# Quantitative Aspects of Speed-Up and Gap Phenomena.

Conference PaperinMathematical Structures in Computer Science 20(5):88-97 · January 2009with5 Reads
Impact Factor: 0.45 · DOI: 10.1017/S0960129510000174 · Source: DBLP
Conference: Theory and Applications of Models of Computation, 6th Annual Conference, TAMC 2009, Changsha, China, May 18-22, 2009. Proceedings
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].

• ##### Comparing Nontriviality for E and EXP
[Hide abstract] 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.
No preview · Article · Jul 2012 · Theory of Computing Systems