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The Sum 2KM(x)−K(x) Over All Prefixes x of Some Binary Sequence Can be Infinite

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Abstract

We consider two quantities that measure complexity of binary strings: K M(x) is defined as the negative logarithm of continuous a priori probability on the binary tree, and K(x) denotes prefix complexity of a binary string x. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence ω such that the sum of 2KM(x)−K(x) over all prefixes x of ω is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.

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Notes

  1. It goes back to P. Gács paper [10], but Gács used a different and rather cumbersome notation there. See [1] for the detailed exposition.

  2. The summation is stopped at the same level N, so the tree height is less by 1 and we can apply the induction assumption. The base of induction is trivial: in the root the ratio m/a ′ is at most 1 for evident reasons.

  3. The weight of vertex 0 in the strategy is equal to the desired increase in k; there are no deep reasons for this choice, but it simplifies the computations.

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  1. Moscow State University, Moscow, Russian Federation

    Mikhail Andreev & Akim Kumok

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  1. Mikhail Andreev
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  2. Akim Kumok
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Correspondence to Mikhail Andreev.

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Andreev, M., Kumok, A. The Sum 2KM(x)−K(x) Over All Prefixes x of Some Binary Sequence Can be Infinite. Theory Comput Syst 58, 424–440 (2016). https://doi.org/10.1007/s00224-014-9604-2

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