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Decidability of Irreducible Tree Shifts of Finite Type

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Abstract

We reveal an algorithm for determining the complete prefix code irreducibility (CPC-irreducibility) of dyadic trees labeled by a finite alphabet. By introducing an extended directed graph representation of tree shift of finite type (TSFT), we show that the CPC-irreducibility of TSFTs is related to the connectivity of its graph representation, which is a similar result to one-dimensional shifts of finite type.

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Acknowledgements

The authors want to express their gratitude to the anonymous referees for their valuable comments and suggestions, which significantly improve the quality of this paper and make the paper more readable.

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Authors and Affiliations

  1. Department of Mathematical Sciences, National Chengchi University, Taipei, 11605, Taiwan, ROC

    Jung-Chao Ban

  2. Math. Division, National Center for Theoretical Science, National Taiwan University, Taipei, 10617, Taiwan, ROC

    Jung-Chao Ban

  3. Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 81148, Taiwan, ROC

    Chih-Hung Chang

  4. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30010, Taiwan, ROC

    Nai-Zhu Huang

  5. Department of Electrical and Computer Engineering, National Chiao Tung University, Hsinchu, 30010, Taiwan, ROC

    Yu-Liang Wu

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  1. Jung-Chao Ban
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  4. Yu-Liang Wu
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Correspondence to Chih-Hung Chang.

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Communicated by Alessandro Giuliani.

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This work is partially supported by the Ministry of Science and Technology, ROC (Contract No MOST 107-2115-M-259 -001 -MY2 and 107-2115-M-390 -002 -MY2).

Appendix: The Complexity of Algorithm

Appendix: The Complexity of Algorithm

In this appendix, we present the pseudo code of our algorithm and estimate the complexity of the algorithm.

The index start from 0 for the following pseudocode.

figure a

Suppose there are k convergent-edges, m divergent-edges, and n vertices in the extended directed graph. It is seen that the complexity of “isReachable” part is at most \(O(m+n+k)\). Since we have m divergent-edges, the complexity of our algorithm is at most

$$\begin{aligned} O((m+n+k) \cdot (m+(m-1)+\cdots +1))=O\left( (m+n+k) \cdot \dfrac{m(m-1)}{2}\right) = O(m^3). \end{aligned}$$

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Ban, JC., Chang, CH., Huang, NZ. et al. Decidability of Irreducible Tree Shifts of Finite Type. J Stat Phys 177, 1043–1062 (2019). https://doi.org/10.1007/s10955-019-02407-z

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