On time-symmetry in cellular automata

The notion of reversibility has been intensively studied in the field of cellular automata (CA), for several reasons. However, a related notion found in physical theories has been so far neglected, not only in CA, but generally in discrete dynamical systems. This is the notion of time-symmetry, which refers to the inability of distinguishing between backward and forward time directions. Here we formalize it in the context of CA, and study some of its basic properties. We also show how some well-known CA fit into the class of time-symmetric CA, and provide a number of results on the relation between this and other classes of CA. The existence of an intrinsically universal time-symmetric CA within the class of reversible CA is proved. Finally, we show the undecidability of time-symmetry for CA of dimension 2 or higher, even within the class of reversible CA. The case of dimension 1 is one of several open questions discussed in the conclusions.

► We introduce the notion of time-symmetry to the context of cellular automata (CA).
► Some well-known CA (Margolus billiard, Langtonʼs ant) are time-symmetric.
► Time-symmetric CA are shown to be intrinsically universal.
► Time-symmetry is undecidable for 2D CA, even if the CA is known to be reversible.
► We show some results relating the class of time-symmetric CA to other CA classes.