The Topological Directional Entropy of Z^2-actions Generated by Linear Cellular Automata

In this paper we study the topological and metric directional entropy of $\mathbb{Z}^2$-actions by generated additive cellular automata (CA hereafter), defined by a local rule $f[l, r]$, $l, r\in \mathbb{Z}$, $l\leq r$, i.e. the maps $T_{f[l, r]}: \mathbb{Z}^\mathbb{Z}_{m} \to \mathbb{Z}^\mathbb{Z}_{m}$ which are given by $T_{f[l, r]}(x) =(y_n)_ {-\infty}^{\infty}$, $y_{n} = f(x_{n+l}, ..., x_{n+r}) = \sum_{i=l}^r\lambda_{i}x_{i+n}(mod m)$, $x=(x_n)_ {n=-\infty}^{\infty}\in \mathbb{Z}^\mathbb{Z}_{m}$, and $f: \mathbb{Z}_{m}^{r-l+1}\to \mathbb{Z}_{m}$, over the ring $\mathbb{Z}_m (m \geq 2)$, and the shift map acting on compact metric space $\mathbb{Z}^\mathbb{Z}_{m}$, where $m$ $(m \geq2)$ is a positive integer. Our main aim is to give an algorithm for computing the topological directional entropy of the $\mathbb{Z}^2$-actions generated by the additive CA and the shift map. Thus, we ask to give a closed formula for the topological directional entropy of $\mathbb{Z}^2$-action generated by the pair $(T_{f[l, r]}, \sigma)$ in the direction $\theta$ that can be efficiently and rightly computed by means of the coefficients of the local rule f as similar to [Theor. Comput. Sci. 290 (2003) 1629-1646]. We generalize the results obtained by Ak\i n [The topological entropy of invertible cellular automata, J. Comput. Appl. Math. 213 (2) (2008) 501-508] to the topological entropy of any invertible linear CA.Comment: 9 pages. submitte

The algebraic entropy of one-dimensional finitary linear cellular automata

The aim of this paper is to present one-dimensional finitary linear cellular automata $S$ on $\mathbb Z_m$ from an algebraic point of view. Among various other results, we: (i) show that the Pontryagin dual $\widehat S$ of $S$ is a classical one-dimensional linear cellular automaton $T$ on $\mathbb Z_m$; (ii) give several equivalent conditions for $S$ to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of $S$, which coincides with the topological entropy of $T=\widehat S$ by the so-called Bridge Theorem. In order to better understand and describe the entropy we introduce the degree $\mathrm{deg}(S)$ and $\mathrm{deg}(T)$ of $S$ and $T$.Comment: 21 page

A Topological Classification of D-Dimensional Cellular Automata

We give a classification of cellular automata in arbitrary dimensions and on arbitrary subshift spaces from the point of view of symbolic and topological dynamics. A cellular automaton is a continuous, shift-commuting map on a subshift space; these objects were first investigated from a purely mathematical point of view by Hedlund in 1969. In the 1980's, Wolfram categorized one-dimensional cellular automata based on features of their asymptotic behavior which could be seen on a computer screen. Gilman's work in 1987 and 1988 was the first attempt to mathematically formalize these characterizations of Wolfram's, using notions of equicontinuity, expansiveness, and measure-theoretic analogs of each. We introduce a topological classification of cellular automata in dimensions two and higher based on the one-dimensional classification given by Kurka. We characterize equicontinuous cellular automata in terms of periodicity, investigate the occurrence of blocking patterns as related to points of equicontinuity, demonstrate that topologically transitive cellular automata are both surjective and have sensitive dependence on initial conditions, and construct subshift spaces in all dimensions on which there exists an expansive cellular automaton. We provide numerous examples throughout and conclude with two diagrams illustrating the interaction of topological properties in all dimensions for the cases of an underlying full shift space and of an underlying subshift space with dense shift-periodic points

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We settle two long-standing open problems about Conway’s Life, a two-dimensional cellular automaton. We solve the Generalized grandfather problem: for all n ≥ 0, there exists a configuration that has an nth predecessor but not an (n+1)st one. We also solve (one interpretation of) the Unique father problem: there exists a finite stable configuration that contains a finite subpattern that has no predecessor patterns except itself. In particular this gives the first example of an unsynthesizable still life. The new key concept is that of a spatiotemporally periodic configuration (agar) that has a unique chain of preimages; we show that this property is semidecidable, and find examples of such agars using a SAT solver.Our results about the topological dynamics of Game of Life are as follows: it never reaches its limit set; its dynamics on its limit set is chain-wandering, in particular it is not topologically transitive and does not have dense periodic points; and the spatial dynamics of its limit set is non-sofic, and does not admit a sublinear gluing radius in the cardinal directions (in particular it is not block-gluing). Our computability results are that Game of Life’s reachability problem, as well as the language of its limit set, are PSPACE-hard.</p

Subshifts with Simple Cellular Automata

A subshift is a set of infinite one- or two-way sequences over a fixed finite set, defined by a set of forbidden patterns. In this thesis, we study subshifts in the topological setting, where the natural morphisms between them are ones defined by a (spatially uniform) local rule. Endomorphisms of subshifts are called cellular automata, and we call the set of cellular automata on a subshift its endomorphism monoid. It is known that the set of all sequences (the full shift) allows cellular automata with complex dynamical and computational properties. We are interested in subshifts that do not support such cellular automata. In particular, we study countable subshifts, minimal subshifts and subshifts with additional universal algebraic structure that cellular automata need to respect, and investigate certain criteria of ‘simplicity’ of the endomorphism monoid, for each of them. In the case of countable subshifts, we concentrate on countable sofic shifts, that is, countable subshifts defined by a finite state automaton. We develop some general tools for studying cellular automata on such subshifts, and show that nilpotency and periodicity of cellular automata are decidable properties, and positive expansivity is impossible. Nevertheless, we also prove various undecidability results, by simulating counter machines with cellular automata. We prove that minimal subshifts generated by primitive Pisot substitutions only support virtually cyclic automorphism groups, and give an example of a Toeplitz subshift whose automorphism group is not finitely generated. In the algebraic setting, we study the centralizers of CA, and group and lattice homomorphic CA. In particular, we obtain results about centralizers of symbol permutations and bipermutive CA, and their connections with group structures.Siirretty Doriast

Discrete scale invariance and complex dimensions

We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear ``spontaneously'' in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.Comment: significantly extended version (Oct. 27, 1998) with new examples in several domains of the review paper with the same title published in Physics Reports 297, 239-270 (1998

Cellular Automata

Modelling and simulation are disciplines of major importance for science and engineering. There is no science without models, and simulation has nowadays become a very useful tool, sometimes unavoidable, for development of both science and engineering. The main attractive feature of cellular automata is that, in spite of their conceptual simplicity which allows an easiness of implementation for computer simulation, as a detailed and complete mathematical analysis in principle, they are able to exhibit a wide variety of amazingly complex behaviour. This feature of cellular automata has attracted the researchers' attention from a wide variety of divergent fields of the exact disciplines of science and engineering, but also of the social sciences, and sometimes beyond. The collective complex behaviour of numerous systems, which emerge from the interaction of a multitude of simple individuals, is being conveniently modelled and simulated with cellular automata for very different purposes. In this book, a number of innovative applications of cellular automata models in the fields of Quantum Computing, Materials Science, Cryptography and Coding, and Robotics and Image Processing are presented

Proceedings of AUTOMATA 2010: 16th International workshop on cellular automata and discrete complex systems

International audienceThese local proceedings hold the papers of two catgeories: (a) Short, non-reviewed papers (b) Full paper