Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Descriptional complexity of cellular automata and decidability questions

We study the descriptional complexity of cellular automata (CA), a parallel model of computation. We show that between one of the simplest cellular models, the realtime-OCA. and "classical" models like deterministic finite automata (DFA) or pushdown automata (PDA), there will be savings concerning the size of description not bounded by any recursive function, a so-called nonrecursive trade-off. Furthermore, nonrecursive trade-offs are shown between some restricted classes of cellular automata. The set of valid computations of a Turing machine can be recognized by a realtime-OCA. This implies that many decidability questions are not even semi decidable for cellular automata. There is no pumping lemma and no minimization algorithm for cellular automata

A complex network approach to urban growth

The economic geography can be viewed as a large and growing network of interacting activities. This fundamental network structure and the large size of such systems makes complex networks an attractive model for its analysis. In this paper we propose the use of complex networks for geographical modeling and demonstrate how such an application can be combined with a cellular model to produce output that is consistent with large scale regularities such as power laws and fractality. Complex networks can provide a stringent framework for growth dynamic modeling where concepts from e.g. spatial interaction models and multiplicative growth models can be combined with the flexible representation of land and behavior found in cellular automata and agent-based models. In addition, there exists a large body of theory for the analysis of complex networks that have direct applications for urban geographic problems. The intended use of such models is twofold: i) to address the problem of how the empirically observed hierarchical structure of settlements can be explained as a stationary property of a stochastic evolutionary process rather than as equilibrium points in a dynamics, and, ii) to improve the prediction quality of applied urban modeling.evolutionary economics, complex networks, urban growth

MFCS\u2798 Satellite Workshop on Cellular Automata

For the 1998 conference on Mathematical Foundations of Computer Science (MFCS\u2798) four papers on Cellular Automata were accepted as regular MFCS\u2798 contributions. Furthermore an MFCS\u2798 satellite workshop on Cellular Automata was organized with ten additional talks. The embedding of the workshop into the conference with its participants coming from a broad spectrum of fields of work lead to interesting discussions and a fruitful exchange of ideas. The contributions which had been accepted for MFCS\u2798 itself may be found in the conference proceedings, edited by L. Brim, J. Gruska and J. Zlatuska, Springer LNCS 1450. All other (invited and regular) papers of the workshop are contained in this technical report. (One paper, for which no postscript file of the full paper is available, is only included in the printed version of the report). Contents: F. Blanchard, E. Formenti, P. Kurka: Cellular automata in the Cantor, Besicovitch and Weyl Spaces K. Kobayashi: On Time Optimal Solutions of the Two-Dimensional Firing Squad Synchronization Problem L. Margara: Topological Mixing and Denseness of Periodic Orbits for Linear Cellular Automata over Z_m B. Martin: A Geometrical Hierarchy of Graph via Cellular Automata K. Morita, K. Imai: Number-Conserving Reversible Cellular Automata and Their Computation-Universality C. Nichitiu, E. Remila: Simulations of graph automata K. Svozil: Is the world a machine? H. Umeo: Cellular Algorithms with 1-bit Inter-Cell Communications F. Reischle, Th. Worsch: Simulations between alternating CA, alternating TM and circuit families K. Sutner: Computation Theory of Cellular Automat

Aspects of algorithms and dynamics of cellular paradigms

Els paradigmes cel·lulars, com les xarxes neuronals cel·lulars (CNN, en anglès) i els autòmats cel·lulars (CA, en anglès), són una eina excel·lent de càlcul, al ser equivalents a una màquina universal de Turing. La introducció de la màquina universal CNN (CNN-UM, en anglès) ha permès desenvolupar hardware, el nucli computacional del qual funciona segons la filosofia cel·lular; aquest hardware ha trobat aplicació en diversos camps al llarg de la darrera dècada. Malgrat això, encara hi ha moltes preguntes a obertes sobre com definir els algoritmes d'una CNN-UM i com estudiar la dinàmica dels autòmats cel·lulars. En aquesta tesis es tracten els dos problemes: primer, es demostra que es possible acotar l'espai dels algoritmes per a la CNN-UM i explorar-lo gràcies a les tècniques genètiques; i segon, s'expliquen els fonaments de l'estudi dels CA per mitjà de la dinàmica no lineal (segons la definició de Chua) i s'il·lustra com aquesta tècnica ha permès trobar resultats innovadors.Los paradigmas celulares, como las redes neuronales celulares (CNN, eninglés) y los autómatas celulares (CA, en inglés), son una excelenteherramienta de cálculo, al ser equivalentes a una maquina universal deTuring. La introducción de la maquina universal CNN (CNN-UM, eninglés) ha permitido desarrollar hardware cuyo núcleo computacionalfunciona según la filosofía celular; dicho hardware ha encontradoaplicación en varios campos a lo largo de la ultima década. Sinembargo, hay aun muchas preguntas abiertas sobre como definir losalgoritmos de una CNN-UM y como estudiar la dinámica de los autómatascelular. En esta tesis se tratan ambos problemas: primero se demuestraque es posible acotar el espacio de los algoritmos para la CNN-UM yexplorarlo gracias a técnicas genéticas; segundo, se explican losfundamentos del estudio de los CA por medio de la dinámica no lineal(según la definición de Chua) y se ilustra como esta técnica hapermitido encontrar resultados novedosos.Cellular paradigms, like Cellular Neural Networks (CNNs) and Cellular Automata (CA) are an excellent tool to perform computation, since they are equivalent to a Universal Turing machine. The introduction of the Cellular Neural Network - Universal Machine (CNN-UM) allowed us to develop hardware whose computational core works according to the principles of cellular paradigms; such a hardware has found application in a number of fields throughout the last decade. Nevertheless, there are still many open questions about how to define algorithms for a CNN-UM, and how to study the dynamics of Cellular Automata. In this dissertation both problems are tackled: first, we prove that it is possible to bound the space of all algorithms of CNN-UM and explore it through genetic techniques; second, we explain the fundamentals of the nonlinear perspective of CA (according to Chua's definition), and we illustrate how this technique has allowed us to find novel results