We illustrate a way of representating finite languages in a geometrical way, that comes from distributed computed [1].

By a language, we mean a set of strings of a given length \(n\) on a finite alphabet \(A\). It can be represented by a simplicial complex whose maximal simplices have \(n\) vertices and represent the elements of the language. Each vertex has a color, which is a position in \(\{0,\ldots,n-1\}\), and a label, which is a letter in \(A\). Each string in \(x\in L\) gives rise to a simplex containing, for each position \(i\), the vertex with color \(i\) and label \(x_i\).

We show the simplicial complex representations of a few languages (for the moment there is only one language, the list will progressively expand...).

[1] Maurice Herlihy, Dmitry Kozlov, and Sergio Rajsbaum. Distributed Computing Through Combinatorial Topology. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1st edition, 2013.