# 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].

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**ABSTRACT:**A set A is nontrivial for the linear-exponential-time class E=DTIME(2lin ) if for any k≥1 there is a set B k ∈E such that B k is (p-m-)reducible to A and $B_{k} \not\in \mathrm{DTIME}(2^{k\cdot n})$ . I.e., intuitively, A is nontrivial for E if there are arbitrarily complex sets in E which can be reduced to A. Similarly, a set A is nontrivial for the polynomial-exponential-time class EXP=DTIME(2poly ) if for any k≥1 there is a set $\hat{B}_{k} \in \mathrm {EXP}$ such that $\hat{B}_{k} $ is reducible to A and $\hat{B}_{k} \not\in \mathrm{DTIME}(2^{n^{k}})$ . We show that these notions are independent, namely, there are sets A 1 and A 2 in E such that A 1 is nontrivial for E but trivial for EXP and A 2 is nontrivial for EXP but trivial for E. In fact, the latter can be strengthened to show that there is a set A∈E which is weakly EXP-hard in the sense of Lutz (SIAM J. Comput. 24:1170–1189, 11) but E-trivial.Theory of Computing Systems 07/2012; 51(1). DOI:10.1007/s00224-011-9370-3 · 0.45 Impact Factor