# DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems

This volume contains all the full papers presented at AUTOMATA 2011, the 17th international workshop on cellular automata and discrete complex systems. The workshop was held on November 21-23, 2011, at the Center for Mathematical Modeling, University of Chile, Santiago, Chile.

AUTOMATA is an annual workshop on the fundamental aspects of cellular automata and related discrete dynamical systems. The spirit of the workshop is to foster collaborations and exchanges between researchers on these areas. The workshop series was started in 1995 by members of the Working Group 1.5 of IFIP, the International Federation for Information Processing.

The volume contains the « full » papers selected by the program committee, which consisted of 27 international experts on cellular automata and related models, and the selection was based on 3 peer reviews on each paper.

Papers in this volume represent a rich sample of current research topics on cellular automata and related models. The papers include theoretical studies of the classical cellular automata model, but also many investigations into various variants and generalizations of the basic concept. The versatile nature and the flexibility of the model is evident from the presented papers, making it a rich source of new research problems for scientists representing a variety of disciplines.

As the editors of these proceedings, we thank all contributors to the scientific program of the workshop. We are especially indebted to the invited speakers and the authors of the contributed papers.

Nazim Fatès, Eric Goles, Alejandro Maass, Iván Rapaport

Program Committee

• Andrew Adamatzky, University of West England, UK
• Stefania Bandini, Università degli Studi di Milano-Bicocca, Italy
• Marie-Pierre Béal, Université Paris-Est, France
• Bruno Durand, Université de Provence, France
• Nazim Fatès, Inria Nancy Grand-Est, France, co-chair
• Paola Flocchini, University of Ottawa, Canada
• Enrico Formenti, Université de Nice-Sophia Antipolis, France
• Henryk Fuks, Brock University, Canada
• Anahí Gajardo, Universidad de Concepción, Chile
• Martin Kutrib, University of Giessen, Germany
• Alejandro Maass, Universidad de Chile, co-chair
• Andrés Moreira, Universidad Técnica Federico Santa María, Chile
• Kenichi Morita, Hiroshima University, Japan
• Pedro de Oliveira, Universidade Presbiteriana Mackenzie, Brazil
• Nicolas Ollinger, Université de Provence, France
• Ronnie Pavlov, Denver University, USA
• Marcus Pivato, Trent University, Canada
• Ivan Rapaport, Universidad de Chile, co-chair
• Dipanwita Roychowdhury, Indian Institute of Technology, India
• Mathieu Sablik, Université de Provence
• Michael Schraudner, Universidad de Chile
• Klaus Sutner, Carnegie Mellon, USA
• Guillaume Theyssier, CNRS, Université de Savoie, France
• Edgardo Ugalde, Universidad Autónoma de San Luis Potosí, Mexico
• Hiroshi Umeo, Osaka Electro-Communication University, Japan
• Thomas Worsch, Karlsruhe University, Germany

### 1. Conservation Laws and Invariant Measures in Surjective Cellular Automata

We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.
Section: Proceedings

### 2. Projective subdynamics and universal shifts

We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.
Section: Proceedings

### 3. NOCAS : A Nonlinear Cellular Automata Based Stream Cipher

LFSR and NFSR are the basic building blocks in almost all the state of the art stream ciphers like Trivium and Grain-128. However, a number of attacks are mounted on these type of ciphers. Cellular Automata (CA) has recently been chosen as a suitable structure for crypto-primitives. In this work, a stream cipher is presented based on hybrid CA. The stream cipher takes 128 bit key and 128 bit initialization vector (IV) as input. It is designed to produce $\mathrm{2^{128}}$ random keystream bits and initialization phase is made faster 4 times than that of Grain-128. We also analyze the cryptographic strength of this cipher. Finally, the proposed cipher is shown to be resistant against known existing attacks.
Section: Proceedings

### 4. Product decomposition for surjective 2-block NCCA

In this paper we define products of one-dimensional Number Conserving Cellular Automata (NCCA) and show that surjective NCCA with 2 blocks (i.e radius 1/2) can always be represented as products of shifts and identites. In particular, this shows that surjective 2-block NCCA are injective.
Section: Proceedings

### 5. Orbits of the Bernoulli measure in single-transition asynchronous cellular automata

We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.
Section: Proceedings

### 6. Selfsimilarity, Simulation and Spacetime Symmetries

We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse.
Section: Proceedings

### 7. On the set of Fixed Points of the Parallel Symmetric Sand Pile Model

Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of $\textit{Self-Organized Criticality}$. From a configuration composed of a finite number of stacked grains, we apply on every possible positions (in parallel) two grain moving transition rules. The transition rules permit one grain to fall to its right or left (symmetric) neighboring column if the difference of height between those columns is larger than 2. The model is nondeterministic and grains always fall downward. We propose a study of the set of fixed points reachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparison with the Symmetric Sand Pile Model (SSPM) on which rules are applied once at each iteration, we get a continuity property. This property states that within PSSPM we can't reach every fixed points of SSPM, but a continuous subset according to the lexicographic order. Moreover we define a successor relation to browse exhaustively the sets of fixed points of those models.
Section: Proceedings

### 8. Bifurcations in Boolean Networks

This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow$1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.
Section: Proceedings

### 9. Asymptotic distribution of entry times in a cellular automaton with annihilating particles

This work considers a cellular automaton (CA) with two particles: a stationary particle $1$ and left-going one $\overline{1}$. When a $\overline{1}$ encounters a $1$, both particles annihilate. We derive asymptotic distribution of appearence of particles at a given site when the CA is initialized with the Bernoulli measure with the probabilities of both particles equal to $1/2$.
Section: Proceedings

### 10. The structure of communication problems in cellular automata

Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various "communication problems'', or "questions'' one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations.
Section: Proceedings

### 11. A fixed point theorem for Boolean networks expressed in terms of forbidden subnetworks

We are interested in fixed points in Boolean networks, $\textit{i.e.}$ functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the hypercubes contained in $\{0,1\}^n$, and we exhibit a class $\mathcal{F}$ of Boolean networks, called even or odd self-dual networks, satisfying the following property: if a network $f$ has no subnetwork in $\mathcal{F}$, then it has a unique fixed point. We then discuss this "forbidden subnetworks theorem''. We show that it generalizes the following fixed point theorem of Shih and Dong: if, for every $x$ in $\{0,1\}^n$, there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. We also show that $\mathcal{F}$ contains the class $\mathcal{F'}$ of networks whose the interaction graph is a directed cycle, but that the absence of subnetwork in $\mathcal{F'}$ does not imply the existence and the uniqueness of a fixed point.
Section: Proceedings