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16
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 ...
Program termination and well partial orderings
, 2006
"... The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many wellfounded relations U1,..., Un then R is wellfounded. A question arises how to bound the ordinal height R  of the relation R in terms of the ordinals αi = Ui. ..."
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The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many wellfounded relations U1,..., Un then R is wellfounded. A question arises how to bound the ordinal height R  of the relation R in terms of the ordinals αi = Ui. We introduce the notion of the stature ‖P ‖ of a well partial ordering P and show that R  ≤ ‖α1 × · · · × αn ‖ and that this bound is tight. The notion of stature is of considerable independent interest. We define ‖P ‖ as the ordinal height of the forest of nonempty bad sequences of P, but it has many other natural and equivalent definitions. In particular, ‖P ‖ is the supremum, and in fact the maximum, of the lengths of linearizations of P. And ‖α1 × · · · × αn ‖ is equal to the natural product α1 ⊗ · · · ⊗ αn.
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
COMPUTABLE LINEARIZATIONS OF WELLPARTIALORDERINGS
, 2007
"... We analyze results on wellpartialorderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every wellpartialorder has a linearization of maximal order type. We show that such a lineariz ..."
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We analyze results on wellpartialorderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every wellpartialorder has a linearization of maximal order type. We show that such a linearization can be found computably. We also show that the process of finding such a linearization is not computably uniform, not even hyperarithmetically.
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...
On Fraïssé’s conjecture for linear orders of finite Hausdorff rank
, 2007
"... We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered ..."
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We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is wellordered”.
Ordinal Theory for Expressiveness of Well Structured Transition Systems
"... Abstract. To the best of our knowledge, we characterize for the first time the importance of resources (counters, channels, alphabets) when measuring expressiveness of WSTS. We establish, for usual classes of wpos, the equivalence between the existence of order reflections (nonmonotonic order embedd ..."
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Abstract. To the best of our knowledge, we characterize for the first time the importance of resources (counters, channels, alphabets) when measuring expressiveness of WSTS. We establish, for usual classes of wpos, the equivalence between the existence of order reflections (nonmonotonic order embeddings) and the simulations with respect to coverability languages. We show that the nonexistence of order reflections can be proved by the computation of order types. This allows us to solve some open problems and to unify the existing proofs of the WSTS classification.
THE MAXIMAL LINEAR EXTENSION THEOREM IN SECOND ORDER ARITHMETIC
, 2011
"... We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0. ..."
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We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0.