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101
Dimension is compression
 In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science
, 2005
"... Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomialspace and constructive dimension can be precisely cha ..."
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Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomialspace and constructive dimension can be precisely characterized in terms of Kolmogorov complexity, while analogous results for polynomialtime dimension haven’t been found. In this paper we remedy the situation by using the natural concept of reversible timebounded compression for finite strings. We completely characterize polynomialtime dimension in terms of polynomialtime compressors. 1
Constructive dimension and weak truthtable degrees
 In Computation and Logic in the Real World  Third Conference of Computability in Europe. SpringerVerlag Lecture Notes in Computer Science #4497
, 2007
"... Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with ..."
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Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1−ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the wtt degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor.
Gales and supergales are equivalent for defining constructive Hausdorff dimension
, 2002
"... We show that for a wide range of probability measures, constructive gales are interchangable with constructive supergales for defining constructive Hausdorff dimension, thus generalizing a previous independent result of Hitchcock [2] and partially answering an open question of Lutz [5]. 1 ..."
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We show that for a wide range of probability measures, constructive gales are interchangable with constructive supergales for defining constructive Hausdorff dimension, thus generalizing a previous independent result of Hitchcock [2] and partially answering an open question of Lutz [5]. 1
Diagonally nonrecursive functions and effective Hausdorff dimension
"... Abstract. We prove that every sufficiently slow growing DNR function computes a real with effective Hausdorff dimension one. Using a proof recently published by Kumabe and Lewis, it follows that there is a real of dimension one and minimal degree. Note that such a real cannot compute a MartinLöf ra ..."
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Abstract. We prove that every sufficiently slow growing DNR function computes a real with effective Hausdorff dimension one. Using a proof recently published by Kumabe and Lewis, it follows that there is a real of dimension one and minimal degree. Note that such a real cannot compute a MartinLöf random real. 1. Introducion Reimann and Terwijn asked the dimension extraction problem: can one effectively increase the information density of a sequence with positive information density? For a formal definition of information density, they used the notion of effective Hausdorff dimension. This effective version of the classical Hausdorff dimension of geometric measure theory was first defined by Lutz [9], using a martingale definition of Hausdorff dimension. Unlike classical dimension, it is possible for singletons to have positive dimension, and so Lutz defined the dimension dim(A) of a binary sequence A ∈ 2 ω to be the effective dimension of the singleton {A}. Later, Mayordomo [11] (but implicit in Ryabko [14]), gave a characterisation using Kolmogorov complexity: for all A ∈ 2 ω, K(A ↾ n) C(A ↾ n) dim(A) = lim inf = lim inf, n→ ∞ n n→ ∞ n where C is plain Kolmogorov complexity and K is the prefixfree version. Given this formal notion, the dimension extraction problem is the following: if dim(A)> 0, is there necessarily a B �T A such that dim(B)> dim(A)? The problem was recently solved by the second author [12], who showed that there is an A ∈ 2ω such that dim(A) = 1/2 and if B �T A, then dim(B) � 1/2. Even while it was still open, the dimension extraction problem spawned variations. The one we consider in the present paper is: Question 1.1. Is there an A ∈ 2ω such that dim(A) = 1 and A computes no MartinLöf random set? One motivation for this, and for the dimension extraction problem, is that the obvious ways to obtain nonrandom sets of positive dimension allow us to extract random sets. For example, we could take a random set X (whose dimension is always 1) and water it down by inserting zeros between the bits of X. This would give us a sequence of dimension 1/2; inserting zeros sparsely would result in a sequence of dimension 1. As long as the insertion positions are computable, we
Martingale Families and Dimension in P
"... We introduce a new measure notion on small complexity classes (called Fmeasure), 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 wi ..."
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We introduce a new measure notion on small complexity classes (called Fmeasure), 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. On larger complexity classes (E and above), Fmeasure is equivalent to Lutz resourcebounded measure. As applications to Fmeasure, 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 nonuniform complexity. We compare Fmeasure to previous notions and prove that martingale families are strictly stronger than Γmeasure [1], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial manyone reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resourcebounded dimension [15] on P, which meets the intuition behind Lutz’s notion. We show that Pdimension lies between finitestate dimension and dimension on E. We prove an analogue of a 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.
Points on computable curves
 In Proceedings of the FortySeventh Annual IEEE Symposium on Foundations of Computer Science
, 1999
"... The “analyst’s traveling salesman theorem ” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higherdimensional Euclidean spaces by Okikiolu (1992), says that a ..."
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The “analyst’s traveling salesman theorem ” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higherdimensional Euclidean spaces by Okikiolu (1992), says that a bounded set K is contained in some curve of finite length if and only if a certain “square beta sum”, involving the “width of K ” in each element of an infinite system of overlapping “tiles” of descending size, is finite. In this paper we characterize those points of Euclidean space that lie on computable curves of finite length. We do this by formulating and proving a computable extension of the analyst’s traveling salesman theorem. Our extension, the computable analyst’s traveling salesman theorem, says that a point in Euclidean space lies on some computable curve of finite length if and only if it is “permitted ” by some computable “Jones constriction”. A Jones constriction here is an explicit assignment of a rational cylinder to each of the abovementioned tiles in such a way that, when the radius of the cylinder corresponding to a tile is used in place of the “width of K ” in each tile, the square beta sum is finite. A point is permitted by a Jones constriction if it is
Every sequence is decompressible from a random one
 In Logical Approaches to Computational Barriers, Proceedings of the Second Conference on Computability in Europe, Springer Lecture Notes in Computer Science, volume 3988 of Computability in Europe
, 2006
"... ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of b ..."
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ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. It is shown that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, a new characterization of constructive dimension is given in terms of Turing reduction compression ratios.
Constructive dimension and Turing degrees
"... This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) ..."
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1 − ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dimH(S)/dimP(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truthtable and bounded Turing reductions differ in their ability to extract dimension.
A note on dimensions of polynomial size circuits
 Electronic Colloquium on Computational Complexity
, 2004
"... In this paper, we use resourcebounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3strong dimension 0. We also show that P/poly i.o. has p ..."
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In this paper, we use resourcebounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3strong dimension 0. We also show that P/poly i.o. has p
KolmogorovLoveland stochasticity and Kolmogorov
"... Abstract. Merkle et al. [11] showed that all KolmogorovLoveland stochastic ..."
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Abstract. Merkle et al. [11] showed that all KolmogorovLoveland stochastic