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Issue title: Special issue on DLT'04
Subtitle: Horse Power Problem
Article type: Research Article
Authors: Nishio, Hidenosuke | Margenstern, Maurice | von Haeseler, Friedrich
Affiliations: Iwakura Miyake-cho 204, Sakyo-ku, 606-0022 Kyoto, Japan. E-mail: [email protected] | LITA, EA 3097, UFR MIM, University of Metz, Île du Saulcy, 57045 Metz, France. E-mail: [email protected] | KU Leuven, Dep. of Electrical Engineering, Kasteelpark Arenberg 10, 3001 Leuven, Belgium. E-mail: [email protected]
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.
Keywords: cellular automaton, neighborhood, horse power problem, finitely generated group, semigroup, hexagonal grid, hyperbolic plane
Journal: Fundamenta Informaticae, vol. 78, no. 3, pp. 397-416, 2007
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