1,544 research outputs found

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

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    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.Comment: 16 page

    Causal graph dynamics

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    We extend the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time - but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalized. The definition we provide for these "causal graph dynamics" is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of Cellular Automata theory carry through: causal graph dynamics admit a characterization as continuous functions, and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics. KEYWORDS: Dynamical networks, Boolean networks, Generative networks automata, Cayley cellular automata, Graph Automata, Graph rewriting automata, Parallel graph transformations, Amalgamated graph transformations, Time-varying graphs, Regge calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3: Typos corrected, figure adde

    Von Neumann Regular Cellular Automata

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    For any group GG and any set AA, a cellular automaton (CA) is a transformation of the configuration space AGA^G defined via a finite memory set and a local function. Let CA(G;A)\text{CA}(G;A) be the monoid of all CA over AGA^G. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element τ∈CA(G;A)\tau \in \text{CA}(G;A) is von Neumann regular (or simply regular) if there exists σ∈CA(G;A)\sigma \in \text{CA}(G;A) such that τ∘σ∘τ=τ\tau \circ \sigma \circ \tau = \tau and σ∘τ∘σ=σ\sigma \circ \tau \circ \sigma = \sigma, where ∘\circ is the composition of functions. Such an element σ\sigma is called a generalised inverse of τ\tau. The monoid CA(G;A)\text{CA}(G;A) itself is regular if all its elements are regular. We establish that CA(G;A)\text{CA}(G;A) is regular if and only if ∣G∣=1\vert G \vert = 1 or ∣A∣=1\vert A \vert = 1, and we characterise all regular elements in CA(G;A)\text{CA}(G;A) when GG and AA are both finite. Furthermore, we study regular linear CA when A=VA= V is a vector space over a field F\mathbb{F}; in particular, we show that every regular linear CA is invertible when GG is torsion-free elementary amenable (e.g. when G=Zd, d∈NG=\mathbb{Z}^d, \ d \in \mathbb{N}) and V=FV=\mathbb{F}, and that every linear CA is regular when VV is finite-dimensional and GG is locally finite with Char(F)∤o(g)\text{Char}(\mathbb{F}) \nmid o(g) for all g∈Gg \in G.Comment: 10 pages. Theorem 5 corrected from previous versions, in A. Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer, 201

    On the structure of Clifford quantum cellular automata

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    We study reversible quantum cellular automata with the restriction that these are also Clifford operations. This means that tensor products of Pauli operators (or discrete Weyl operators) are mapped to tensor products of Pauli operators. Therefore Clifford quantum cellular automata are induced by symplectic cellular automata in phase space. We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin. In the one dimensional case we also find that every uniquely determined and translationally invariant stabilizer state can be prepared from a product state by a single Clifford cellular automaton timestep, thereby characterizing these class of stabilizer states, and we show that all 1D Clifford quantum cellular automata are generated by a few elementary operations. We also show that the correspondence between translationally invariant stabilizer states and translationally invariant Clifford operations holds for periodic boundary conditions.Comment: 28 pages, 2 figures, LaTe

    Phase Space Invertible Asynchronous Cellular Automata

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    While for synchronous deterministic cellular automata there is an accepted definition of reversibility, the situation is less clear for asynchronous cellular automata. We first discuss a few possibilities and then investigate what we call phase space invertible asynchronous cellular automata in more detail. We will show that for each Turing machine there is such a cellular automaton simulating it, and that it is decidable whether an asynchronous cellular automaton has this property or not, even in higher dimensions.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
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