Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T14:12:59.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Expansion trees with cut

Published online by Cambridge University Press:  08 October 2019

Federico Aschieri ,
Stefan Hetzl Open the ORCID record for Stefan Hetzl [Opens in a new window]  and
Daniel Weller
Federico Aschieri
Affiliation:
Institut für Logic and Computation, Technische Universität Wien, Austria
Stefan Hetzl*
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien
Daniel Weller
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien
*
*Corresponding author. Email: stefan.hetzl@tuwien.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

Herbrand’s theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logics by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller’s expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas.

In this paper, we present a new syntactic approach that directly extends Miller’s expansion trees by cuts and also covers non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and shows that it is weakly normalizing.

Keywords

Proof theorycut-eliminationHerbrand’s theoremexpansion trees
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Special Issue 8: A special issue on structural proof theory, automated reasoning and computation in celebration of Dale Miller’s 60th birthday , September 2019 , pp. 1009 - 1029
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Funded by FWF Lise Meitner Grant M 1930–N35 and START project Y544–N23

‡

Funded by the WWTF Vienna Research Group (VRG) 12-004.

References

Aschieri, F. (2017). Game semantics and the geometry of backtracking: a new complexity analysis of interaction. Journal of Symbolic Logic 82 (2) 672–708.CrossRefGoogle Scholar
Aschieri, F. and Zorzi, M. (2016). On natural deduction in classical first-order logic: Curry-Howard correspondence, strong normalization and Herbrand’s theorem. Theoretical Computer Science 625, 125–146.CrossRefGoogle Scholar
Baaz, M. and Hetzl, S. (2011). On the non-confluence of cut-elimination. Journal of Symbolic Logic 76(1), 313–340.CrossRefGoogle Scholar
Baaz, M., Hetzl, S., Leitsch, A., Richter, C. and Spohr, H. (2005). Cut-elimination: experiments with CERES. In: Baader, F. and Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning (LPAR) 2004, Lecture Notes in Computer Science, vol. 3452, Springer, 481–495.CrossRefGoogle Scholar
Baaz, M. and Leitsch, A. (1999). Cut normal forms and proof complexity. Annals of Pure and Applied Logic 97, 127–177.CrossRefGoogle Scholar
Baaz, M. and Leitsch, A. (2000). Cut-elimination and redundancy-elimination by resolution. Journal of Symbolic Computation 29(2), 149–176.CrossRefGoogle Scholar
Buss, S. R. (1995). On Herbrand’s theorem. In: Logic and Computational Complexity, Lecture Notes in Computer Science, vol. 960, Springer, 195–209.Google Scholar
Coquand, T. (1995). A semantics of evidence for classical arithmetic. Journal of Symbolic Logic 60(1), 325–337.CrossRefGoogle Scholar
Curien, P.-L. and Herbelin, H. (2000). The duality of computation. In: Proceedings of the Fifth ACM SIGPLAN International Conference on Functional Programming (ICFP’00), ACM, 233–243.CrossRefGoogle Scholar
Danos, V., Joinet, J.-B. and Schellinx, H. (1997). A new deconstructive logic: linear logic. Journal of Symbolic Logic 62(3), 755–807.CrossRefGoogle Scholar
Ebner, G., Hetzl, S., Reis, G., Riener, M., Wolfsteiner, S. and Zivota, S. (2016). System description: GAPT 2.0. In: Olivetti, N. and Tiwari, A. (eds.) 8th International Joint Conference on Automated Reasoning (IJCAR), Lecture Notes in Computer Science, vol. 9706, Springer, 293–301.CrossRefGoogle Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50(1), 1–101.CrossRefGoogle Scholar
Heijltjes, W. (2010). Classical proof forestry. Annals of Pure and Applied Logic 161(11), 1346–1366.CrossRefGoogle Scholar
Heijltjes, W. (2011). Graphical Representation of Canonical Proof: Two Case Studies. Phd thesis, University of Edinburgh.Google Scholar
Herbrand, J. (1930). Recherches sur la théorie de la démonstration. Phd thesis, Université de Paris.Google Scholar
Hetzl, S. (2012). The computational content of arithmetical proofs. Notre Dame Journal of Formal Logic 53(3), 289–296.CrossRefGoogle Scholar
Hetzl, S. and Straßburger, L. (2012). Herbrand-confluence for cut-elimination in classical first-order logic. In: Cégielski, P. and Durand, A. (eds.) Computer Science Logic (CSL) 2012, Leibniz International Proceedings in Informatics (LIPIcs), vol. 16, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 320–334.Google Scholar
Hilbert, D. and Bernays, P. (1939). Grundlagen der Mathematik II, Springer.Google Scholar
Korovin, K. (2009). Instantiation-based automated reasoning: from theory to practice. In: Schmidt, R. A. (ed.) 22nd International Conference on Automated Deduction (CADE), Lecture Notes in Computer Science, vol. 5663, Springer, 163–166.CrossRefGoogle Scholar
McKinley, R. (2013). Proof nets for Herbrand’s theorem. ACM Transactions on Computational Logic 14(1), 5:1–5:31.CrossRefGoogle Scholar
Miller, D. (1987). A compact representation of proofs. Studia Logica 46(4), 347–370.CrossRefGoogle Scholar
Moser, G. and Zach, R. (2006). The epsilon calculus and Herbrand complexity. Studia Logica 82(1), 133–155.CrossRefGoogle Scholar
Parigot, M. (1992). λμ-Calculus: an algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) Logic Programming and Automated Reasoning, International Conference LPAR’92, Proceedings, Lecture Notes in Computer Science, vol. 624, Springer, 190–201.CrossRefGoogle Scholar
Ratiu, D. and Trifonov, T. (2012). Exploring the computational content of the infinite pigeonhole principle. Journal of Logic and Computation 22(2), 329–350.CrossRefGoogle Scholar
Urban, C. (2000). Classical Logic and Computation. Phd thesis, University of Cambridge.Google Scholar
Urban, C. and Bierman, G. (2000). Strong normalization of cut-elimination in classical logic. Fundamenta Informaticae 45, 123–155.Google Scholar