When--and how--can a cellular automaton be rewritten as a lattice gas?

TL;DR

This paper investigates the conditions under which cellular automata can be represented as lattice gases, proving that noninvertible CA are generally representable as LGs except for a small class, with implications for computational thermodynamics.

Contribution

It establishes that most noninvertible cellular automata, including the Game of Life, can be represented as lattice gases, extending previous results limited to invertible CA.

Findings

Invertible CA can be rewritten as LGs, confirming previous conjectures.

Most noninvertible CA, including common examples like the Game of Life, are representable as LGs.

The results suggest a tradeoff between dissipation and complexity affecting microscopic thermodynamics.

Abstract

Both cellular automata (CA) and lattice-gas automata (LG) provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics; it is customary to use the term `cellular automaton' or `lattice gas' for the dynamic system itself as well as for its presentation. The two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability. Following a conjecture by Toffoli and Margolus, it had been proved by Kari (and by Durand--Lose for more than two dimensions) that any invertible CA can be rewritten as an LG (with a possibly much more complex ``unit cell''). But until now it was not known whether this is possible in general for noninvertible CA--which comprise ``almost all'' CA and represent the bulk of examples in theory and applications. Even circumstantial evidence--whether in favor or against--was lacking. Here, for noninvertible CA, (a) we prove that an LG presentation is out of the question for the vanishingly small class of surjective ones. We then turn our attention to all the rest--noninvertible and nonsurjective--which comprise all the typical ones, including Conway's `Game of Life'. For these (b) we prove by explicit construction that all the one-dimensional ones are representable as LG, and (c) we present and motivate the conjecture that this result extends to any number of dimensions. The tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale.