Abstract
We will show that the class of reversible cellular automata (CA) with right Lyapunov exponent 2 cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most \(2-\delta \) for some absolute constant \(\delta >0\). Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA F and a positive rational number \(\epsilon >0\), outputs the Lyapunov exponents of F with accuracy \(\epsilon \).
The work was partially supported by the Academy of Finland grant 296018 and by the Vilho, Yrjö and Kalle Väisälä Foundation.
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Notes
- 1.
The arrow markings are used as a shorthand for some coloring such that the heads and tails of the arrows in neighboring tiles match in a valid tiling.
- 2.
By performing more careful estimates it can be shown that \(\lambda ^{+}=1\), but we will not attempt to formalize the argument for this.
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Kopra, J. (2019). The Lyapunov Exponents of Reversible Cellular Automata Are Uncomputable. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_15
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