Dimension Characterizations of Complexity Classes

  • Xiaoyang Gu
  • Jack H. Lutz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


We use derandomization to show that sequences of positive pspace-dimension – in fact, even positive Δ\(^{\rm p}_{\rm k}\)-dimension for suitable k – have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose Δ\(^{\rm p}_{\rm 3}\)-dimension is positive, then BPP ⊆ P S and, moreover, every BPP promise problem is P S -separable. We prove analogous results at higher levels of the polynomial-time hierarchy.

The dimension-almost-class of a complexity class \(\mathcal{C}\), denoted by dimalmost-\(\mathcal{C}\), is the class consisting of all problems A such that \(A \in \mathcal{C}^S\) for all but a Hausdorff dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmost-P and AM = dimalmost-NP, that refine previously known results on almost-classes. They also yield results, such as Promise-BPP = almost-P-Sep = dimalmost-P-Sep, in which even the almost-class appears to be a new characterization.


Complexity Class Binary Sequence SIAM Journal Random Oracle Kolmogorov Complexity 
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 2006

Authors and Affiliations

  • Xiaoyang Gu
    • 1
  • Jack H. Lutz
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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