Rewriting theory for the Hydra battle and the extended Grzegorczyk hierarchy

A subrecursive rewriting framework for the classical Kirby and Paris hydra battle is introduced. The termination of a natural rewrite system RH for the Hydra battle is shown by using ordinals and additionally by proving an upper bound on the derivation lengths in terms of a fast growing function of ordinal index ε0. It is shown that the RH -derivation lengths cannot be bounded by a fast growing function of ordinal index less that ε0, hence the termination of RH cannot be proved in first order Peano arithmetic. This yields that any natural pointwise termination ordering for the hydra battle rewrite system RH must have order type equal to the Howard Bachmann ordinal, as conjectured by E.A. Cichon. Rewrite systems for various levels of the extended Grzegorczyk hierarchy (up to ordinal level ε0) are introduced and their derivation lengths are classified with appropriate functions from the fast growing hierarchy. 1 A subrecursive rewriting framework for the hydra battle We begin with recalling the definition of the classical Kirby and Paris hydra battle [cf.[12]]. Since the geometrical formulation of this battle is well known we will not repeat it here. Instead we will directly deal with the underlying set of involved ordinal notations for the ordinals less than ε0. The (commutative) natural sum of ordinals is denoted by #. See, for example, [14, 18] for a definition. Due to the Cantor normal form theorem every ordinal less than ε0 can be ∗This work has been supported by HCM-grant 293F0740051507126. The author would like to thank P. Lescanne and E.A. Cichon for their hospitality during his stay at Nancy as a member of the Eureca group. He also would like to thank S. Feferman for his hospitality while visiting Stanford University from 6/2/95 3/3/95 and S.S. Wainer for his hospitality while visiting Leeds University from 19/3/95-25/3/95. The results of this paper have been presented 1995 at talks in Nancy, Stanford, Leeds, Oberwolfach and La Bresse.