The Hydra battle and Cichon’s principle

  • Georg MoserEmail author


In rewriting the Hydra battle refers to a term rewrite system \({\mathcal{H}}\) proposed by Dershowitz and Jouannaud. To date, \({\mathcal{H}}\) withstands any attempt to prove its termination automatically. This motivates our interest in term rewrite systems encoding the Hydra battle, as a careful study of such systems may prove useful in the design of automatic termination tools. Moreover it has been an open problem, whether any termination order compatible with \({\mathcal{H}}\) has to have the Howard–Bachmann ordinal as its order type, i.e., the proof theoretic ordinal of the theory of one inductive definition. We answer this question in the negative, by providing a reduction order compatible with \({\mathcal{H}}\) , whose order type is at most \({\epsilon_0}\) , the proof theoretic ordinal of Peano arithmetic.


Function Symbol Reduction Order Order Type Notation System Winning Strategy 
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.
    Arts T., Giesl J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236, 133–178 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baader F., Nipkow T.: Term Rewriting and All That. Cambridge University Press, London (1998)Google Scholar
  3. 3.
    Beklemishev, L.: Representing worms as a term rewriting system. In: Mini-Workshop: Logic, Combinatorics and Independence Results, pp. 3093–3095. Mathematisches Forschungsinstitut Oberwolfach, Report No. 52/2006, 2006, 2006 (abstract)Google Scholar
  4. 4.
    Buchholz W.: Proof-theoretical analysis of termination proofs. Ann. Pure Appl. Log. 75, 57–65 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Buchholz, W.: Another rewrite system for the standard Hydra battle. In: Mini-Workshop: Logic, Combinatorics and Independence Results, pp. 3011–3099. Mathematisches Forschungsinstitut Oberwolfach, Report No. 52/2006, 2006, 2006 (abstract)Google Scholar
  6. 6.
    Buchholz W., Cichon E.-A., Weiermann A.: A uniform approach to fundamental sequences and hierarchies. Math. Log. Q. 40, 273–286 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Buss, S.-R. (eds): Handbook of Proof Theory, vol. 137. Elsevier Science, London (1998)Google Scholar
  8. 8.
    Cichon, E.-A.: Termination orderings and complexity characterisations. In: Aczel, P., Simmons, H., Wainer, S.S. (eds.) Proof Theory, pp. 171–193 (1992)Google Scholar
  9. 9.
    Dershowitz, N.: 33 examples of termination. In: French Spring School of Theoretical Computer Science Advanced Course on Term Rewriting, Font Romeux, France, May 1993, LNCS, vol. 909, pp. 16–26 (1995)Google Scholar
  10. 10.
    Dershowitz N., Jouannaud J.-P.: Rewrite systems. In: Leeuwen, J. (eds) Handbook of Theoretical Computer Science, pp. 245–319. Elsevier Science, London (1990)Google Scholar
  11. 11.
    Dershowitz N., Manna Z.: Proving termination with multiset orderings. Commun. ACM 22(8), 465–476 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dershowitz, N., Moser, G.: The Hydra battle revisited. In: Rewriting, Computation and Proof. LNCS, vol. 4600, pp. 1–27. Springer, Heidelberg (2007) (Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60th Birthday)Google Scholar
  13. 13.
    Ferreira, M.C.F.: Termination of Term Rewriting. Well-Foundedness, Totality and Transformations. Ph.D. Thesis, University of Utrecht, November (1995)Google Scholar
  14. 14.
    Hofbauer D.: Termination proofs by multiset path orderings imply primitive recursive derivation lengths. Theor. Comput. Sci. 105, 129–140 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jech T.: Set Theory. Springer, Heidelberg (2002)Google Scholar
  16. 16.
    Kirby L., Paris J.: Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 4, 285–293 (1982)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Lepper I.: Derivation lengths and order types of Knuth-Bendix orders. Theor. Comput. Sci. 269, 433–450 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lepper, I.: Simplification Orders in Term Rewriting. PhD thesis, WWU Münster, 2002.
  19. 19.
    Lepper I.: Simply terminating rewrite systems with long derivations. Arch. Math. Log. 43, 1–18 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Manna, Z., Ness, S.: On the termination of Markov algorithms. In: Proceedings of the Third Hawaii International Conference on System Science, pp. 789–792 (1970)Google Scholar
  21. 21.
    Moser, G.: Derivational complexity of Knuth Bendix orders revisited. In: Proceedings of the 13th International Conference on Logic for Programming Artificial Intelligence and Reasoning. LNCS, vol. 4246, pp. 75–89. Springer, Heidelberg (2006)Google Scholar
  22. 22.
    Moser, G., Weiermann, A.: Relating derivation lengths with the slow-growing hierarchy directly. In: Proceedings of the 14th International Conference on Rewriting Techniques and Applications, number 2706 in LNCS, pages 296–310. Springer Verlag, 2003Google Scholar
  23. 23.
    Pohlers W.: Proof Theory: The First Step into Impredicativity. Universitext. Springer, Heidelberg (2008)Google Scholar
  24. 24.
    Takeuti G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987)Google Scholar
  25. 25.
    TeReSe. Term Rewriting Systems. Cambridge Tracks in Theoretical Computer Science, vol. 55. Cambridge University Press, London (2003)Google Scholar
  26. 26.
    Touzet, H.: Encoding the Hydra battle as a rewrite system. In: Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science. LNCS 1450, pp. 267–276. Springer, Heidelberg (1998)Google Scholar
  27. 27.
    Weiermann A.: Termination proofs for term rewriting systems with lexicographic path ordering imply multiply recursive derivation lengths. Theor. Comput. Sci. 139, 355–362 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Zantema H.: Termination of term rewriting: interpretation and type elimination. J. Symb. Comput. 17(1), 23–50 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of InnsbruckInnsbruckAustria

Personalised recommendations