We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. We then show that on sofic groups, where it is known that injective cellular automata are surjective, post-surjectivity implies pre-injectivity. As no non-sofic groups are currently known, we conjecture that this implication always holds. This mirrors Gottschalk's conjecture that every injective cellular automaton is surjective.

Automata, Logic and Semantics International audience Fekete's lemma is a well-known combinatorial result on number sequences: we extend it to functions defined on d-tuples of integers. As an application of the new variant, we show that nonsurjective d-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.

a b s t r a c tBoth 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 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 that any invertible CA, at least up to two dimensions, can be rewritten as an 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 an LG presentation is out of the question for the vanishingly small class of surjective ones. We then turn our attention to all the restnoninvertible 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.

No abstract

The dynamics of a physical system is linked to its phase-space geometry by Noether's theorem, which holds under standard hypotheses including continuity. Does an analogous theorem hold for discrete systems? As a testbed, we take the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that-and why-energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant.