The Hydra Battle and Cichon’s Principle

Abstract In rewriting the Hydra battle refers to a term rewrite system H proposed by Dershowitz and Jouannaud. To date, H withstands any attempt to prove its termination automatically. This motivates our interest in term rewrite systems encoding the Hydra battle, as a careful study of such systems may prove useful in the design of automatic termination tools. Moreover it has been an open problem, whether any termination order compatible with H has to have the Howard-Bachmann ordinal as its order type, i.e., the proof theoretic ordinal of the theory of one inductive denition. We answer this question in the negative, by providing a reduction order compatible with H, whose order type is at most 0 , the proof theoretic ordinal of Peano arithmetic