Quantitative Aspects of Speed-Up and Gap Phenomena.
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].
- Journal of Symbolic Logic 01/1972; · 0.54 Impact Factor
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ABSTRACT: We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuit-size and space-bounded Kolmogorov complexity almost everywhere. (The circuit-size lower bound actually exceeds, and thereby strengthens, the Shannon 2n/n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity.Journal of Computer and System Sciences. 01/1992;
- Journal of the ACM 01/1972; 19(1):158-174. · 2.37 Impact Factor