Scaled dimension and the kolmogorov complexity of turing-hard sets

Abstract

Scaled dimension has been introduced by Hitchcock et al. (2003) in order to quantitatively distinguish among classes such as SIZE(2αn) and SIZE(2nα) that have trivial dimension and measure in ESPACE. This paper gives an exact characterization of effective scaled dimension in terms of resource-bounded Kolmogorov complexity. We can now view each result on the scaled dimension of a class of languages as upper and lower bounds on the Kolmogorov complexity of the languages in the class. We prove a Small Span Theorem for Turing reductions that implies the class of ≤TP/poly-hard sets for ESPACE has (-3)rd-pspace dimension 0. As a consequence we have a nontrivial upper bound on the Kolmogorov complexity of all hard sets for ESPACE for this very general nonuniform reduction, ≤TP/poly. This is, to our knowledge, the first such bound. We also show that this upper bound does not hold for most decidable languages, so ≤TP/poly -hard languages are unusually simple. © Springer-Verlag 2004.