On surjunctive monoids

A monoid $M$ is called surjunctive if every injective cellular automata with finite alphabet over $M$ is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive

Cellular automata with complicated dynamics

A subshift is a collection of bi-infinite sequences (configurations) of symbols where some finite patterns of symbols are forbidden to occur. A cellular automaton is a transformation that changes each configuration of a subshift into another one by using a finite look-up table that tells how any symbol occurring at any possible context is to be changed. A cellular automaton can be applied repeatedly on the configurations of the subshift, thus making it a dynamical system. This thesis focuses on cellular automata with complex dynamical behavior, with some different definitions of the word “complex”. First we consider a naturally occurring class of cellular automata that we call multiplication automata and we present a case study with the point of view of symbolic, topological and measurable dynamics. We also present an application of these automata to a generalized version of Mahler’s problem. For different notions of complex behavior one may also ask whether a given subshift or class of subshifts has a cellular automaton that presents this behavior. We show that in the class of full shifts the Lyapunov exponents of a given reversible cellular automaton are uncomputable. This means that in the dynamics of reversible cellular automata the long term maximal propagation speed of a perturbation made in an initial configuration cannot be determined in general from short term observations. In the last part we construct, on all mixing sofic shifts, diffusive glider cellular automata that can decompose any finite configuration into two distinct components that shift into opposing direction under repeated action of the automaton. This implies that every mixing sofic shift has a reversible cellular automaton all of whose directions are sensitive in the sense of the definition of Sablik. We contrast this by presenting a family of synchronizing subshifts on which all reversible cellular automata always have a nonsensitive direction

On some one-sided dynamics of cellular automata

A dynamical system consists of a space of all possible world states and a transformation of said space. Cellular automata are dynamical systems where the space is a set of one- or two-way infinite symbol sequences and the transformation is defined by a homogenous local rule. In the setting of cellular automata, the geometry of the underlying space allows one to define one-sided variants of some dynamical properties; this thesis considers some such one-sided dynamics of cellular automata. One main topic are the dynamical concepts of expansivity and that of pseudo-orbit tracing property. Expansivity is a strong form of sensitivity to the initial conditions while pseudo-orbit tracing property is a type of approximability. For cellular automata we define one-sided variants of both of these concepts. We give some examples of cellular automata with these properties and prove, for example, that right-expansive cellular automata are chain-mixing. We also show that left-sided pseudo-orbit tracing property together with right-sided expansivity imply that a cellular automaton has the pseudo-orbit tracing property. Another main topic is conjugacy. Two dynamical systems are conjugate if, in a dynamical sense, they are the same system. We show that for one-sided cellular automata conjugacy is undecidable. In fact the result is stronger and shows that the relations of being a factor or a susbsystem are undecidable, too

Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid G induced by G, and representations of G. Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for G to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the "out-degrees of vertices". From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.Comment: 69 page

Symmetry structure in discrete models of biochemical systems : natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

© 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.Interaction Computing (IC) is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are (1) to identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this, and (2) to use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in Systems Biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, Krebs cycle, and p53-mdm2 genetic regulation constructed from Systems Biology models have canonically associated algebraic structures { transformation semigroups. These contain permutation groups (local substructures exhibiting symmetry) that correspond to "pools of reversibility". These natural subsystems are related to one another in a hierarchical manner by the notion of "weak control ". We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-abelian groups (SNAGs) are found in biological examples and can be harnessed to realize nitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.Peer reviewe

Reversible Computation: Extending Horizons of Computing

This open access State-of-the-Art Survey presents the main recent scientific outcomes in the area of reversible computation, focusing on those that have emerged during COST Action IC1405 "Reversible Computation - Extending Horizons of Computing", a European research network that operated from May 2015 to April 2019. Reversible computation is a new paradigm that extends the traditional forwards-only mode of computation with the ability to execute in reverse, so that computation can run backwards as easily and naturally as forwards. It aims to deliver novel computing devices and software, and to enhance existing systems by equipping them with reversibility. There are many potential applications of reversible computation, including languages and software tools for reliable and recovery-oriented distributed systems and revolutionary reversible logic gates and circuits, but they can only be realized and have lasting effect if conceptual and firm theoretical foundations are established first