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A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies

Published online by Cambridge University Press:  12 September 2008

Walter A. Deuber  and
Wolfgang Thumser
Walter A. Deuber
Affiliation:
University of Bielefeld, Faukultät Mathematik, Postfach 10 01 31 33501 Bielefeld 1, Germany
Wolfgang Thumser
Affiliation:
University of Bielefeld, Faukultät Mathematik, Postfach 10 01 31 33501 Bielefeld 1, Germany
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Abstract

Long regressive sequences in well-quasi-ordered sets contain ascendingsubsequences of length n. The complexity of the corresponding function H(n) is studied in the Grzegorczyk-Wainer hierarchy. An extension to regressive canonical colourings is indicated.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 2 , June 1994 , pp. 177 - 190
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Erdőos, P., Hainal, A., Máté, A. and Rado, R. (1984) Combinatorial set theory: Partition relations for cardinals. Studies in Logic and the Foundations of Mathematics 106, North-Holland.Google Scholar
[2]Erdos, P. and Mills, G. (1981) Some Bounds for the Ramsey-Harrington Numbers. J. of Comb. Theory, Ser. A 30, 53–70.CrossRefGoogle Scholar
[3]Erdős, R. and Rado, R. (1950) A combinatorial Theorem. Journal of the London Mathematical Society 25, 249–255.CrossRefGoogle Scholar
[4]Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Composito Math. 2, 464–470.Google Scholar
[5]Gentzen, G. (1936) Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen 112, 493–565.CrossRefGoogle Scholar
[6]Gordan's, P. (1885) Vorlesungen über Invariantentheorie, Hrsg. v. Geo. Kerschensteiner. 1. Bd. Determinanten (XI, 201S.). Teubner, Leibzig.Google Scholar
[7]Gödel, K. (1931) Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatschefte fur Mathematik und Physik 38, 173–198.CrossRefGoogle Scholar
[8]Grzegorczyk, A. (1933) Some classes of recursive functions. Rozprawy matematiczne 4, Instytut Matematyczny Polskiej Akademie Nauk, Warsaw.Google Scholar
[9]Graham, R., Rothschild, B. and Spencer, J. (1990) Ramsey theory, Wiley, New York.Google Scholar
[10]Harzheim, E. (1967) Eine kombinatorische Frage zahlentheoretischer Art. Publicationes Mathematicae Debrecen 14, 45–51.CrossRefGoogle Scholar
[11]Harzheim, E. (1982) Combinatorial theorems on contractive mappings in power sets. Discrete math. 40, 193–201.CrossRefGoogle Scholar
[12]Higman, G. (1952) Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2, 326–336.CrossRefGoogle Scholar
[13]Jullien, P. (1968) Analyse combinatoire – Sur un théorème d'extension dans la théorie des mots. C.R. Acad. Sci. Paris, Ser. A 266, 851–854.Google Scholar
[14]Kanamori, A. and McAloon, K. (1987) On Gödel incompleteness and finite combinatorics. Annals of Pure and Applied Logic 33, 23–41.CrossRefGoogle Scholar
[15]Kreisel, G. (1952) On the interpretation of nonfinitistic proofs. Journal of Symbolic Logic 17, II, 43–58.CrossRefGoogle Scholar
[16]Kruskal, J. B. (1972) The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept. Journal of Combinatorial Theory (A) 13, 297–305.CrossRefGoogle Scholar
[17]Leeb, K. (1973) Vorlesungen über Pascaltheorie. Arbeitsbericht des Instituts für mathematische Maschinen und Datenverarbeitung, Friedrich Alexander Universität Erlangen Nürnberg, Bd. 6 Nr. 7.Google Scholar
[18]Leeb, K. Personal communications.Google Scholar
[19]Loebl, M. and Nešsetfil, J. (1991) Unprovable combinatorial statements. In: Keedwell, A. D.(ed.) Surveys in Combinatorics.CrossRefGoogle Scholar
[20]Mills, G. (1980) A tree analysis of unprovable combinatorial statements. Model theory of Algebra and Arithmetic. Springer-Verlag Lecture Notes in Mathematics 834, 248–311.CrossRefGoogle Scholar
[21]Nešsetřil, J. and Rödl, V. (1990) Mathematics of Ramsey Theory, Springer-Verlag, Berlin, Heidelberg.CrossRefGoogle Scholar
[22]Paris, J. and Harrington, L. (1977) A mathematical incompleteness in Peano Arithmetic. Handbook of Mathematical Logic. In: Barwise, J. (ed.) North-Holland Publishing Company, 1133–1142.CrossRefGoogle Scholar
[23]Prömel, H. J., Thumser, W. and Voigt, B. (1989) Fast growing functions based on Ramsey theorems. Forschungsinstitut fur Diskrete Mathematik, Bonn (preprint).Google Scholar
[24]Prömel, H. J., Thumser, W. and Voigt, B. (1991) Fast growing functions based on Ramsey theorems. Discrete Mathematics 95, 341–358.CrossRefGoogle Scholar
[25]Prömel, H. J. and Voigt, B. (1989) Aspects of Ramsey Theory I: Sets, Report number 87495-OR, Forschungsinstitut fur Diskrete Mathematik, Universität Bonn, Germany.Google Scholar
[26]Prömel, H. J. and Voigt, B. (1993) Aspects of Ramsey Theory, Springer Verlag, Berlin.Google Scholar
[27]Ramsey, F. P. (1930) On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264–286.CrossRefGoogle Scholar
[28]Simpson, S. G. (1987) Unprovable theorems and fast growing functions. In: Simpson, S. G. (ed.) Logic and Combinatorics. Contemporary Mathematics 65, 359–394.CrossRefGoogle Scholar
[29]Thumser, W. (1989) On upper Bounds for Kanamori McAloon Function, preprint 89–10, Sonderforschungsbereich 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld.Google Scholar
[30]Thumser, W. (1992) On the well-order type of certain combinatorial structures, Bielefeld (manuscript, submitted).Google Scholar
[31]Wainer, S. S. (1972) Ordinal recursion and a refinement of the extended Grzegorczyk hierarchy. Journal of Symbolic Logic 37 281–292.CrossRefGoogle Scholar