Quantitative aspects of speed-up and gap phenomena

Authors:
Klaus Ambos-spies;Thorsten KrÄ/ling
Affiliations:
Institut fü/r informatik, university of heidelberg, inf 294, d-69120 heidelberg, germany email: ambos@math.uni-heidelberg.de/ thorsten.kraeling@informatik.uni-heidelberg.de;Institut fü/r informatik, university of heidelberg, inf 294, d-69120 heidelberg, germany email: ambos@math.uni-heidelberg.de/ thorsten.kraeling@informatik.uni-heidelberg.de
Venue:
Mathematical Structures in Computer Science
Year:
2010

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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 that 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 non-speedable computable problems does not have effective measure 0. These results answer some questions raised by Calude and Zimand. We also give a quantitative analysis of Borodin and Trakhtenbrot's Gap Theorem, which corrects a claim by Calude and Zimand.