Quantitative Aspects of Speed-Up and Gap Phenomena
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].
KeywordsQuantitative Aspect Computable Function Category Concept Baire Space Abstract Complexity
Unable to display preview. Download preview PDF.
- [AS96]Ambos-Spies, K.: Resource-bounded genericity. In: Cooper, S., Slaman, T., Wainer, S. (eds.) Computability, enumerability, unsolvability. Directions in recursion theory, pp. 1–60. Cambridge University Press, Cambridge (1996)Google Scholar
- [ASM97]Ambos-Spies, K., Mayordomo, E.: Resource-bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory, pp. 1–47. Dekker, New York (1997)Google Scholar
- [ASR97]Ambos-Spies, K., Reimann, J.: Effective Baire category concepts. In: Proc. Sixth Asian Logic Conference 1996, pp. 13–29. Singapore University Press (1997)Google Scholar
- [Me73]Mehlhorn, K.: On the size of sets of computable functions. In: Proceedings of the 14th IEEE Symposium on Switching and Automata Theory, pp. 190–196 (1973)Google Scholar
- [Tra67]Trakhtenbrot, B.A.: Complexity of algorithms and computations. Course Notes, Novosibirsk (in Russian) (1967)Google Scholar
- [Zim06]Zimand, M.: Computational Complexity: A Quantitative Perspective. Elsevier, Amsterdam (2006)Google Scholar