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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Inverse and injectivity of parallel relations induced by cellular automata


Author: Takeo Yaku
Journal: Proc. Amer. Math. Soc. 58 (1976), 216-220
MSC: Primary 94A30
DOI: https://doi.org/10.1090/S0002-9939-1976-0444346-6
MathSciNet review: 0444346
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Abstract: Moore and Myhill showed that Garden-of-Eden theorem [2], [3]. A binary relation over the configurations is said to be ``parallel'' if it is induced by a cellular (tessellation) automaton. Richardson showed the equivalence between a parallel relation (a nondeterministic parallel map) with the quiescent state to be injective and its inverse to be parallel by the Garden-of-Eden theorem plus compactness of product topology [4]. This paper deals with the inverse and the injectivity when a cellular automaton is given that induces a parallel relation. We give an equivalent condition, concerning only the local map, for the inverse of a parallel relation to be parallel. Furthermore we show an equivalent condition, concerning only the local map, for the injectivity of a parallel map. Consequently, we note that these two conditions are represented by semirecursive predicates.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0444346-6
Keywords: Cellular automata, tessellation structures, parallel maps, parallel relations, self-reproducing, Garden-of-Eden
Article copyright: © Copyright 1976 American Mathematical Society