Conference Paper
Quantitative Aspects of Speed-Up and Gap Phenomena.
DOI: 10.1017/S0960129510000174 Conference: Theory and Applications of Models of Computation, 6th Annual Conference, TAMC 2009, Changsha, China, May 18-22, 2009. Proceedings
Source: DBLP
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].
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].
- [Show abstract] [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.
Similar Publications
Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.