On Algebraic Structure of Neighborhoods of Cellular Automata: Horse Power Problem

  • Authors:

  • Hidenosuke Nishio;Maurice Margenstern;Friedrich von Haeseler

  • Affiliations:

  • Iwakura Miyake-cho 204, Sakyo-ku, 606-0022 Kyoto, Japan. E-mail: YRA05762@nifty.ne.jp;LITA, EA 3097, UFR MIM, University of Metz, Île du Saulcy, 57045 Metz, France. E-mail: margens@sciences.univ-metz.fr;KU Leuven, Dep. of Electrical Engineering, Kasteelpark Arenberg 10, 3001 Leuven, Belgium. E-mail: friedrich.vonHaeseler@esat.kuleuven.ac.be

  • Venue:
  • Fundamenta Informaticae - Special issue on DLT'04
  • Year:
  • 2007

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Abstract

In a previous paper we formulated and analyzed the structure of neighborhoods of cellular automata in an algebraic setting such that the cellular space S is represented by the Cayley graph of a finitely generated group and the neighbors are defined as a semigroup generated by the neighborhood N as a subset of S, Nishio and Margenstern 2004 [14,15]. Particularly we discussed the horse power problem whether the motion of a horse (knight) fills the infinite chess board or Z$^2$ - that is, an algebraic problem whether a subset of a group generates it or not. Among others we proved that a horse fills Z$^2$ even when its move is restricted to properly chosen 3 directions and gave a necessary and sufficient condition for a generalized 3-horse to fill Z$^2$. This paper gives further developments of the horse power problem, say, on the higher dimensional Euclidean grid, the hexagonal grid and the hyperbolic plane.