Quantitative Aspects of Speed-Up and Gap Phenomena

Authors:
Klaus Ambos-Spies;Thorsten Kräling
Affiliations:
Institut für Informatik, University of Heidelberg, Heidelberg, Germany D-69120;Institut für Informatik, University of Heidelberg, Heidelberg, Germany D-69120
Venue:
TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
Year:
2009

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Abstract

We show that, for any abstract complexity measure in the sense of Blum and for any computable function f (or computable operator F ), the class of problems which are f -speedable (or F -speedable) does not have effective measure 0. On the other hand, for sufficiently fast growing f (or F ), the class of the nonspeedable problems does not have effective measure 0 too. These results answer some questions raised by Calude and Zimand in [CZ96] and [Zim06]. We also give a short quantitative analysis of Borodin and Trakhtenbrot's Gap Theorem which corrects a claim in [CZ96] and [Zim06].