Weak gardens of Eden for 1-dimensional tessellation automata
If T is the parallel map associated with a 1-dimensional tessellation automaton, then we say a configuration f is a weak Garden of Eden for T if f has no pre-image under T other than a shift of itself. Let WG(T)= the set of weak Gardens of Eden for T and G(T)= the set of Gardens of Eden (i.e., the s...
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| Language: | English |
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Wiley
1985-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171285000631 |
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| _version_ | 1832556265947004928 |
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| author | Michael D. Taylor |
| author_facet | Michael D. Taylor |
| author_sort | Michael D. Taylor |
| collection | DOAJ |
| description | If T is the parallel map associated with a 1-dimensional tessellation automaton, then we say a configuration f is a weak Garden of Eden for T if f has no pre-image under T other than a shift of itself. Let WG(T)= the set of weak Gardens of Eden for T and G(T)= the set of Gardens of Eden (i.e., the set of configurations not in the range of T). Typically members of WG(T)−G(T) satisfy an equation of the form Tf=Smf where Sm is the shift defined by (Smf)(j)=f(j+m). Subject to a mild restriction on m, the equation Tf=Smf always has a solution f, and all such solutions are periodic. We present a few other properties of weak Gardens of Eden and a characterization of WG(T) for a class of parallel maps we call (0,1)-characteristic transformations in the case where there are at least three cell states. |
| format | Article |
| id | doaj-art-35c0f09ed5e44b7cbbc12e83b796e3f8 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1985-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-35c0f09ed5e44b7cbbc12e83b796e3f82025-02-03T05:45:52ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251985-01-018357958710.1155/S0161171285000631Weak gardens of Eden for 1-dimensional tessellation automataMichael D. Taylor0Mathematics Department, University of Central Florida, Orlando, Florida, USAIf T is the parallel map associated with a 1-dimensional tessellation automaton, then we say a configuration f is a weak Garden of Eden for T if f has no pre-image under T other than a shift of itself. Let WG(T)= the set of weak Gardens of Eden for T and G(T)= the set of Gardens of Eden (i.e., the set of configurations not in the range of T). Typically members of WG(T)−G(T) satisfy an equation of the form Tf=Smf where Sm is the shift defined by (Smf)(j)=f(j+m). Subject to a mild restriction on m, the equation Tf=Smf always has a solution f, and all such solutions are periodic. We present a few other properties of weak Gardens of Eden and a characterization of WG(T) for a class of parallel maps we call (0,1)-characteristic transformations in the case where there are at least three cell states.http://dx.doi.org/10.1155/S0161171285000631cellular automatatessellation automataGardens of Edenparallel maps. |
| spellingShingle | Michael D. Taylor Weak gardens of Eden for 1-dimensional tessellation automata International Journal of Mathematics and Mathematical Sciences cellular automata tessellation automata Gardens of Eden parallel maps. |
| title | Weak gardens of Eden for 1-dimensional tessellation automata |
| title_full | Weak gardens of Eden for 1-dimensional tessellation automata |
| title_fullStr | Weak gardens of Eden for 1-dimensional tessellation automata |
| title_full_unstemmed | Weak gardens of Eden for 1-dimensional tessellation automata |
| title_short | Weak gardens of Eden for 1-dimensional tessellation automata |
| title_sort | weak gardens of eden for 1 dimensional tessellation automata |
| topic | cellular automata tessellation automata Gardens of Eden parallel maps. |
| url | http://dx.doi.org/10.1155/S0161171285000631 |
| work_keys_str_mv | AT michaeldtaylor weakgardensofedenfor1dimensionaltessellationautomata |