Quantitative Aspects of Speed-Up and Gap Phenomena

摘要

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].