From quantum cellular automata to quantum lattice gases

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.Comment: 22 pages, plain TeX, 9 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages); minor typographical corrections and journal reference adde

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 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

Boolean Dynamics with Random Couplings

This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical Sciences Serie

Cellular automaton supercolliders

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular-automaton analogous of localizations or quasi-local collective excitations travelling in a spatially extended non-linear medium. They can be considered as binary strings or symbols travelling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyse what types of interaction occur between gliders travelling on a cellular automaton `cyclotron' and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in non-linear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analysed via implementation of cyclic tag systems

Von Neumann Regular Cellular Automata

For any group $G$ and any set $A$, a cellular automaton (CA) is a transformation of the configuration space $A^G$ defined via a finite memory set and a local function. Let $\text{CA}(G;A)$ be the monoid of all CA over $A^G$. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element $\tau \in \text{CA}(G;A)$ is von Neumann regular (or simply regular) if there exists $\sigma \in \text{CA}(G;A)$ such that $\tau \circ \sigma \circ \tau = \tau$ and $\sigma \circ \tau \circ \sigma = \sigma$, where $\circ$ is the composition of functions. Such an element $\sigma$ is called a generalised inverse of $\tau$. The monoid $\text{CA}(G;A)$ itself is regular if all its elements are regular. We establish that $\text{CA}(G;A)$ is regular if and only if $\vert G \vert = 1$ or $\vert A \vert = 1$, and we characterise all regular elements in $\text{CA}(G;A)$ when $G$ and $A$ are both finite. Furthermore, we study regular linear CA when $A= V$ is a vector space over a field $\mathbb{F}$; in particular, we show that every regular linear CA is invertible when $G$ is torsion-free elementary amenable (e.g. when $G=\mathbb{Z}^d, \ d \in \mathbb{N}$) and $V=\mathbb{F}$, and that every linear CA is regular when $V$ is finite-dimensional and $G$ is locally finite with $\text{Char}(\mathbb{F}) \nmid o(g)$ for all $g \in G$.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