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(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 i ^th 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 $SIZE\left( {2^{n^\alpha } } \right)$, $KT^q \left( {2^{n^\alpha } } \right)$, and $KS^q \left( {2^{n^\alpha } } \right)$ have 2^nd-order dimension α in ESPACE.

• 3. The classes KT^q(2ⁿ(1 − 2^−αn)) and KS^q(2ⁿ(1 − 2^−αn) have −1^st-order dimension α in ESPACE.

This research was supported in part by National Science Foundation Grant 9988483.

This research was supported in part by National Science Foundation Grants 9610461 and 9988483.

This research was supported in part by Spanish Government MEC projects PB98-0937-C04-02 and TIC98-0973-C03-02. It was done while visiting Iowa State University.