# Encoding the hydra battle as a rewrite system

## Abstract

In rewriting theory, termination of a rewrite system by Kruskal's theorem implies a theoretical upper bound on the complexity of the system. This bound is, however, far from having been reached by *known* examples of rewrite systems. All known orderings used to establish termination by Kruskal's theorem yield a multiply recursive bound. Furthermore, the study of the order types of such orderings suggests that the class of multiple recursive functions constitutes the least upper bound. Contradicting this intuition, we construct here a rewrite system which reduces by Kruskal's theorem and whose complexity is not multiply recursive. This system is even *totally terminating*. This leads to a new lower bound for the complexity of totally terminating rewrite systems and rewrite systems which reduce by Kruskal's theorem. Our construction relies on the *Hydra battle* using classical tools from ordinal theory and subrecursive functions.

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