The Hydra Battle Revisited

Showing termination of the Battle of Hercules and Hydra is a challenge. We present the battle both as a rewrite system and as an arithmetic while program, provide proofs of their termination, and recall why their termination cannot be proved within Peano arithmetic.

[1]  S. Buss.  Handbook of proof theory , 1998 .

[2]  Nachum Dershowitz.  Orderings for Term-Rewriting Systems , 1979, FOCS.

[3]  Stanley S. Wainer,et al.  Chapter III - Hierarchies of Provably Recursive Functions , 1998 .

[4]  Ingo Lepper,et al.  Simply terminating rewrite systems with long derivations , 2004, Arch. Math. Log..

[5]  Nachum Dershowitz,et al.  Trees, Ordinals and Termination , 1993, TAPSOFT.

[6]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[7]  Robin Milner,et al.  Action Structures and the Pi Calculus , 1995 .

[8]  Cliff B. Jones,et al.  An Early Program Proof by Alan Turing , 1984, Annals of the History of Computing.

[9]  Review: R. L. Goodstein, On the Restricted Ordinal Theorem , 1945 .

[10]  Nachum Dershowitz,et al.  Examples of Termination , 1993, Term Rewriting.

[11]  Jan Willem Klop,et al.  Term Rewriting Systems: From Church-Rosser to Knuth-Bendix and Beyond , 1990, ICALP.

[12]  Hélène Touzet,et al.  Encoding the Hydra Battle as a Rewrite System , 1998, MFCS.

[13]  Nachum Dershowitz,et al.  Proof-theoretic techniques for term rewriting theory , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[14]  Enno Ohlebusch,et al.  Modular Termination Proofs for Rewriting Using Dependency Pairs , 2002, J. Symb. Comput..

[15]  A. M. Turing,et al.  Checking a large routine , 1989 .

[16]  R. L. Goodstein,et al.  On the restricted ordinal theorem , 1944, Journal of Symbolic Logic.

[17]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[18]  Andreas Weiermann,et al.  A Uniform Approach to Fundamental Sequences and Hierarchies , 1994, Math. Log. Q..

[19]  Wilfried Buchholz,et al.  An independence result for (II11-CA)+BI , 1987, Ann. Pure Appl. Log..

[20]  Rudolf Fleischer.  Die Another Day , 2007, FUN.

[21]  Fabrizio Luccio,et al.  Death of a monster , 2000, SIGA.

[22]  Jean H. Gallier,et al.  What's So Special About Kruskal's Theorem and the Ordinal Gamma0? A Survey of Some Results in Proof Theory , 1991, Ann. Pure Appl. Log..

[23]  Nachum Dershowitz,et al.  Natural Termination , 1995, Theor. Comput. Sci..

[24]  David Gries.  Is Sometimes Ever Better Than Always? , 1978, Program Construction.

[25]  J. Paris,et al.  Accessible Independence Results for Peano Arithmetic , 1982 .

[26]  Jozef Gruska,et al.  Mathematical Foundations of Computer Science 1998 , 1998, Lecture Notes in Computer Science.

[27]  Jean-Pierre Jouannaud,et al.  TAPSOFT'93: Theory and Practice of Software Development , 1993, Lecture Notes in Computer Science.

[28]  M. Gardner.  Bulgarian Solitaire and Other Seemingly Endless Tasks , 1997 .

[29]  Hans Zantema,et al.  The Termination Competition , 2007, RTA.

[30]  Nao Hirokawa,et al.  Dependency Pairs Revisited , 2004, RTA.

[31]  J. Paris.  A Mathematical Incompleteness in Peano Arithmetic , 1977 .

[32]  Georg Moser,et al.  Relating Derivation Lengths with the Slow-Growing Hierarchy Directly , 2003, RTA.

[33]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

[34]  Stanley S. Wainer,et al.  Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy , 1972, Journal of Symbolic Logic.

[35]  Nachum Dershowitz,et al.  Orderings for term-rewriting systems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[36]  Jürgen Giesl,et al.  Termination of term rewriting using dependency pairs , 2000, Theor. Comput. Sci..

[37]  Jean-Pierre Jouannaud,et al.  Open Problems in Rewriting , 1991, RTA.

[38]  Jean Gallier,et al.  Ann. Pure Appl. Logic , 1997 .

[39]  Tobias Nipkow,et al.  Term rewriting and all that , 1998 .