Resource-bounded balanced genericity, stochasticity and weak randomness

Ambos-Spies, Klaus; Mayordomo, Elvira; Wang, Yongge; Zheng, Xizhong

doi:10.1007/3-540-60922-9_6

Cited by 17 publications

(73 citation statements)

References 17 publications

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“…The concept of strategy in [3] is different from the above one, but the concept of t(n)-martingale in [3] is equivalent to the above one, provided that we take DTIME(t(n)) in [3] as DTIME(O (t(n))). The concept of t(n)-martingale in [4,5] is not exactly equivalent to the above one, but the following relationship holds [2, [4,5]; and if A is t(n)-random in the sense of [4,5] then A is O ∼ (t(n))-random in the sense of Definition 2.…”

Section: A (Betting) Strategymentioning

confidence: 94%

“…The concept of t(n)-generic set in [4,3,5] is equivalent to that of t(n + 1)-generic set in the above definition.…”

Section: Resource-bounded Genericity Of Ambos-spies Et Almentioning

confidence: 96%

“…A counterpart of Martin-Löf randomness in this context is called t(n)-randomness. Pioneering works of this topic have been done by Lutz [13] (see also [14]), and extended by Ambos-Spies et al [2] (see also [3][4][5]). …”

Section: Resource-bounded Randomness and Genericitymentioning

confidence: 96%

“…A correction is given in [23] as follows. [23,Theorems 2,3].) (1) There exists a primitive recursive 1-Dowd set.…”

Section: Computability Of Dowd-type Generic Setsmentioning

confidence: 99%

“…Lutz [13,14] introduced resource-bounded randomness. Ambos-Spies et al [4,3] have studied resource-bounded randomness under time-bounds. …”

Section: Resource-bounded Randomnessmentioning

confidence: 99%

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Resource-bounded martingales and computable Dowd-type generic sets

Kumabe

Suzuki

2015

Information and Computation

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Martin-Löf defined algorithmic randomness, the use of martingales in this definition and several variants were explored by Schnorr, and Lutz introduced resource-bounded randomness. Ambos-Spies et al. have studied resource-bounded randomness under timebounds and have shown that resource-bounded randomness implies resource-bounded genericity. While the genericity of Ambos-Spies is based on time-bound on finite-extension strategy, the genericity of Dowd, the main topic of this paper, is based on an analogy of the forcing theorem. For a positive integer r, an oracle D is r-generic in the sense of Dowd (r-Dowd) if the following holds: If a certain formula F on an exponential-sized portion of D is a tautology then a polynomial-sized subfunction of D forces F to be a tautology. Here, r is the number of occurrences of query symbols in F . The relationship between resource-bounded randomness and Dowd-type genericity has been not clear so far. We show that there exists a primitive recursive function t(n) with the following property: Every t(n)-random set is r-Dowd for each positive integer r. A proof is done by means of constructing resource-bounded martingales. In our former paper, we left an open problem whether there exists a primitive recursive set which is r-Dowd for every positive integer r. In our recent work, we give an affirmative answer to the problem. The main theorem of the present paper gives an alternative proof of the affirmative answer.

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Section: A (Betting) Strategymentioning

confidence: 94%

“…The concept of t(n)-generic set in [4,3,5] is equivalent to that of t(n + 1)-generic set in the above definition.…”

Section: Resource-bounded Genericity Of Ambos-spies Et Almentioning

confidence: 96%

Section: Resource-bounded Randomness and Genericitymentioning

confidence: 96%

“…A correction is given in [23] as follows. [23,Theorems 2,3].) (1) There exists a primitive recursive 1-Dowd set.…”

Section: Computability Of Dowd-type Generic Setsmentioning

confidence: 99%

“…Lutz [13,14] introduced resource-bounded randomness. Ambos-Spies et al [4,3] have studied resource-bounded randomness under time-bounds. …”

Section: Resource-bounded Randomnessmentioning

confidence: 99%

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Resource-bounded martingales and computable Dowd-type generic sets

Kumabe

Suzuki

2015

Information and Computation

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On unstable and unoptimal prediction

Kalociński

Steifer

2019

Mathematical Logic Qtrly

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We consider the notion of prediction functions (or predictors) studied before in the context of randomness and stochasticity by Ko, and later by Ambos‐Spies and others. Predictor is a total computable function which tries to predict bits of some infinite binary sequence. The prediction error is defined as the limit of the number of incorrect answers divided by the number of answers given so far. We discuss indefiniteness of prediction errors for weak 1‐generics and show that this phenomenon affects certain c.e. sequences as well. On the other hand, a notion of optimal predictor is considered. It is shown that there is a sequence for which increasingly better predictors exist but for which no predictor is optimal.

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Randomness, stochasticity and approximations

Wang

1997

Randomization and Approximation Techniques in Computer Science

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Polynomial time unsafe approximations for intractable sets were introduced by Meyer and Paterson [9] and Yesha [19] respectively. The question of which sets have optimal unsafe approximations has been investigated extensively, see, e.g., [1,5,15,16]. Recently, Wang [15,16] showed that polynomial time random sets are neither optimally unsafe approximable nor -levelable. In this paper, we will show that: (1) There exists a polynomial time stochastic set in E2 which has an optimal unsafe approximation. (2). There exists a polynomial time stochastic set in E2 which i s -levelable. The above t w o results answer a question asked by Ambos-Spies and Lutz et al. [3]: Which kind of natural complexity property can be characterized by p-randomness but not by p-stochasticity? Our above results also extend Ville's [13] historical result. The proof of our rst result shows that, for Ville's stochastic sequence, we can nd an optimal betting strategy (prediction function) such that we will never lose our own money (except the money we h a v e earned), that is to say, if at the beginning we h a v e only one dollar and we always bet one dollar that the next selected bit is 1, then we always have enough money to bet on the next bit. Our second result shows that there is a stochastic sequence for which there is a betting strategy such that we will never lose our own money (except the money we h a v e earned), but there is no such kind of optimal betting strategy. That is to say, for any such kind of betting strategy, w e can nd another betting strategy which could be used to make money more quickly.

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Resource-bounded balanced genericity, stochasticity and weak randomness

Cited by 17 publications

References 17 publications

Resource-bounded martingales and computable Dowd-type generic sets

Resource-bounded martingales and computable Dowd-type generic sets

On unstable and unoptimal prediction

Randomness, stochasticity and approximations

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