Advertisement

Die Another Day

  • Rudolf Fleischer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4475)

Abstract

The Hydra was a many-headed monster from Greek mythology that would immediately replace a head that was cut off by one or two new heads. It was the second task of Hercules to kill this monster. In an abstract sense, a Hydra can be modeled as a tree where the leaves are the heads, and when a head is cut off some subtrees get duplicated. Different Hydra species differ by which subtress can be duplicated in which multiplicity. Using some deep mathematics, it had been shown that two classes of Hydra species must always die, independent of the order in which heads are cut off. In this paper we identify three properties for a Hydra that are necessary and sufficient to make it immortal or force it to die. We also give a simple combinatorial proof for this classification. Now, if Hercules had known this...

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avigad, J.: Ordinal analysis without proofs. In: Sieg, W., Sommer, R., Talcott, C. (eds.) Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman. Lecture Notes in Logic, vol. 15, pp. 1–36. A. K. Peters, Ltd, Wellesley, MA (2002)Google Scholar
  2. 2.
    Beklemishev, L.D.: The worm principle. Technical Report 219, Department of Philosophy, University of Utrecht, Logic Group Preprint Series (2003)Google Scholar
  3. 3.
    Buchholz, W.: An independence result for (Π 1 1− CA) + BI. Annals of Pure and Applied Logic 33, 131–155 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carlucci, L.: A new proof-theoretic proof of the independence of the Kirby-Paris’ Hydra theorem. Theoretical Computer Science 300(1–3), 365–378 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cichon, E.A.: A short proof of two recently discovered independence results using recursion theoretic methods. Proceedings of the American Mathematical Society 87(4), 704–706 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Clote, P., McAloon, K.: Two further combinatorial theorems equivalent to the 1-consistency of Peano arithmetic. The. Journal of Symbolic Logic 48(4), 1090–1104 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daly, B. (no title) (17 August, 2000), Archived in the Mathematical Atlas at: http://www.math.niu.edu/~rusin/known-math/00_incoming/goodstein
  8. 8.
    Dershowitz, N.: Trees, ordinals and termination. In: Gaudel, M.-C., Jouannaud, J.-P. (eds.) CAAP 1993, FASE 1993, and TAPSOFT 1993. LNCS, vol. 668, pp. 243–250. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  9. 9.
    Feferman, S., Friedman, H.M., Maddy, P., Steel, J.R.: Does mathematics need new axioms? The Bulletin of Symbolic Logic 6(4), 401–446 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gardner, M.: Mathematical games: Tasks you cannot help finishing no matter how hard you try to block finishing them. Scientific American, pp. 8–13 (1983)Google Scholar
  11. 11.
    Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112:493–565, 1936. Appendix: Galley proof of sections IV and V, Mathematische Annalen received on 11th August 1935. Translated as The consistency of elementary number theory in [25], pp. 132–213 (1935)Google Scholar
  12. 12.
    Goodstein, R.J.: On the restricted ordinal theorem. The Journal of Symbolic Logic 9, 33–41 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hamano, M., Okada, M.: A relationship among Gentzen’s proof-reduction, Kirby-Paris’ Hydra game and Buchholz’s Hydra game. Mathematical Logic Quarterly 43, 103–120 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hamano, M., Okada, M.: A direct independence proof of Buchholz’s Hydra game on finite labeled trees. Archive for Mathematical Logic 37, 67–89 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ishihara, H.: Weak König’s Lemma implies Brouwer’s Fan Theorem. Notre Dame Journal of Formal Logic 47(2), 249–252 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Keisler, H.J.: Nonstandard arithmetic and reverse mathematics. The. Bulletin of Symbolic Logic 12(1), 100–125 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kirby, L., Paris, J.: Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society 14, 285–293 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kirby, L., Paris, J.: Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society 14, 285–293 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lagarias, J.: The 3x + 1 problem and its generalizations. American Mathematical Monthly 92, 3–23 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Leadbetter, R.: Hydra. In: Encyclopedia Mythica (1999), http://www.pantheon.org/articles/h/hydra.html
  21. 21.
    Luccio, F., Pagli, L.: Death of a monster. ACM SIGACT News 31(4), 130–133 (2000)CrossRefGoogle Scholar
  22. 22.
    Paris, J.: Some independence results for Peano arithmetic. The. Journal of Symbolic Logic 43, 725–731 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    The Lernean Hydra. In: Crane, G., (ed.), The Perseus Project. Tufts University, Department of the Classics (2000), http://www.perseus.tufts.edu/Herakles/hydra.html
  24. 24.
    Smullyan, R.M.: Trees and ball games. Annals of the New York Academy of Sciences 321, 86–90 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Szabo, M.E. (ed.): The Collected Papers of Gerhard Gentzen. North Holland, Amsterdam (1969)zbMATHGoogle Scholar
  26. 26.
    Tait, W.W.: Gödels reformulation of Gentzen’s first consistency proof for arithmetic: the no-counterexample interpretation. The. Bulletin of Symbolic Logic 11(2), 225–238 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tucholsky, K.: Ratschläge für einen schlechten Redner (Advice for a bad speaker). In: Zwischen gestern und morgen, pp. 95–96. Rowohlt Verlag, Hamburg (1952), English translation at http://www.nobel133.physto.se/Programme/tucholsky.htm
  28. 28.
    Wainer, S.S.: Accessible recursive functions. The. Bulletin of Symbolic Logic 5(3), 367–388 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weiermann, A.: Classifying the phase transition of Hydra games and Goodstein sequences (2006) Manuscript, available at http://www.math.uu.nl/people/weierman/goodstein.pdf

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Rudolf Fleischer
    • 1
  1. 1.Fudan University, Shanghai Key Laboratory of Intelligent Information Processing, Department of Computer Science and Engineering, Shanghai 200433China

Personalised recommendations