Scaled dimension and nonuniform complexity

Scaled Dimension and Nonuniform Complexity

Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resource-bounded measure 0. For example, while it has long been known that the Boolean circuit-size complexity class SIZE ( α n

Scaled Dimension of Individual Strings

We define a new discrete version of scaled dimension and we find connections between the scaled dimension of a string and its Kolmogorov complexity and predictability. We give a new characterization of constructive scaled dimension by Kolmogorov complexity, and prove a new result about scaled dimension and prediction.

Scaled Dimension and the Berman-Hartmanis Conjecture

In 1977, L. Berman and J. Hartmanis [BH77] conjectured that all polynomialtime many-one complete sets for NP are are pairwise polynomially isomorphic. It was stated as an open problem in [LM99] to resolve this conjecture under the measure hypothesis from quantitative complexity theory. In this paper we study the polynomial-time isomorphism degrees within degm(SAT ) in the context of polynomial ...

Resource - Bounded Dimension , Nonuniform Complexity , and Approximation of MAX 3 SAT

Almost Everywhere High Nonuniform Complexity

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 n n lower bound for almost every problem with no computability co...