A Generalization of Resource-Bounded Measure, with Application to the BPP vs. EXP Problem

Buhrman, Harry; Melkebeek, Dieter van; Regan, Kenneth W.; Sivakumar, D.; Strauss, Martin J.

doi:10.1137/s0097539798343891

Cited by 18 publications

(26 citation statements)

References 21 publications

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“…If we are successful (this in particular happens whenever w e k = A I k ), our reserved capital for this J e k is multiplied by 2 |J e k | , i.e. we now have for this J e k , a capital of 1 s k · (k + 1) 2 · 2 (|I k |/s k ) Replacing s k by its value (and remembering that |I k | ≥ 2 k−O (1) ), an elementary calculation shows that this quantity is greater than 1 for almost all k. Thus, our betting strategy succeeds on A. Indeed, for infinitely many k, A I k is an element of S k , hence for some e we will be successful in the above sub-strategy, making an amount of money greater than 1 for infinitely many k, hence our capital tends to infinity throughout the game.…”

Section: Understanding the Strength Of Injective Strategies: The Clasmentioning

confidence: 99%

Separations of non-monotonic randomness notions

Bienvenu

Hölzl

Kräling

et al. 2010

Journal of Logic and Computation

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Abstract. In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-Löf randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by betting on bits in order. However, computably random sequences can have properties that one may consider to be incompatible with being random, in particular, there are computably random sequences that are highly compressible. The concept of Martin-Löf randomness is much better behaved in this and other respects, on the other hand its definition in terms of martingales is considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr's model by also allowing non-monotonic strategies, i.e. strategies that do not bet on bits in order. The subsequent "non-monotonic" notion of randomness, now called Kolmogorov-Loveland-randomness, has been shown to be quite close to Martin-Löf randomness, but whether these two classes coincide remains a fundamental open question. In order to get a better understanding of non-monotonic randomness notions, Miller and Nies introduced some interesting intermediate concepts, where one only allows non-adaptive strategies, i.e., strategies that can still bet non-monotonically, but such that the sequence of betting positions is known in advance (and computable). Recently, these notions were shown by Kastermans and Lempp to differ from Martin-Löf randomness. We continue the study of the non-monotonic randomness notions introduced by Miller and Nies and obtain results about the Kolmogorov complexities of initial segments that may and may not occur for such sequences, where these results then imply a complete classification of these randomness notions by order of strength.

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Section: Understanding the Strength Of Injective Strategies: The Clasmentioning

confidence: 99%

Separations of non-monotonic randomness notions

Bienvenu

Hölzl

Kräling

et al. 2010

Journal of Logic and Computation

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“…The measure of the p-T-autoreducible sets in E is not known (see [5]). For more restrictive reducibilities, however, the class of autoreducible sets has measure 0 in E. Examples are the classes of the p-m-autoreducible sets and of the sets that are p-Tautoreducible via order-decreasing reductions, i.e., by reductions that on input x can only query their oracle at places y < x. Dimension in E allows us to distinguish the size of these two measure-0-classes in E. y E N} and let 2 be any set in DTIME(2") that is p-mequivalent to X .…”

Section: Corollary 15mentioning

confidence: 99%

Hausdorff dimension in exponential time

Ambos-Spies

Merkle

Reimann³

et al.

Proceedings 16th Annual IEEE Conference on Computational Complexity

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In this paper we investigate effective versions of Hausdo$ dimension which have been recently introduced byLutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immuniv. By a new general invariance theorem f o r resource-bounded dimension we show that the class of pm-complete sets for E has dimension 1 in E. Moreovel; we show that there are p-m-lower spans in E of dimension X ( P ) for any rational /3 between 0 and 1, where X ( P ) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz's concept of weak completeness. Finally we characterize resourcebounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions.In 1919, HAUSDORFF [8] introduced a theory of dimension which generalizes Lebesgue measure and, in particular, ramifies the structure of measure zero sets. Hausdog dimension, as it was called afterwards, found applications in many areas of mathematics such as number theory or dynamical systems (see for instance [7]). Probably bestknown is the role Hausdorff dimension plays in fractal geometry, where it turned out to be a suitable concept for distinguishing fractal sets from sets of a rather "smooth" geometrical nature.In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4]), a close connection between Hausdorff dimension and Kolmogorov complexity was established [6,16, 17,18,20, 211, and the notion of effective dimension was introduced [ 141. Recently, LUTZ [ 131 has extended the theory of effective dimension to complexity theory by introducing resource-bounded dimension. Like

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“…, X n−1 ), i.e., the σ-algebra generated by the first n projections. For a function f : 4 To be clear, P is the product measure on (2 N ) N of the product measure on 2 N of the fair-coin measure on 2 = {0, 1}. 5 As usual n∈N ωn n,k = (ωn) k where ·, · is a computable bijective pairing function.…”

Section: Doob Random Sequences Of Sequencesmentioning

confidence: 99%

Algorithmic randomness for Doob's martingale convergence theorem in continuous time

Kjos-Hanssen

Nguyen

Rute

2014

Logical Methods in Computer Science

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Abstract. We study Doob's martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, Doob randomness is strictly weaker than computable randomness and is incomparable with Schnorr randomness.

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A Generalization of Resource-Bounded Measure, with Application to the BPP vs. EXP Problem

Cited by 18 publications

References 21 publications

Separations of non-monotonic randomness notions

Separations of non-monotonic randomness notions

Hausdorff dimension in exponential time

Algorithmic randomness for Doob's martingale convergence theorem in continuous time

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