Recursive Computational Depth

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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“…Theorem 4 (Lathrop and Lutz [4], Muchnik [10]). For every computable order h, there is a sequence A ∈ TMR such that, for all n ∈ N,…”

Section: Building a Sequence In Tmr Of Low Complexitymentioning

confidence: 99%

Separations of non-monotonic randomness notions

Bienvenu

Hölzl

Kräling

et al. 2010

Journal of Logic and Computation

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“…Our goal is to study polynomial versions of Bennett's original depth notion [7], and its recursive version called recursive depth [11]. Recursive depth [11] is defined in terms of recursive observers competing against recursive observers, i.e. ∆ and ∆ ′ have the same power.…”

Section: Compression: For Allmentioning

confidence: 99%

“…∆ and ∆ ′ can be the same class e.g. for recursive depth [11], ∆ = ∆ ′ are recursive time bounds, or different classes e.g. for Bennett's depth [7], ∆ are recursive time bounds but ∆ ′ is unbounded Kolmogorov complexity.…”

Section: Introductionmentioning

confidence: 99%

On the polynomial depth of various sets of random strings

Moser

2013

Theoretical Computer Science

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This paper proposes new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions i.e., both trivial and random sequences are not monotone poly deep, monotone poly depth satisfies the slow growth law i.e., no simple process can transform a non deep sequence into a deep one, and monotone poly deep sequences exist (unconditionally).We give two natural examples of deep sets, by showing that both the set of Levinrandom strings and the set of Kolmogorov random strings are monotone poly deep. 1 complexity, nonuniform circuit complexity, and efficient search for satisfying assignments to Boolean formulas. In [9], both a notion of finite-state and polynomial depth were investigated, and the depth of polynomial weakly useful languages was shown.

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“…Intuitively a sequence is strongly deep if no computable time bound is enough to compress infinitely many of its prefixes to within a constant number of bits of its smallest representation. An interpretation of strongly deep objects is given in [LL99]; a strongly deep sequence is analogous to a great work of literature for which no number of readings suffices to exhaust its value. Subsequently Judes, Lathrop, and Lutz [JLL94] extended Bennett's work defining the classes of weakly useful sequences.…”

Section: Introductionmentioning

confidence: 99%

“…Later, Fenner et al [FLMR05] proved that there exist sequences that are weakly useful but not strongly useful. Lathrop and Lutz [LL99] introduced refinements (named recursive weak depth and recursive strong depth) of Bennett's notion of weak and strong depth, and studied its fundamental properties, showing that recursively weakly (resp. strongly) deep sequences form a proper subclass of the class of weakly (resp.…”

Section: Introductionmentioning

confidence: 99%

Depth as Randomness Deficiency

et al. 2009

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Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program in absence of auxiliary information. Variants known as logical depth and computational depth are expressed in Kolmogorov complexity theory.We derive quantitative relation between logical depth and computational depth and unify the different depth notions by relating them to A. Kolmogorov and L. Levin's fruitful notion of randomness deficiency. Subsequently, we revisit the computational depth of infinite strings, introducing the notion of super deep sequences and relate it with other approaches.

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Recursive Computational Depth

Cited by 28 publications

References 21 publications

Separations of non-monotonic randomness notions

Separations of non-monotonic randomness notions

On the polynomial depth of various sets of random strings

Depth as Randomness Deficiency

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