The most general conservation law for a cellular automaton

Authors:
Enrico Formenti;Jarkko Kari;Siamak Taati
Affiliations:
Université de Nice-Sophia Antipolis, Départment d'Informatique, Nice Cedex 2, France;Department of Mathematics, University of Turku, Turku, Finland;Department of Mathematics, University of Turku, Turku, Finland and Turku Centre for Computer Science, Turku, Finland
Venue:
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Year:
2008

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Abstract

We study the group-valued and semigroup-valued conservation laws in cellular automata (CA). We provide examples to distinguish between semigroup-valued, group-valued and real-valued conservation laws. We prove that, even in one-dimensional case, it is undecidable if a CA has any non-trivial conservation law of each type. For a fixed range, each CA has a most general (group-valued or semigroup-valued) conservation law, encapsulating all conservation laws of that range. For one-dimensional CA the semigroup corresponding to such a most general conservation law has an effectively constructible finite presentation, while for higher-dimensional ones no such effective construction exists.