Abstract
The notions of a Garden-of-Eden (GOE) configuration and mutually erasable configurations were discovered and investigated in the framework of Euclidean tessellations in the 1960s and 1970s. In 1993, Machi and Mignosi extended the GOE theorem for tessellation automata on Cayley graphs of non-exponential growth. We give an alternative proof to the GOE theorem for the discrete Heisenberg groups. Our proof has the advantage that it fully exploits the particular group structure so that a Moore-Myhill like tiling can be explicitly constructed. The tiling, as the wording suggests, is a spatially economic covering. The non-uniform packing method is introduced as a key technique for the construction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.C.S. Amoroso and Y. Patt, Some clarifications of the concept of a Garden-of-Eden configuration. J. Computer and System Sciences,10 (1975), 77–82.
M. Garzon, Models of Massive Parallelism, Springer, 1995.
A. Machì and F. Mignosi, Garden of Eden configurations for cellular automata on Cayley graphs of groups. SIAM J. Discrete Math.,6 (1993), 44–56.
E.F. Moore, Machine models of self-reproduction. Proceedings of a Symposium of the Applied Mathematical Society, Providence, R.I., 1962, 17–33.
J. Myhill, The converse to Moore’s Garden-of-Eden theorem. Proc. Amer. Math. Soc,14 (1963), 685–686.
J. Von Neumann, Theory of Self-reproducing Automata (Edited and completed by A.W. Burks). Univ. of Illinois Press, Urbana, 1966.
S. Yukita, Tessellation automata on free groups. Hiroshima Math. J.,25 (1995), 561–570. (first appeared as a Tech. Rep. 94-1-044, Department of Computer Software, The University of Aizu, 1994.)
S. Yukita, Tessellation automata on Fuchsian groups. Tech. Rep. 94-1-045, Department of Computer Software, The University of Aizu, 1994.
Author information
Authors and Affiliations
About this article
Cite this article
Yukita, S. The Moore-Myhill pseudo tiling for the Heisenberg tessellation automata. Japan J. Indust. Appl. Math. 16, 47 (1999). https://doi.org/10.1007/BF03167524
Received:
Revised:
DOI: https://doi.org/10.1007/BF03167524