31 research outputs found
The Quantitative Structure of Exponential Time
Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E = DTIME(2^linear) and E2 = DTIME(2^polynomial) are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed
The Quantitative Structure of Exponential Time
Department of Computer Science Iowa State University Ames, Iowa 50010 Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed
Genericity and measure for exponential time
AbstractRecently, Lutz [14, 15] introduced a polynomial time bounded version of Lebesgue measure. He and others (see e.g. [11, 13–18, 20]) used this concept to investigate the quantitative structure of Exponential Time (E = DTIME(2lin)). Previously, Ambos-Spies et al. [2, 3] introduced polynomial time bounded genericity concepts and used them for the investigation of structural properties of NP (under appropriate assumptions) and E. Here we relate these concepts to each other. We show that, for any c ⩾ 1, the class of nc-generic sets has p-measure 1. This allows us to simplify and extend certain p-measure 1-results. To illustrate the power of generic sets we take the Small Span Theorem of Juedes and Lutz [11] as an example and prove a generalization for bounded query reductions
Resource-bounded Measure on Probabilistic Classes
We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain
notions of resource-bounded measure on probabilistic complexity classes such as
BPE and BPEXP. Unlike former attempts, our resource bounded measure notions
satisfy all three basic measure properties, that is every singleton {L} has measure
zero, the whole space has measure one, and "enumerable infinite unions" of measure
zero sets have measure zero
Resource-bounded Measure on Probabilistic Classes
We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain
notions of resource-bounded measure on probabilistic complexity classes such as
BPE and BPEXP. Unlike former attempts, our resource bounded measure notions
satisfy all three basic measure properties, that is every singleton {L} has measure
zero, the whole space has measure one, and "enumerable infinite unions" of measure
zero sets have measure zero