Scaled dimension and nonuniform complexity

  • Authors:

  • John M. Hitchcock;Jack H. Lutz;Elvira Mayordomo

  • Affiliations:

  • Department of Computer Science, University of Wyoming, Laramie, WY;Department of Computer Science, Iowa State University, Ames, IA;Departamento de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, 50015 Zaragoza, Spain

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2004

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Abstract

Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (in: Proceedings of the 15th Annual IEEE Conference on Computational Complexity, 2000, pp. 158-169) 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(α2n/n) has measure 0 in ESPACE for all 0≤α≤1, we now know that SIZE(α2n/n) has dimension α in ESPACE for all 0≤α≤1. The present paper furthers this program by developing a natural hierarchy of "rescaled" resource-bounded dimensions. For each integer i and each set X of decision problems, we define the ith-order dimension of X in suitable complexity classes. The zeroth-order dimension is precisely the dimension of Hausdorff (Math. Ann. 79 (1919) 157-179) and Lutz (2000). Higher and lower orders are useful for various sets X. For example, we prove the following for 0≤α≤1 and any polynomial q(n)≥n2: 1. The class SIZE(2αn) and the time- and space-bounded Kolmogorov complexity classes KTq(2αn) and Ksq(2αn) have first-order dimension α in ESPACE. 2. The classes SIZE(2nα), KTq(2nα), and KSq(2nα) have second-order dimension α in ESPACE. 3. The classes KTq(2n(1-2-αn)) and KSq(2n(1-2αn) have negative-first-order dimension α in ESPACE.