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

arXiv:0709.1173v1 [nlin.CG] 7 Sep 2007 calg November 2, 2018 When—and how—can a cellular automaton be rewritten as a lattice gas? aTommaso Toffoli, bSilvio Capobianco, and cPatrizia Mentrasti aElectrical and Computer Engineering, Boston University bSchool. Comp. Sci., Reykjavik University, Iceland cDip. di Matematica, Universit`a di Roma “La Sapienza” tt@bu.edu, silvio@ru.is, mentrasti@mat.uniroma1.it 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 tradeoffbetween dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale. I do not know of any single instance where something useful for the work on lattice gases has been borrowed from the cellular automata field. . . . Lattice gases differ in essence from cellular automata. A confusion of the two fields dis- torts our thinking, hides the special properties of lattice gases, and makes it harder to develop a good intuition. (Michel H´enon[17], with specific reference to Wolfram[42]) 1 Introduction Cellular automata (CA) provide a quick modeling route to phenomenological aspects of nature—especially the emer- gence of complex behavior in dissipative systems. But lattice-gas automata (LG) are unmatched as a source of fine-grained models of fundamental aspects of physics, es- pecially for expressing the dynamics of conservative1 sys- tems. In the quote at the beginning of this paper, one may well sympathize with H´enon’s annoyance: it turns out that dy- namical behavior that is synthesized with the utmost nat- uralness when using lattice gases as a “programming lan- guage” become perversely hard to express in the cellular automata language. Yet, H´enon’s are visceral feelings, not 1A dynamics is called ‘conservative’ if it is the manifest expression of an invertible microscopic mechanism. It is called ‘dissipative’ if the underlying mechanism is not invertible to begin with, or if its invertibility is de facto irrelevant because one is not capable or willing to maintain a strict accounting of microscopic states—perhaps owing to lack of precise knowledge of the initial state and the evolution laws, impredictable influences on the part of the environment, or the sheer size of the task. argued conclusions. With as much irritation one could re- tort, “How can lattice gases differ ‘in essence’ from cellular automata if they are merely a subset of them? What are these CA legacies that may ‘distort our thinking’ and ‘hide the special properties of lattice gases’? And aren’t there dy- namical systems that are much more naturally and easily modeled as cellular automata?” Today, with the benefit of twenty years’ hindsight—and especially after the results of the present paper—we are in a position to defuse the argument. H´enon’s appeal could less belligerantly be reworded as follows: “Even though CA and LG describe essentially the same class of objects, for sound pedagogical reasons it may be expedient to deal with them in separate chapters—or even in separate books for different audiences and applications. What is ox in the stable may well be beef on the table.” The bottom-line message is that these two modeling approaches do not reflect mutually exclusive strategies, but just opposite tradeoffs between the structural complexity of a piece of computing machinery and its thermodynamic efficiency. 2 Preview Let C and L be the sets of dynamical systems representable, respectively, as cellular automata and lattice gases. Our overall question is, How are these two sets related? On one hand, we shall see

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