ESPACE

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(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 dimension of X in suitable complexity classes. The 0th-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) ≥ n2. 1. The class SIZE(2αn) and the time- and space-bounded Kolmogorov complexity classes KTq(2αn) and KSq(2αn) have 1st-order dimension α in ESPACE. 2. The classes SIZE(2nα), KTq(2nα), and KSq(2nα) have 2nd-order dimension α in ESPACE. 3. The classes KTq(2n(1-2-αn)) and KSq(2n(1-2-αn have -1st- order dimension α in ESPACE.