Classification of two-dimensional binary cellular automata with respect to surjectivity

TL;DR

This paper develops a classification method for two-dimensional binary cellular automata based on surjectivity, using tests like the balance theorem and introduces slice permutivity, achieving complete classification for small neighborhoods.

Contribution

It introduces the concept of slice permutivity and applies a systematic testing approach to classify 2D CA rules by surjectivity, covering neighborhoods up to five sites.

Findings

Complete classification for neighborhoods with fewer than five sites.

Surjectivity can be determined for certain five-site neighborhoods, including specific pentomino shapes.

Slice permutivity implies surjectivity in 2D cellular automata.

Abstract

While the surjectivity of the global map in two-dimensional cellular automata (2D CA) is undecidable in general, in specific cases one can often decide if the rule is surjective or not. We attempt to classify as many 2D CA as possible by using a sequence of tests based on the balance theorem, injectivity of the restriction to finite configurations, as well as permutivity. We introduce the notion of slice permutivity which is shown to imply surjectivity in 2D CA. The tests are applied to 2D binary CA with neighbourhoods consisting of up to five sites, considering all possible contiguous shapes of the neighbourhood. We find that if the size of the neighbourhood is less than five, complete classification of all rules is possible. Among 5-site rules, those with von Neuman neighbourhoods as well as neighbourhoods corresponding to T, V, and Z pentominos can also be completely classified.