A New Dimension Sensitive Property for Cellular Automata

In this paper we study number-decreasing cellular automata. They form a super-class of standard number-conserving cellular automata. It is well-known that the property of being number-conserving is decidable in quasi-linear time. In this paper we prove that being number-decreasing is dimension sensitive i.e. it is decidable for one-dimensional cellular automata and undecidable for dimension 2 or greater. There are only few known examples of dimension sensitive properties for cellular automata and this denotes some rich panel of phenomena in this class. Introduction Number-conserving cellular automata (NCA) were introduced in [13] as an alternative to differential equations for modeling highway car traffic. The huge literature (for instance [4, 5, 6, 2]) published since then is a testimony to the great interest they received from the scientific community gathering together physicists, computer scientists and mathematicians. In fact, the NCA model applies to all phenomena governed by conservation laws of mass or energy. In [5], the authors proved that the property of being number-conserving is decidable in all dimensions. This is quite surprising since almost all non-trivial properties on cellular automata are known to be undecidable [12, 15, 16, 8, 14]. In this paper we focus on number-decreasing cellular automata, a variant of NCA that can be used for modeling systems in which the total mass or energy do not increase during the evolution. Number-decreasing cellular automata and how they relate to particle automata has been studied in [3]. ∗A shorter version of this paper was presented at the MFCS 2004 conference, see [1]. †LIF, CNRS Université de Provence, 39 rue F. Joliot Curie 13453 Marseille Cedex 13, France. Email: {Vincent.Bernardi,Bruno.Durand}@lif.univ-mrs.fr ‡I3S, UNSA CNRS, 2000, route des lucioles, Les Algorithmes bat Euclide B, BP 121, 06903 Sophia Antipolis Cedex, France. Email: [email protected] §Department of Mathematics, FIN-20014, University of Turku, Finland. Email: [email protected] ¶Research supported by the Academy of Finland grant 54102