Scaled Dimension and Nonuniform Complexity

Abstract

Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (2000) 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 α 2 n n has measure 0 in ESPACE for all 0 ≤ α ≤ 1, we now know that SIZE α 2 n 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 i th -order dimension of X in suitable complexity classes. The 0 th -order dimension is precisely the dimension of Hausdorff (1919) 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) ≥ n 2 .1. The class SIZE(2 αn ) and the time-and space-bounded Kolmogorov complexity classes KT q (2 αn ) and KS q (2 αn ) have 1 st -order dimension α in ESPACE.2. The classes SIZE(2 n α ), KT q (2 n α ), and KS q (2 n α ) have 2 nd -order dimension α in ESPACE.3. The classes KT q (2 n (1 − 2 −αn )) and KS q (2 n (1 − 2 −αn ) have −1 st -order dimension α in ESPACE.

Scaled Dimension and the Kolmogorov Complexity of Turing-Hard Sets

“…These functions are a rescaled version of the more familiar concept of gales [25]. The main concept in the definition of scaled gales is a scale, which is a function g : (a, ∞) × [0, ∞) → R. A scale must satisfy certain properties that are given in [12] and will not be discussed here. The most important family of scale functions…”

“…The first one states that dimension is smaller than K or KS (depending on the case) and it only holds for i ≤ j in the space-bounded case. The proof is based in the following scaled dimension version of the Borel-Cantelli Lemma [12].…”

“…For example, classes such as SIZE(2 αn ) or SIZE (2 n α ) are not distinguished by unscaled dimension because they all have dimension 0. Scaled dimension precisely quantifies the difference among these circuit-size classes [12] and has several other applications in complexity theory [7,10,11,13,14,15].…”

Two Open Problems on Effective Dimension

“…Scaled dimension [9] are versions of resource-bounded dimension that have been "rescaled" to better fit the phenomena that they are measuring. They correspond to the concept of generalized dimension already suggested by Hausdorff.…”

“…We refer to [9] for a justification of the choice of these scales, related for instance to complexity classes such as SIZE(2 nα ) and SIZE(2 n α ).…”