Finite-state dimension

Dai, Jack Jie; Lathrop, James I.; Lutz, Jack H.; Mayordomo, Elvira

doi:10.1016/s0304-3975(03)00244-5

Cited by 84 publications

(128 citation statements)

References 17 publications

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“…we prove that the class of languages with asymptotic frequency α have strong dimension the Shannon entropy of α in P. Analogue version of the theorem of Eggleston have been proved for various resource bounds [4,13].…”

Section: Application: Connecting Frequency and Shannon Entropymentioning

confidence: 77%

“…Finite-state dimension is defined via martingale computable by finite-state machines (called FSG: finite-state gamblers), see [4] for more details.…”

Section: A8 Proof Of Theorem 42mentioning

confidence: 99%

“…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%

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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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Section: Application: Connecting Frequency and Shannon Entropymentioning

confidence: 77%

“…Finite-state dimension is defined via martingale computable by finite-state machines (called FSG: finite-state gamblers), see [4] for more details.…”

Section: A8 Proof Of Theorem 42mentioning

confidence: 99%

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Martingale Families and Dimension in P

Moser

2006

Logical Approaches to Computational Barriers

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“…[10]. The finite-state dimension of S is dim FS (S) = inf {s | there is an FSG that s-succeeds on S } .…”

Section: Definition a Finite-state Gambler (Fsg) Is A 5-tuplementioning

confidence: 99%

“…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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Dimension Measure for Complex Networks

Shanker

2013

Advances in Network Complexity

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Finite-state dimension

Cited by 84 publications

References 17 publications

Martingale Families and Dimension in P

Martingale Families and Dimension in P

Dimensions of Copeland-Erdös Sequences

Dimension Measure for Complex Networks

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