Phase transition thresholds for some natural subclasses of the computable functions

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 3988 LNCS, Issue , 2006, Pages 556-570

  Weiermann, Andreas ^a  

^a Fakulteit Bètawetenschappen Departement Wiskunde ^*  (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

FUNCTION EVALUATION; PARAMETER ESTIMATION; PHASE TRANSITIONS; RECURSIVE FUNCTIONS;

COMPUTABLE FUNCTIONS; RAMSEY FUNCTIONS; WELL-PARTIAL ORDERS;

COMPUTABILITY AND DECIDABILITY;

EID: 33746080425     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/11780342_57     Document Type: Conference Paper

References (31)

1. T. Arai, On the slowly well orderedness of ε0, Math. Log. Q., 48 (2002), 125-130.
2. W. Buchholz, A. Cichon and A. Weiermann, A uniform approach to fundamental sequences and hierarchies, Math. Log. Q., 40 (1994), 273-286.
3. L. Carlucci, G. Lee and A. Weiermann Classifying the phase transition threshold for regressive Ramsey functions. Preprint 2006 (submitted to TAMS).
4. L.E. Dickson: Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. American Journal of Mathematics, Vol. 35 (1913), 413-422.
5. P. Erdös and G. Mills, Some bounds for the Ramsey-Paris-Hamngton numbers, J. Combin. Theory Ser. A, 30 (1981), no. 1, 53-70.
6. P. Erdös, A. Hajnal, A. Máté, and R. Rado. Combinatorial set theory: Partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, 106. North-Holland, 1984.
7. P. Erdös and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. (3), 2 (1952), 417-439.
8. R. L. Graham, B. L. Rotschild and J. H. Spencer, Ramsey Theory, Wiley, 1980.
9. Graham Higman: Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society 3(2) (1952) pp. 326-336.
10. A. Kanamori and K. McAloon, On Gödel incompleteness and finite combinatorics, Ann. Pure Appl. Logic, 33 (1987), no. 1, 23-41.
11. L. Kirby, J.B Paris. Initial segments of models of Peano's axioms. In: Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976), pp. 211-226. Lecture Notes in Math., Vol. 619. Springer, Berlin (1977).
12. M. Kojman, G. Lee, E. Omri and A. Weiermann, Sharp thresholds for the Phase Transition between Primitive Recursive and Ackermaniann Ramsey Numbers, Preprint, 2005.
13. M. Kojman and S. Shelah, Regressive Ramsey numbers are Ackermannian, J. Comb. Theory Ser. A, 86 (1999), no.1, 177-181.
14. S. Kripke: The problem of entailment. Journal of Symbolic Logic. Vol. 24 (1959), 324 (abstract).
15. G. Lee, Phase Transitions in Axiomatic Thought, PhD thesis (written under the supervision of A. Weiermann), Münster 2005, 121 pages.
16. G. Moreno Sodas: An Ackermannian polynomial ideal. Lecture Notes in Computer Science 539 (1991), 269-280.
17. F. P. Ramsey, On a problem of Formal Logic, Proc. London Math. Soc., ser. 2, 30 (1930), 338-384.
18. H. Schwichtenberg, Eine Klassifikation der ε0-rekursiven Funktionen, Z. Math. Logik Grundlagen Math., 17 (1971), 61-74.
19. S.G. Simpson. Ordinal numbers and the Hilbert basis theorem. J. Symbolic Logic 53 (1988), no. 3, 961-974.
20. Rick L. Smith: The consistency strength of some finite forms of the Higman and Kruskal theorems. In Harvey Friedman's Research on the Foundations of Mathematics, L. A. Harrington et al. (editors), (1985), pp. 119-136.
21. S. S. Wainer, A classification of the ordinal recursive functions, Archiv für Mathematische Logik und Grundlagenforschung, 13 (1970), 136-153.
22. A. Weiermann, How to characterize provably total functions by local predicativity, J. Symbolic Logic, 61 (1996), no.1, 52-69.
23. A. Weiermann, An application of graphical enumeration to PA, J. Symbolic Logic 68 (2003), no. 1, 5-16.
24. A. Weiermann, An application of results by Hardy, Ramanujan and Karamata to Ackermannian functions, Discrete Mathematics and Computer Science, 6 (2003), 133-142.
25. A. Weiermann: A very slow growing hierarchy for Λ0. Logic Colloquium '99, 182-199, Lect. Notes Log., 17, Assoc. Symbol. Logic, Urbana, IL, 2004.
26. A. Weiermann, A classification of rapidly growing Ramsey functions, Proc. Amer. Math. Soc., 132 (2004), no. 2, 553-561.
27. A. Weiermann, Analytic Combinatorics, proof-theoretic ordinals and phase transitions for independence results, Ann. Pure Appl. Logic, 136 (2005), 189-218.
28. A. Weiermann:An extremely sharp phase transition threshold for the slow growing hierarchy. Mathematical Structures in Computer Science (to appear).
29. A. Weiermann, Classifying the phase transition for Paris Harrington numbers, Preprint, 2005.
30. A. Weiermann: Phase transitions for some Friedman style independence results Preprint 2006, to appear in MLQ.
31. H.S. Wilf: generatingfunctionology. Second edition. Academic Press, Inc., Boston, MA, 1994. x+228 pp.