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Open AccessArticle

Statistical Complexity Analysis of Turing Machine tapes with Fixed Algorithmic Complexity Using the Best-Order Markov Model

1
Institute of Electronics and Informatics Engineering of Aveiro, University of Aveiro, 3810-193 Aveiro, Portugal
2
Department of Electronics, Telecommunications and Informatics, University of Aveiro, 3810-193 Aveiro, Portugal
3
Department of Virology, University of Helsinki, 00100 Helsinki, Finland
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(1), 105; https://doi.org/10.3390/e22010105
Received: 11 November 2019 / Revised: 7 January 2020 / Accepted: 13 January 2020 / Published: 16 January 2020
(This article belongs to the Special Issue Shannon Information and Kolmogorov Complexity)
Sources that generate symbolic sequences with algorithmic nature may differ in statistical complexity because they create structures that follow algorithmic schemes, rather than generating symbols from a probabilistic function assuming independence. In the case of Turing machines, this means that machines with the same algorithmic complexity can create tapes with different statistical complexity. In this paper, we use a compression-based approach to measure global and local statistical complexity of specific Turing machine tapes with the same number of states and alphabet. Both measures are estimated using the best-order Markov model. For the global measure, we use the Normalized Compression (NC), while, for the local measures, we define and use normal and dynamic complexity profiles to quantify and localize lower and higher regions of statistical complexity. We assessed the validity of our methodology on synthetic and real genomic data showing that it is tolerant to increasing rates of editions and block permutations. Regarding the analysis of the tapes, we localize patterns of higher statistical complexity in two regions, for a different number of machine states. We show that these patterns are generated by a decrease of the tape’s amplitude, given the setting of small rule cycles. Additionally, we performed a comparison with a measure that uses both algorithmic and statistical approaches (BDM) for analysis of the tapes. Naturally, BDM is efficient given the algorithmic nature of the tapes. However, for a higher number of states, BDM is progressively approximated by our methodology. Finally, we provide a simple algorithm to increase the statistical complexity of a Turing machine tape while retaining the same algorithmic complexity. We supply a publicly available implementation of the algorithm in C++ language under the GPLv3 license. All results can be reproduced in full with scripts provided at the repository. View Full-Text
Keywords: turing machines; information theory; statistical complexity; algorithmic complexity; computational complexity; Markov models; compression-based analysis turing machines; information theory; statistical complexity; algorithmic complexity; computational complexity; Markov models; compression-based analysis
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MDPI and ACS Style

Silva, J.M.; Pinho, E.; Matos, S.; Pratas, D. Statistical Complexity Analysis of Turing Machine tapes with Fixed Algorithmic Complexity Using the Best-Order Markov Model. Entropy 2020, 22, 105. https://doi.org/10.3390/e22010105

AMA Style

Silva JM, Pinho E, Matos S, Pratas D. Statistical Complexity Analysis of Turing Machine tapes with Fixed Algorithmic Complexity Using the Best-Order Markov Model. Entropy. 2020; 22(1):105. https://doi.org/10.3390/e22010105

Chicago/Turabian Style

Silva, Jorge M.; Pinho, Eduardo; Matos, Sérgio; Pratas, Diogo. 2020. "Statistical Complexity Analysis of Turing Machine tapes with Fixed Algorithmic Complexity Using the Best-Order Markov Model" Entropy 22, no. 1: 105. https://doi.org/10.3390/e22010105

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