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14
A uniform approach to fundamental sequences and hierarchies
 Math. Logic Quart
, 1994
"... In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of numbertheoretic functions and we show the equivalence of the new approach with the classical one. ..."
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Cited by 28 (7 self)
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In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of numbertheoretic functions and we show the equivalence of the new approach with the classical one.
Classifying the provably total functions of PA
 BSL
, 2006
"... We give a selfcontained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for ..."
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Cited by 5 (4 self)
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We give a selfcontained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for teaching purposes and just requires basic familiarity with PA and the ordinals below ε0. (Familiarity with a cut elimination theorem for a Gentzen or Tait calculus is helpful but not presupposed). 1
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& ..."
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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...
A Relationship among Gentzen’s ProofReduction
 KirbyParis’ Hydra Game, and Buchholz’s Hydra Game, Math. Logic Quarterly
, 1997
"... KirbyParis [9] found a certain combinatorial game called Hydra Game whose termination is true but cannot be proved in $PA $. Cichon [4] gave a new proof based on Wainer’s finite characterization of the $\mathrm{P}\mathrm{A}$provably recursive functions by the use of Hardy functions. Both KirbyP ..."
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KirbyParis [9] found a certain combinatorial game called Hydra Game whose termination is true but cannot be proved in $PA $. Cichon [4] gave a new proof based on Wainer’s finite characterization of the $\mathrm{P}\mathrm{A}$provably recursive functions by the use of Hardy functions. Both KirbyParis and Cichon’s proofs on the unprovability result were obtained by a certain investigation on the ordinals less than $\epsilon_{0} $ , the critical ordinal of $\mathrm{P}\mathrm{A} $ , with the help of the fast and slow growing hierarchies, respectively. On the other hand, Jervell [7] proposed a combinatorial game, called Gentzen Game, on finite binary labeled trees. His Game was defined by abstracting some of the proof reduction procedure of Gentzen’s consistency proof of PA [5], which directly implies Gentzen Game’s unprovability of PA via G\"odel’s incompleteness theorem. The rules of Gentzen Game look rather artificial and complicated while those of KirbyParis ’ are more natural and simpler as a combinatorial game. Moreover, the modified proof reduction Jervell considered ignores Gentzen’s notion of potential (or height), which makes the Gentzen Game less complicated but the natural termination proof requires much larger than $\epsilon_{0} $. Hence, the resulting Gentzen Game is much stronger than $\mathrm{P}\mathrm{A} $ , while KirbyParis ’ Game is considered an optimal game, in the sense that any subgame restricted to the hydras with an upper bound size turns out to be provable in $\mathrm{P}\mathrm{A} $. On the other hand, Jervell’s unprovability proof (on Gentzen Game) is direct and clear (thanks to the fact that the Game is directly connected to Gentzen’s consistency proof) while the unprovability proofs of KirbyParis Game are more involved and complicated. Hence, it is very natural to ask if one can find another way of interpreting Gentzen’s proofreduction procedure into a more natural combinatorial game such as KirbyParis’, (so that one can get a natural combinatorial game and a direct unprovabil
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
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.
Theory Comput Syst (2009) 44: 205–214 DOI 10.1007/s002240089109y Die Another Day
, 2008
"... 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 ..."
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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...
NorthHolland AN INDEPENDENCE RESULT FOR (n:CA) + BI
, 1984
"... In Kirby and Paris [5] it was shown that a certain combinatorial statement (conceming finite trees) is independent of Peano Arithmetic. Here we present a not too complicated extension of tbis statement and prove its independence from ..."
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In Kirby and Paris [5] it was shown that a certain combinatorial statement (conceming finite trees) is independent of Peano Arithmetic. Here we present a not too complicated extension of tbis statement and prove its independence from