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.


Simple Proof Production Rule Proof Theory Tree Automaton Tree Language 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baaz, M., Leitsch, A.: Cut Normal Forms and Proof Complexity. Annals of Pure and Applied Logic 97, 127–177 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buss, S.R.: On Herbrand’s Theorem. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 195–209. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  3. 3.
    Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata: Techniques and Applications (2007), (release October 12, 2007)
  4. 4.
    Cook, S., Nguyen, P.: Logical Foundations of Proof Complexity. Perspectives in Logic. Cambridge University Press (2010)Google Scholar
  5. 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)CrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 176–210, 405–431 (1934-1935)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Herbrand, J.: Recherches sur la théorie de la démonstration. Ph.D. thesis, Université de Paris (1930)Google Scholar
  10. 10.
    Hetzl, S.: Proofs as Tree Languages (preprint),
  11. 11.
    Hetzl, S.: On the form of witness terms. Archive for Mathematical Logic 49(5), 529–554 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hetzl, S., Leitsch, A., Weller, D.: Towards Algorithmic Cut-Introduction (submitted)Google Scholar
  13. 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)CrossRefGoogle Scholar
  14. 14.
    Jacquemard, F., Klay, F., Vacher, C.: Rigid tree automata and applications. Information and Computation 209, 486–512 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  16. 16.
    Miller, D.: A Compact Representation of Proofs. Studia Logica 46(4), 347–370 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)zbMATHGoogle Scholar
  18. 18.
    Pudlák, P.: The Lengths of Proofs. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 547–637. Elsevier (1998)Google Scholar
  19. 19.
    Schwichtenberg, H., Troelstra, A.S.: Basic Proof Theory. Cambridge University Press (1996)Google Scholar
  20. 20.
    Shallit, J., Breitbart, Y.: Automaticity I: Properties of a Measure of Descriptional Complexity. Journal of Computer and System Sciences 53, 10–25 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shallit, J., Wang, M.W.: Automatic complexity of strings. Journal of Automata, Languages and Combinatorics 6(4), 537–554 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Shoenfield, J.R.: Mathematical Logic, 2nd edn. AK Peters (2001)Google Scholar
  23. 23.
    Statman, R.: Lower bounds on Herbrand’s theorem. Proceedings of the American Mathematical Society 75, 104–107 (1979)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Hetzl
    • 1
  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria

Personalised recommendations