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Encoding the Hydra Battle as a rewrite system
, 1998
"... . 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 the ..."
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. 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. Introduction One of the main questions in rewriting theory i...
Die another day
 Proceedings of the 4th International Conference ‘FUN with Algorithms 4’, Lecture Notes in Computer Science
, 2007
"... Abstract. The Hydra was a manyheaded monster from Greek mythology that would immediately replace a head that was cut off by one or two new heads. It was the second task of Hercules to kill this monster. In an abstract sense, a Hydra can be modeled as a tree where the leaves are the heads, and when ..."
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Abstract. The Hydra was a manyheaded monster from Greek mythology that would immediately replace a head that was cut off by one or two new heads. It was the second task of Hercules to kill this monster. In an abstract sense, a Hydra can be modeled as a tree where the leaves are the heads, and when a head is cut off some subtrees get duplicated. Different Hydra species differ by which subtress can be duplicated in which multiplicity. Using some deep mathematics, it had been shown that two classes of Hydra species must always die, independent of the order in which heads are cut off. In this paper we identify three properties for a Hydra that are necessary and sufficient to make it immortal or force it to die. We also give a simple combinatorial proof for this classification. Now, if Hercules had known this... 1
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.
doi:10.1006/inco.2002.3160 A Characterisation of Multiply Recursive Functions with Higman’s Lemma
, 1999
"... We prove that string rewriting systems which reduce by Higman’s lemma exhaust the multiply recursive functions. This result provides a full characterisation of the expressiveness of Higman’s lemma when applied to rewriting theory. The underlying argument of our construction is to connect the order t ..."
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We prove that string rewriting systems which reduce by Higman’s lemma exhaust the multiply recursive functions. This result provides a full characterisation of the expressiveness of Higman’s lemma when applied to rewriting theory. The underlying argument of our construction is to connect the order type and the derivation length via the Hardy hierarchy. C ○ 2002 Elsevier Science (USA) 1.