On Factor Universality in Symbolic Spaces

The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T, the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes

Decidability and Universality in Symbolic Dynamical Systems

Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a model-checking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2: minor orthographic changes v3: section 5.2 (collatz functions) mathematically improved v4: orthographic corrections, one reference added v5:27 pages. Important modifications. The formalism is strengthened: temporal logic replaced by finite automata. New results. Submitte

Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences

We consider infinite sequences of superstable orbits (cascades) generated by systematic substitutions of letters in the symbolic dynamics of one-dimensional nonlinear systems in the logistic map universality class. We identify the conditions under which the topological entropy of successive words converges as a double exponential onto the accumulation point, and find the convergence rates analytically for selected cascades. Numerical tests of the convergence of the control parameter reveal a tendency to quantitatively universal double-exponential convergence. Taking a specific physical example, we consider cascades of stable orbits described by symbolic sequences with the symmetries of quasilattices. We show that all quasilattices can be realised as stable trajectories in nonlinear dynamical systems, extending previous results in which two were identified.Comment: This version: updated figures and added discussion of generalised time quasilattices. 17 pages, 4 figure

Isomorphism and embedding of Borel systems on full sets

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy at least t by removing the periodic points and "restricting" to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.Comment: 17 pages, v2: correction to bibliograph

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to the fundamental problems in computational complexity with numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings (Harvey et al SODA 2005, Harvey et al SODA 2006). We design efficient deterministic algorithms for common generalizations of the results of Lovasz and Geelen on this problem by allowing linear functions in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. We present also a deterministic polynomial time algorithm for finding the minimal number of generators of a given module structure given by matrices. We establish further several hardness results related to matrix algebras and modules. As a result we connect the classical problem of polynomial identity testing with checking surjectivity (or injectivity) between two given modules. One of the elements of our algorithm is a construction of a greedy algorithm for finding a maximum rank element in the more general setting of the problem. The proof methods used in this paper could be also of independent interest.Comment: 14 pages, preliminar

Epistemologies of Spaces and Places: An Introduction

The article introduces a special themed issue of Theory of Science on epistemologies of spaces and places. It provides a disciplinary context of the theme and reviews some of the key arguments that led to the so-called spatial turn in social sciences and the humanities. Science studies in the broad sense have also been affected by this shift of research interest to spatial aspects of science at both micro- and macro-levels. Scientific knowledge has been subject to analyses that stress its local contingencies, mobility and dependencies on spatial arrangements. The ensuing new epistemologies require novel concepts or reconsideration of the older terms, such as universality or objectivity