The Most General Conservation Law for a Cellular Automaton

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.

• Type: Conference
• Year: 2008
• Where: CSR
• Authors: Enrico Formenti, Jarkko Kari, Siamak Taati
• Keywords: Conservation Law, CSR 2008, Real-valued Conservation Laws, Semigroup-valued Conservation Laws, Theoretical Computer Science