A weakly universal weighted cellular automaton in the heptagrid with 6 states

In this paper we prove that there is a weakly universal weighted cellular automaton in the heptagrid, the tessellation {7,3} of the hyperbolic plane, with 6 states. The present paper improves the same result deposited on arXiv:2301.10691v1 and also arXiv:2301.10691v2. In the deposited papers, the result is proved with 7 states. In the present replacement the number of states is reduced to 6. Such a reducing is not trivial and requires substantial changes in the implementation. The maximal weight is now 34, a very strong reduction with the best result with 7 states. Also, the table has 137 entries, signifcantly less than the 160 entries of the paper with 7 states. The reduction is obtained by a new implementation of the tracks which play a key role as far as without tracks there is no computational universality result.Comment: 37 pages, 21 figures. arXiv admin note: substantial text overlap with arXiv:2108.13094, arXiv:2104.01561, arXiv:2107.0484

A weakly universal cellular automaton in the pentagrid with five states

In this paper, we construct a cellular automaton on the pentagrid which is planar, weakly universal and which have five states only. This result much improves the best result which was with nine statesComment: 23 pages, 21 figure

A strongly universal cellular automaton on the heptagrif with seven states, new proof

In this paper, we prove that there is a strongly universal cellular automaton on the heptagrid with seven states which is rotation invariant. This improves a previous paper of the author with the same number of states. Here, the structures are simpler and the number of rules is much less.Comment: 33 pages, 26 figure