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Exact Constructive and Computable Dimensions

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Abstract

In this paper we derive several results which generalise the constructive dimension of (sets of) infinite strings to the case of exact dimension. We start with proving a martingale characterisation of exact Hausdorff dimension. Then using semi-computable super-martingales we introduce the notion of exact constructive dimension of (sets of) infinite strings. This allows us to derive several bounds on the complexity functions of infinite strings, that is, functions assigning to every finite prefix its Kolmogorov complexity. In particular, it is shown that the exact Hausdorff dimension of a set of infinite strings lower bounds the maximum complexity function of strings in this set. Furthermore, we show a result bounding the exact Hausdorff dimension of a set of strings having a certain computable complexity function as upper bound. Obviously, the Hausdorff dimension of a set of strings alone without any computability constraints cannot yield upper bounds on the complexity of strings in the set. If we require, however, the set of strings to be Σ2-definable several results upper bounding the complexity by the exact Hausdorff dimension hold true. First we prove that for a Σ2-definable set with computable dimension function one can construct a computable – not only semi-computable – martingale succeeding on this set. Then, using this result, a tight upper bound on the prefix complexity function for all strings in the set is obtained.

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Notes

  1. In the recent monograph [6] ω-words are referred to as sets and ω-languages as classes. In this paper we reserve the term “set” to the original meaning as introduced by Georg Cantor.

  2. In fact, since we are only interested in the values \(h(r^{-n}),\ n\in \mathbb {N}\), the requirement of right continuity is just to conform with the usual meaning (cf. [11, 27]).

  3. Observe that h ′≤ h implies h ≼ h ′.

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  1. Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, von-Seckendorff-Platz 1, 06099, Halle (Saale), Germany

    Ludwig Staiger

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This article is part of the Topical Collection on Special Issue on Computability, Complexity and Randomness (CCR 2015)

The results of this paper were presented at the conference “Varieties of Algorithmic Information 2015”, June 15–18, 2015, Heidelberg, Germany.

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Staiger, L. Exact Constructive and Computable Dimensions. Theory Comput Syst 61, 1288–1314 (2017). https://doi.org/10.1007/s00224-017-9790-9

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