Almost everywhere high nonuniform complexity
Journal of Computer and System Sciences
Completeness and weak completeness under polynomial-size circuits
Information and Computation
MAX3SAT is exponentially hard to approximate if NP has positive dimension
Theoretical Computer Science
COLT '02 Proceedings of the 15th Annual Conference on Computational Learning Theory
COCO '98 Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity
Hausdorff Dimension in Exponential Time
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Note: fractal dimension and logarithmic loss unpredictability
Theoretical Computer Science
Dimension in Complexity Classes
SIAM Journal on Computing
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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.