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22
Prooftheoretic investigations on Kruskal's theorem
 Ann. Pure Appl. Logic
, 1993
"... In this paper we calibrate the exact prooftheoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his ..."
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In this paper we calibrate the exact prooftheoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents prooftheoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the prooftheoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of \Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...
Foundations of BQO Theory and Subsystems of SecondOrder Arithmetic
 Pennsylvania State University
, 1993
"... Abstract. We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open pro ..."
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Abstract. We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open problems about the axiomatic strength of both elementary and advanced results. A quasiordering (i.e. a reflexive and transitive binary relation) is awqo (well quasiordering) if it contains no infinite descending chains and no infinite sets of pairwise incomparable elements. This concept is very natural, and has been introduced several times, as documented in [19]. The usual working definition of wqo is obtained from the one given above with an application of Ramsey’s theorem: a quasiordering on the setQ is wqo if for every sequence {xn  n ∈ N}of elements ofQ there existm < n such thatxm xn. The notion of bqo (better quasiordering) is a strengthening of wqo which was introduced byNashWilliams in the 1960’s in a sequence of papers culminating in [30] and [31]. This notion has proved to be very useful in showing that specific quasiorderings are indeed wqo. Moreover the property of being bqo is preserved
The Power of Priority Channel Systems ∗
, 1301
"... We introduce Priority Channel Systems, a new class of channel systems where messages carry a numeric priority and where higherpriority messages can supersede lowerpriority messages preceding them in the fifo communication buffers. The decidability of safety and inevitability properties is shown vi ..."
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We introduce Priority Channel Systems, a new class of channel systems where messages carry a numeric priority and where higherpriority messages can supersede lowerpriority messages preceding them in the fifo communication buffers. The decidability of safety and inevitability properties is shown via the introduction of a priority embedding, a wellquasiordering that has not previously been used in wellstructured systems. We then show how Priority Channel Systems can compute FastGrowing functions and prove that the aforementioned verification problems are Fε0complete. 1
An Inductive Version of NashWilliams’ MinimalBadSequence Argument for Higman’s Lemma
 IN P. CALLAGHAN, E.AL., TYPES FOR PROOFS AND PROGRAMS, LECTURE NOTES IN COMPUTER SCIENCE 2277
, 2001
"... Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquan ..."
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Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders.
Worms, gaps and hydras
 Mathematical Logic Quarterly
, 2005
"... We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the wellknown KirbyParis ’ Hydra Game. We further show that the elements in a reduction sequence of the Wo ..."
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We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the wellknown KirbyParis ’ Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the wellquasiordering of finite sequences of natural numbers with respect to Friedman’s gapembeddability. 1
Applications of inductive definitions and choice principles to program synthesis
"... Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive defi ..."
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Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive definitions, the other applies a combination of Gödel’s negativeand Friedman’s Atranslation. Both approaches are explained by means of a case study on Higman’s Lemma and its wellknown classical proof due to NashWilliams. We also discuss some prooftheoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1