Effective Strong Dimension in Algorithmic Information and Computational Complexity

Athreya, Krishna B.; Hitchcock, John M.; Lutz, Jack H.; Mayordomo, Elvira

doi:10.1007/978-3-540-24749-4_55

Cited by 94 publications

(244 citation statements)

References 37 publications

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“…More properties of these finite-state dimensions, including their relationships to classical Hausdorff and packing dimensions, respectively, may be found in [10,3].…”

Section: It Is Easy To Verify Thatmentioning

confidence: 99%

Dimensions of Copeland-Erdös Sequences

Lutz

Moser

2005

Lecture Notes in Computer Science

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The base-k Copeland-Erdös sequence given by an infinite set A of positive integers is the infinite sequence CE k (A) formed by concatenating the base-k representations of the elements of A in numerical order. This paper concerns the following four quantities.• The finite-state dimension dim FS (CE k (A)), a finite-state version of classical Hausdorff dimension introduced in 2001.• The finite-state strong dimension Dim FS (CE k (A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of dim• The zeta-dimension Dim ζ (A), a kind of discrete fractal dimension discovered many times over the past few decades.• The lower zeta-dimension dim ζ (A), a dual of Dim ζ (A) satisfying dim ζ (A) ≤ Dim ζ (A).We prove the following.. This extends the 1946 proof by Copeland and Erdös that the sequence CE k (PRIMES) is Borel normal. Dim FS (CE3. These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in [0, 1] satisfying the four above-mentioned inequalities.

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“…More properties of these finite-state dimensions, including their relationships to classical Hausdorff and packing dimensions, respectively, may be found in [10,3].…”

Section: It Is Easy To Verify Thatmentioning

confidence: 99%

Dimensions of Copeland-Erdös Sequences

Lutz

Moser

2005

Lecture Notes in Computer Science

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“…In particular, for any polynomial q(n) ≥ n 2 , dim (1) (KT q (2 αn )|ESPACE) = dim (1) (KS q (2 αn )|ESPACE) = α, and dim (2) (KT q (2 n α )|ESPACE) = dim (2) (KS q (2 n α )|ESPACE) = α.…”

Section: Nonuniform Complexitymentioning

confidence: 99%

“…These preliminary results suggest new relationships between information and complexity and open the way for investigating the fractal structure of complexity classes. More recent work has already used resource-bounded dimension to illuminate a variety of topics in computational complexity [1,2,6,8,9,3].…”

Section: Introductionmentioning

confidence: 99%

Scaled Dimension and Nonuniform Complexity

Hitchcock

Lutz

Mayordomo³

2003

Automata, Languages and Programming

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

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“…Recently resource-bounded measure has been refined via effective dimension which is an effectivization of Hausdorff dimension, yielding applications in a variety of topics, including algorithmic information theory, computational complexity, prediction, and data compression [13,17,14,4,2,6].…”

Section: Introductionmentioning

confidence: 99%

Martingale Families and Dimension in P

Moser

2006

Logical Approaches to Computational Barriers

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We introduce a new measure notion on small complexity classes (called F -measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0's and 1's. As applications to F -measure, we answer a question raised in [1] by improving their result to: for almost every language A decidable in subexponential time, P A = BPP A . We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F -measure to previous notions and prove that martingale families are strictly stronger than Γ-measure [1], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [13] on P, which meets the intuition behind Lutz's notion. We show that P-dimension lies between finitestate dimension and dimension on E. We prove an analogue to the Theorem of Eggleston in P, i.e. the class of languages whose characteristic sequence contains 1's with frequency α, has dimension the Shannon entropy of α in P.

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Effective Strong Dimension in Algorithmic Information and Computational Complexity

Cited by 94 publications

References 37 publications

Dimensions of Copeland-Erdös Sequences

Dimensions of Copeland-Erdös Sequences

Scaled Dimension and Nonuniform Complexity

Martingale Families and Dimension in P

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