A4 Article in conference proceedings

Conjugacy of one-dimensional one-sided cellular automata is undecidable

List of Authors: Joonatan Jalonen, Jarkko Kari

Conference name: International Conference on Current Trends in Theory and Practice of Computer Science

Publisher: Springer Verlag

Publication year: 2018

Journal: Lecture Notes in Computer Science

Book title *: SOFSEM 2018: Theory and Practice of Computer Science

Journal name in source: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volume number: 10706

ISBN: 978-3-319-73116-2

eISBN: 978-3-319-73117-9

ISSN: 0302-9743

DOI: http://dx.doi.org/10.1007/978-3-319-73117-9_16

Two cellular automata are strongly conjugate if there exists a shift-commuting conjugacy between them. We prove that the following two sets of pairs (

*F*,

*G*) of one-dimensional one-sided cellular automata over a full shift are recursively inseparable:

(i) pairs where *F* has strictly larger topological entropy than *G*, and

(ii) pairs that are strongly conjugate and have zero topological entropy.

Because there is no factor map from a lower entropy system to a higher entropy one, and there is no embedding of a higher entropy system into a lower entropy system, we also get as corollaries that the following decision problems are undecidable: Given two one-dimensional one-sided cellular automata *F* and *G* over a full shift: Are *F* and *G* conjugate? Is *F* a factor of *G*? Is *F* a subsystem of *G*? All of these are undecidable in both strong and weak variants (whether the homomorphism is required to commute with the shift or not, respectively). It also immediately follows that these results hold for one-dimensional two-sided cellular automata.

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