Kolmogorov Complexity and Algorithmic
                    Randomness

Shen, Alexander; Uspensky, Vladimir; Vereshchagin, Nikolai K.

doi:10.1090/surv/220

Cited by 92 publications

(121 citation statements)

References 51 publications

(78 reference statements)

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“…One area where both p-values and e-values have been used for a long time is the algorithmic theory of randomness (see, e.g., Shen et al (2017)), which originated in Kolmogorov's work on the algorithmic foundations of probability and information (Kolmogorov, 1965(Kolmogorov, , 1968. Martin-Löf (1966) introduced an algorithmic version of p-values, and then Levin (1976) introduced an algorithmic version of e-values.…”

Section: Introductionmentioning

confidence: 99%

Combining e-values and p-values

Vovk¹,

Wang

2019

SSRN Journal

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Section: Introductionmentioning

confidence: 99%

Combining e-values and p-values

Vovk¹,

Wang

2019

SSRN Journal

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“…For this reason, we define algorithmic randomness in CTMCs using the martingale (betting strategy) approach of Schnorr [18]. This approach extends to other levels of randomness in a straightforward manner, while our present state (i.e., lack) of knowledge in computational complexity theory does not allow us to extend other approaches (e.g., Martin-Löf tests or Kolmogorov complexity, which are known to be equivalent to the martingale approach at the algorithmic level [13,4,17,22]) to time-bounded complexity classes.…”

Section: Introductionmentioning

confidence: 91%

Algorithmic Randomness in Continuous-Time Markov Chains

Huang

Lutz

Migunov

2019

2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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In this paper we develop the elements of the theory of algorithmic randomness in continuous-time Markov chains (CTMCs). Our main contribution is a rigorous, useful notion of what it means for an individual trajectory of a CTMC to be random. CTMCs have discrete state spaces and operate in continuous time. This, together with the fact that trajectories may or may not halt, presents challenges not encountered in more conventional developments of algorithmic randomness.Although we formulate algorithmic randomness in the general context of CTMCs, we are primarily interested in the computational power of stochastic chemical reaction networks, which are special cases of CTMCs. This leads us to embrace situations in which the long-term behavior of a network depends essentially on its initial state and hence to eschew assumptions that are frequently made in Markov chain theory to avoid such dependencies.After defining the randomness of trajectories in terms of a new kind of martingale (algorithmic betting strategy), we prove equivalent characterizations in terms of constructive measure theory and Kolmogorov complexity. As a preliminary application we prove that, in any stochastic chemical reaction network, every random trajectory with bounded molecular counts has the non-Zeno property that infinitely many reactions do not occur in any finite interval of time.

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“…where U is a fixed universal prefix Turing machine, and π is the length of a binary "program π for x." Extensive discussions of the history and intuition behind this notion, including its essential invariance with respect to the choice of the universal Turing machine U , may be found in any of the standard texts [23,11,40,41]. By routine encoding we extend this notion to let x range over various countable sets, so that K(x) is well defined when x is an element of N, Q, Q n , etc.…”

Section: Algorithmic Information and Algorithmic Dimensionsmentioning

confidence: 99%