Dimension in complexity classes

Abstract

We prove three results on the dimension structure of complexity classes.1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set X of languages in terms of the relativized resourcebounded dimensions of the individual elements of X, provided that the former resource bound is large enough to parameterize the latter. Thus for example, the dimension of a class X of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of X.2. Every language that is ≤ P m -reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is ≤ P m -hard for NP. 3. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.

Scaled Dimension and Nonuniform Complexity

“…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.…”

“…In early 2000, Lutz [14] developed resource-bounded dimension in order to remedy this situation. Just as resource-bounded measure is a complexity-theoretic generalization of classical Lebesgue measure, resource-bounded dimension is a complexity-theoretic generalization of classical Hausdorff dimension.…”

“…Moreover, just as classical Hausdorff dimension enables us to quantify the structures of many sets of Lebesgue measure 0, resource-bounded dimension enables us to quantify the structures of some sets that have measure 0 in complexity classes. For example, Lutz [14] showed that for every real number α ∈ [0, 1], the class SIZE α 2 n n has dimension α in ESPACE. He also showed that for every p-computable α ∈ [0, 1], the class of languages with limiting frequency α has dimension H(α) in E, where H is the binary entropy function of Shannon information theory.…”

“…Many classes that occur naturally in computational complexity are parametrized in such a way as to remain out of reach of the resource-bounded dimension of [14]. For example, when discussing cryptographic security or derandomization, one is typically interested in circuit-size bounds of the form 2 αn or 2 n α , rather than the α 2 n n bound of the above-cited result.…”

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AbstractResource-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.

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