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Quantitative aspects of speed-up and gap phenomena

Published online by Cambridge University Press:  27 October 2010

KLAUS AMBOS-SPIES  and
THORSTEN KRÄLING
KLAUS AMBOS-SPIES
Affiliation:
Institut für Informatik, University of Heidelberg, INF 294, D-69120 Heidelberg, Germany Email: ambos@math.uni-heidelberg.de; thorsten.kraeling@informatik.uni-heidelberg.de
THORSTEN KRÄLING
Affiliation:
Institut für Informatik, University of Heidelberg, INF 294, D-69120 Heidelberg, Germany Email: ambos@math.uni-heidelberg.de; thorsten.kraeling@informatik.uni-heidelberg.de
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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.

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