Number-conserving cellular automata with a von Neumann neighborhood of range one

Abstract

We present necessary and sufficient conditions for a cellular automaton with a von Neumann neighborhood of range one to be number-conserving. The conditions are formulated for any dimension and for any set of states containing zero. The use of the geometric structure of the von Neumann neighborhood allows for computationally tractable conditions even in higher dimensions.

A split-and-perturb decomposition of number-conserving cellular automata

“…In [27], using a novel approach based on a geometric analysis of the structure of the von Neumann neighborhood in higher dimensions, necessary and sufficient conditions for a d-dimensional CA to be number-conserving are formulated in terms of the local rule in a similar way as in [4]. These conditions apply for any state set Q Ă R, whether it is finite or not.…”

“…These conditions apply for any state set Q Ă R, whether it is finite or not. The main result presented in [27] allows to find all two-dimensional three-state number-conserving CAs [10] and all two-dimensional six-state rotation-symmetric number-conserving CAs [11]. Moreover, it allows to describe all affine continuous density-conserving CAs with state set Q " r0, 1s, which is infinite [8].…”

Exploração do Espaço de Regras Conservativas Binárias com Vizinhança de Moore

“…Tambémé demonstrado o que as regras devem respeitar para serem consideradas conservativas em dimensões d > 1 com vizinhança hiper-retangular. Em [Wolnik et al 2017], foram apresentadas as condições necessárias e suficientes para que ACs de dimensões e de estados arbitrários em vizinhança de von Neumann sejam conservativos. Em [Wolnik and De Baets 2019a], foi provado que todos os ACs binários conservativos são intrinsecamente unidimensionais em vizinhança de von Neumann, independente de suas dimensões, istoé, sempre existirá 4d + 1 regras conservativas, sendo elas a identidade, os deslocamentos e as regras de tráfego.…”

Coarse Graining of Partitioned Cellular Automata

“…Since the focus here is to study emergence in physics, where the properties of conservation and reversibility play an important role, we employ a cellular automata class known as partitioned cellular automata (PCA). Although the notions of reversibility and/or conservation are present in the context of CAs [14][15][16][17] these properties can be achieved more easily in the PCA or block automaton, as proposed by Toffoli and Margolus [18] and further developed by Morita [14]. By employing a PCA, the concepts of reversibility and conservation become straightforward.…”