Skip to main content
SpringerLink
Log in
Book cover

International Conference on Language and Automata Theory and Applications

LATA 2012: Language and Automata Theory and Applications pp 301–312Cite as

Applying Tree Languages in Proof Theory

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7183))

Abstract

We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both directions, from proofs to grammars and from grammars to proofs, are provided. This correspondence allows theoretical as well as practical applications.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baaz, M., Leitsch, A.: Cut Normal Forms and Proof Complexity. Annals of Pure and Applied Logic 97, 127–177 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buss, S.R.: On Herbrand’s Theorem. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 195–209. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  3. Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata: Techniques and Applications (2007), http://www.grappa.univ-lille3.fr/tata (release October 12, 2007)

  4. Cook, S., Nguyen, P.: Logical Foundations of Proof Complexity. Perspectives in Logic. Cambridge University Press (2010)

    Google Scholar 

  5. Filiot, E., Talbot, J.-M., Tison, S.: Satisfiability of a Spatial Logic with Tree Variables. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 130–145. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Filiot, E., Talbot, J.-M., Tison, S.: Tree Automata with Global Constraints. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 314–326. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Filiot, E., Talbot, J.M., Tison, S.: Tree Automata With Global Constraints. International Journal of Foundations of Computer Science 21(4), 571–596 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 176–210, 405–431 (1934-1935)

    Article  MathSciNet  MATH  Google Scholar 

  9. Herbrand, J.: Recherches sur la théorie de la démonstration. Ph.D. thesis, Université de Paris (1930)

    Google Scholar 

  10. Hetzl, S.: Proofs as Tree Languages (preprint), http://hal.archives-ouvertes.fr/hal-00613713/

  11. Hetzl, S.: On the form of witness terms. Archive for Mathematical Logic 49(5), 529–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hetzl, S., Leitsch, A., Weller, D.: Towards Algorithmic Cut-Introduction (submitted)

    Google Scholar 

  13. Jacquemard, F., Klay, F., Vacher, C.: Rigid Tree Automata. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 446–457. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  14. Jacquemard, F., Klay, F., Vacher, C.: Rigid tree automata and applications. Information and Computation 209, 486–512 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  16. Miller, D.: A Compact Representation of Proofs. Studia Logica 46(4), 347–370 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Orevkov, V.P.: Lower bounds for increasing complexity of derivations after cut elimination. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta 88, 137–161 (1979)

    MATH  Google Scholar 

  18. Pudlák, P.: The Lengths of Proofs. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 547–637. Elsevier (1998)

    Google Scholar 

  19. Schwichtenberg, H., Troelstra, A.S.: Basic Proof Theory. Cambridge University Press (1996)

    Google Scholar 

  20. Shallit, J., Breitbart, Y.: Automaticity I: Properties of a Measure of Descriptional Complexity. Journal of Computer and System Sciences 53, 10–25 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shallit, J., Wang, M.W.: Automatic complexity of strings. Journal of Automata, Languages and Combinatorics 6(4), 537–554 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Shoenfield, J.R.: Mathematical Logic, 2nd edn. AK Peters (2001)

    Google Scholar 

  23. Statman, R.: Lower bounds on Herbrand’s theorem. Proceedings of the American Mathematical Society 75, 104–107 (1979)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040, Vienna, Austria

    Stefan Hetzl

Authors
  1. Stefan Hetzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Avinguda Catalunya, 35, 43002, Tarragona, Spain

    Adrian-Horia Dediu  & Carlos Martín-Vide  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hetzl, S. (2012). Applying Tree Languages in Proof Theory. In: Dediu, AH., Martín-Vide, C. (eds) Language and Automata Theory and Applications. LATA 2012. Lecture Notes in Computer Science, vol 7183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28332-1_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-28332-1_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28331-4

  • Online ISBN: 978-3-642-28332-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation