Abstract
Scaled dimension has been introduced by Hitchcock et al. (2003) in order to quantitatively distinguish among classes such as SIZE(2αn) 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.
This research was supported in part by Spanish Government MEC project TIC 2002-04019-C03-03.
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Hitchcock, J.M., López-Valdés, M., Mayordomo, E. (2004). Scaled Dimension and the Kolmogorov Complexity of Turing-Hard Sets. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds) Mathematical Foundations of Computer Science 2004. MFCS 2004. Lecture Notes in Computer Science, vol 3153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28629-5_36
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DOI: https://doi.org/10.1007/978-3-540-28629-5_36
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