On the Induction Operation for Shift Subspaces and Cellular Automata as Presentations of Dynamical Systems

We consider continuous, translation-commuting transformations of compact, translation-invariant families of mappingsfrom finitely generated groups into finite alphabets. It is well-known that such transformations and spaces can be described "locally" via families of patterns and finitary functions; such descriptions can be re-used on groups larger than the original, usually defining non-isomorphic structures. We show how some of the properties of the "induced" entities can be deduced from those of the original ones, and vice versa; then, we show how to "simulate" the smaller structure into the larger one, and obtain a characterization in terms of group actions for the dynamical systems admitting of presentations via structures as such. Special attention is given to the class of sofic shifts.Comment: 20 pages, no figures. Presented at LATA 2008. Extended version, submitted to Information and Computatio

Post-surjectivity and balancedness of cellular automata over groups

We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. Moreover, on sofic groups, post-surjectivity alone implies reversibility. We also prove that reversible cellular automata over arbitrary groups are balanced, that is, they preserve the uniform measure on the configuration space.Comment: 16 pages, 3 figures, LaTeX "dmtcs-episciences" document class. Final version for Discrete Mathematics and Theoretical Computer Science. Prepared according to the editor's request

Topological Conjugacies Between Cellular Automata

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant

The Differential Scheme and Quantum Computation

It is well-known that standard models of computation are representable as simple dynamical systems that evolve in discrete time, and that systems that evolve in continuous time are often representable by dynamical systems governed by ordinary differential equations. In many applications, e.g., molecular networks and hybrid Fermi-Pasta-Ulam systems, one must work with dynamical systems comprising both discrete and continuous components. Reasoning about and verifying the properties of the evolving state of such systems is currently a piecemeal affair that depends on the nature of major components of a system: e.g., discrete vs. continuous components of state, discrete vs. continuous time, local vs. distributed clocks, classical vs. quantum states and state evolution. We present the Differential Scheme as a unifying framework for reasoning about and verifying the properties of the evolving state of a system, whether the system in question evolves in discrete time, as for standard models of computation, or continuous time, or a combination of both. We show how instances of the differential scheme can accommodate classical computation. We also generalize a relatively new model of quantum computation, the quantum cellular automaton, with an eye towards extending the differential scheme to accommodate quantum computation and hybrid classical/quantum computation. All the components of a specific instance of the differential scheme are Convergence Spaces. Convergence spaces generalize notions of continuity and convergence. The category of convergence spaces, Conv, subsumes both simple discrete structures (e.g., digraphs), and complex continuous structures (e.g., topological spaces, domains, and the standard fields of analysis: R and C). We present novel uses for convergence spaces, and extend their theory by defining differential calculi on Conv. It is to the use of convergence spaces that the differential scheme owes its generality and flexibility

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast

Gibbs and Quantum Discrete Spaces

Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure

Learning Structure in Time Series for Neuroscience and Beyond

Advances in neuroscience are producing data at an astounding rate - data which are fiendishly complex both to process and to interpret. Biological neural networks are high-dimensional, nonlinear, noisy, heterogeneous, and in nearly every way defy the simplifying assumptions of standard statistical methods. In this dissertation we address a number of issues with understanding the structure of neural populations, from the abstract level of how to uncover structure in generic time series, to the practical matter of finding relevant biological structure in state-of-the-art experimental techniques. To learn the structure of generic time series, we develop a new statistical model, which we dub the probabilistic deterministic infinite automata (PDIA), which uses tools from nonparametric Bayesian inference to learn a very general class of sequence models. We show that the models learned by the PDIA often offer better predictive performance and faster inference than Hidden Markov Models, while being significantly more compact than models that simply memorize contexts. For large populations of neurons, models like the PDIA become unwieldy, and we instead investigate ways to robustly reduce the dimensionality of the data. In particular, we adapt the generalized linear model (GLM) framework for regres- sion to the case of matrix completion, which we call the low-dimensional GLM. We show that subspaces and dynamics of neural activity can be accurately recovered from model data, and with only minimal assumptions about the structure of the dynamics can still lead to good predictive performance on real data. Finally, to bridge the gap between recording technology and analysis, particularly as recordings from ever-larger populations of neurons becomes the norm, automated methods for extracting activity from raw recordings become a necessity. We present a number of methods for automatically segmenting biological units from optical imaging data, with applications to light sheet recording of genetically encoded calcium indicator fluorescence in the larval zebrafish, and optical electrophysiology using genetically encoded voltage indicators in culture. Together, these methods are a powerful set of tools for addressing the diverse challenges of modern neuroscience

Interacting Hopf Algebras: the theory of linear systems

Scientists in diverse fields use diagrammatic formalisms to reason about various kinds of networks, or compound systems. Examples include electrical circuits, signal flow graphs, Penrose and Feynman diagrams, Bayesian networks, Petri nets, Kahn process networks, proof nets, UML specifications, amongst many others. Graphical languages provide a convenient abstraction of some underlying mathematical formalism, which gives meaning to diagrams. For instance, signal flow graphs, foundational structures in control theory, are traditionally translated into systems of linear equations. This is typical: diagrammatic languages are used as an interface for more traditional mathematics, but rarely studied per se. Recent trends in computer science analyse diagrams as first-class objects using formal methods from programming language semantics. In many such approaches, diagrams are generated as the arrows of a PROP — a special kind of monoidal category — by a two-dimensional syntax and equations. The domain of interpretation of diagrams is also formalised as a PROP and the (compositional) semantics is expressed as a functor preserving the PROP structure. The first main contribution of this thesis is the characterisation of SVk, the PROP of linear subspaces over a field k. This is an important domain of interpretation for diagrams appearing in diverse research areas, like the signal flow graphs mentioned above. We present by generators and equations the PROP IH of string diagrams whose free model is SVk. The name IH stands for interacting Hopf algebras: indeed, the equations of IH arise by distributive laws between Hopf algebras, which we obtain using Lack’s technique for composing PROPs. The significance of the result is two-fold. On the one hand, it offers a canonical string diagrammatic syntax for linear algebra: linear maps, kernels, subspaces and the standard linear algebraic transformations are all faithfully represented in the graphical language. On the other hand, the equations of IH describe familiar algebraic structures — Hopf algebras and Frobenius algebras — which are at the heart of graphical formalisms as seemingly diverse as quantum circuits, signal flow graphs, simple electrical circuits and Petri nets. Our characterisation enlightens the provenance of these axioms and reveals their linear algebraic nature. Our second main contribution is an application of IH to the semantics of signal processing circuits. We develop a formal theory of signal flow graphs, featuring a string diagrammatic syntax for circuits, a structural operational semantics and a denotational semantics. We prove soundness and completeness of the equations of IH for denotational equivalence. Also, we study the full abstraction question: it turns out that the purely operational picture is too concrete — two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised — rewritten, using the equations of IH, into an executable form where the operational behaviour and the denotation coincide. This realisability theorem — which is the culmination of our developments — suggests a reflection about the role of causality in the semantics of signal flow graphs and, more generally, of computing devices