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Scaled Dimension and the Kolmogorov Complexity of Turing-Hard Sets

  • John M. Hitchcock
  • María López-Valdés
  • Elvira Mayordomo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)

Abstract

Scaled dimension has been introduced by Hitchcock et al. (2003) in order to quantitatively distinguish among classes such as SIZE(2 αn ) and SIZE(\( 2^{n^{\alpha}}\)) 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 ≤ P/poly T-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, ≤ P/poly T. This is, to our knowledge, the first such bound. We also show that this upper bound does not hold for most decidable languages, so \(\leq^{\rm P/poly}_{\rm T}\)-hard languages are unusually simple.

Keywords

Kolmogorov Complexity Lower Span Exact Characterization Turing Reduction Trivial Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • John M. Hitchcock
    • 1
  • María López-Valdés
    • 2
  • Elvira Mayordomo
    • 2
  1. 1.Department of Computer ScienceUniversity of WyomingUSA
  2. 2.Departamento de Informática e Ingeniería de SistemasUniversidad de ZaragozaZaragozaSPAIN

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