Dresden; Technische Universitaet Wien, Dresden, Germany, Wien, Austria (2007)
Paleo, B.W.: Herbrand Sequent Extraction. VDM-Verlag. Saarbruecken, Germany (2008)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hetzl, S., Leitsch, A., Weller, D., Woltzenlogel Paleo, B. (2008). Herbrand Sequent Extraction. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds) Intelligent Computer Mathematics. CICM 2008. Lecture Notes in Computer Science(), vol 5144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85110-3_38
Download citation
DOI: https://doi.org/10.1007/978-3-540-85110-3_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85109-7
Online ISBN: 978-3-540-85110-3
eBook Packages: Computer ScienceComputer Science (R0)