Abstract
Topological entropy is an important invariant of topological dynamical systems. It is often regarded as the measure of complexity of the system and can be used to tell non-conjugate systems apart from each other. We will show that the decision problem that asks whether the topological entropy is zero or not is undedicable in the class of reversible one-dimensional cellular automata. We will also show that some related decision problems are also undecidable in the setting of reversible cellular automata and reversible and complete Turing machines.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Boyle, M.: Open problems in symbolic dynamics. Contemp. Math. 469 (2008). https://doi.org/10.1090/conm/469/09161
Gajardo, A., Ollinger, N., Torres-Avilés, R.: Some undecidable problems about the trace-subshift associated to a Turing machine. Discrete Math. Theor. Comput. Sci. 17(2), 267–284 (2015). https://doi.org/10.46298/dmtcs.2137, https://hal.inria.fr/hal-01349052
Gajardo, A., Ollinger, N., Torres-Avilés, R.: The Transitivity Problem of Turing Machines (2015)
Guillon, P., Salo, V.: Distortion in one-head machines and cellular automata. In: Dennunzio, A., Formenti, E., Manzoni, L., Porreca, A.E. (eds.) AUTOMATA 2017. LNCS, vol. 10248, pp. 120–138. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58631-1_10
Hurd, L.P., Kari, J., Culik, K.: The topological entropy of cellular automata is uncomputable. Ergodic Theory Dynam. Systems 12(2), 255–265 (1992). https://doi.org/10.1017/S0143385700006738
Jeandel, E.: Computability of the entropy of one-tape Turing machines. In: Leibniz International Proceedings in Informatics, vol. 25. LIPIcs, February 2013. https://doi.org/10.4230/LIPIcs.STACS.2014.421
Kari, J., Ollinger, N.: Periodicity and Immortality in Reversible Computing. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 419–430. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85238-4_34
Kopra, J.: The Lyapunov exponents of reversible cellular automata are uncomputable. In: McQuillan, I., Seki, S. (eds.) UCNC 2019. LNCS, vol. 11493, pp. 178–190. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19311-9_15
Kůrka, P.: On topological dynamics of Turing machines. Theor. Comput. Sci. 174(1), 203–216 (1997). https://doi.org/10.1016/S0304-3975(96)00025-4, http://www.sciencedirect.com/science/article/pii/S0304397596000254
Salo, V.: Conjugacy of reversible cellular automata and one-head machines (2020). https://doi.org/10.48550/ARXIV.2011.07827, https://arxiv.org/abs/2011.07827
Acknowledgements
The author acknowledges the emmy.network foundation under the aegis of the Fondation de Luxembourg for its financial support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Hotanen, T. (2023). Undecidability of the Topological Entropy of Reversible Cellular Automata and Related Problems. In: Genova, D., Kari, J. (eds) Unconventional Computation and Natural Computation. UCNC 2023. Lecture Notes in Computer Science, vol 14003. Springer, Cham. https://doi.org/10.1007/978-3-031-34034-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-34034-5_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-34033-8
Online ISBN: 978-3-031-34034-5
eBook Packages: Computer ScienceComputer Science (R0)