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

TL;DR

This paper demonstrates a weakly universal weighted cellular automaton in the hyperbolic heptagrid with only 6 states, improving previous results that required 7 states, through a new implementation of tracks.

Contribution

It reduces the number of states needed for weak universality in the heptagrid from 7 to 6, with significant improvements in implementation complexity.

Findings

Reduced states from 7 to 6

Lowered maximum weight to 34

Decreased table entries from 160 to 137

Abstract

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.