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

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 terms 'cellular automaton' and 'lattice gas' for a dynamic system it self as well as for its presentation. The two kinds of presentation share many traits but also display profound differences oil issues ranging from decidability to modeling convenience and physical implementability. Following a conjecture by Toffoli and Margolus, it had been proved by Kari that any invertible CA, at least Up to two dimensions, can be rewritten as all isomorphic LG. 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 ail 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. (C) 2008 Elsevier B.V. All rights reserved.