29,467 research outputs found
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
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