On some slowly terminating term rewriting systems

We formulate some term rewriting systems in which the num- ber of computation steps is finite for each output, but this number can- not be bounded by a provably total computable function in Peano arith- metic PA. Thus, the termination of such systems is unprovable in PA. These systems are derived from an independent combinatorial result known as the Worm principle; they can also be viewed as versions of the well-known Hercules-Hydra game introduced by Paris and Kirby. Bibliography: 16 titles.

[1]  Tobias Nipkow,et al.  Term Rewriting and All That by Franz Baader , 1998 .

[2]  Lev D. Beklemishev,et al.  Reflection principles and provability algebras in formal arithmetic , 2005 .

[3]  Лев Дмитриевич Беклемишев,et al.  Схемы рефлексии и алгебры доказуемости в формальной арифметике@@@Reflection principles and provability algebras in formal arithmetic , 2005 .

[5]  Lev D. Beklemishev.  Representing Worms as a term rewriting system , 2006 .

[6]  Lev D. Beklemishev.  Calibrating Provability Logic: From Modal Logic to Reflection Calculus , 2012, Advances in Modal Logic.

[7]  Mitsuhiro Okada,et al.  A Relationship Among Gentzen's Proof‐Reduction, Kirby‐Paris' Hydra Game and Buchholz's Hydra Game , 1995, Math. Log. Q..

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

[9]  D Beklemishev Lev.  Calibrating Provability Logic: From Modal Logic to Re ection Calculus , 2012, AiML 2012.

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

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

[12]  Jean H. Gallier,et al.  WHAT'S SO SPECIAL ABOUT KRUSKAL'S THEOREM AND THE ORDINAL 0? A SURVEY OF SOME RESULTS IN PROOF THEORY This paper has appeared in Annals of Pure and Applied Logic, 53 (1991), 199-260. , 2012 .

[13]  Aart Middeldorp,et al.  Beyond polynomials and Peano arithmetic - automation of elementary and ordinal interpretations , 2015, J. Symb. Comput..

[14]  Paliath Narendran.  Book review: Term Rewriting and all that by Franz Baader and Tobias Nipkow (Cambridge Univ . Press, 313 pages) , 2000, SIGA.

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

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

[17]  Nachum Dershowitz,et al.  The Hydra Battle Revisited , 2007, Rewriting, Computation and Proof.

[18]  Lev D. Beklemishev.  The Worm principle , 2003 .

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