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 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) ≥ n 2.
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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.
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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 2nd-order dimension α in ESPACE.
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3.
The classes KTq(2n(1 − 2−αn)) and KSq(2n(1 − 2−αn) have −1st-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.
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Hitchcock, J.M., Lutz, J.H., Mayordomo, E. (2003). Scaled Dimension and Nonuniform Complexity. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_24
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DOI: https://doi.org/10.1007/3-540-45061-0_24
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