Abstract

Kirby and Paris (1982) proved in a celebrated paper that a theorem of Goodstein (1944) cannot be established in Peano (1889) arithmetic. We present an encoding of Goodstein’s theorem as a termination problem of a finite rewrite system. Using a novel implementation of ordinal interpretations, we are able to automatically prove termination of this system, resulting in the first automatic termination proof for a system whose derivational complexity is not multiple recursive. Our method can also cope with the encoding by Touzet (1998) of the battle of Hercules and Hydra, yet another system which has been out of reach for automated tools, until now.

BibTeX Entry

@inproceedings{WZM-RTA13,
 author    = "Sarah Winkler and Harald Zankl and Aart Middeldorp",
 title     = "Beyond {P}eano Arithmetic---{A}utomatically Proving
              Termination of the {G}oodstein Sequence",
 booktitle = "Proceedings of the 24th International Conference on Rewriting
              Techniques and Applications",
 editor    = "Femke van Raamsdonk",
 series    = "Leibniz International Proceedings in Informatics",
 volume    = 21,
 pages     = "335--351",
 year      = 2013,
 doi       = "10.4230/LIPIcs.RTA.2013.335"
}