954 research outputs found
Coarse-graining of cellular automata, emergence, and the predictability of complex systems
We study the predictability of emergent phenomena in complex systems. Using
nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show
how to construct local coarse-grained descriptions of CA in all classes of
Wolfram's classification. The resulting coarse-grained CA that we construct are
capable of emulating the large-scale behavior of the original systems without
accounting for small-scale details. Several CA that can be coarse-grained by
this construction are known to be universal Turing machines; they can emulate
any CA or other computing devices and are therefore undecidable. We thus show
that because in practice one only seeks coarse-grained information, complex
physical systems can be predictable and even decidable at some level of
description. The renormalization group flows that we construct induce a
hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity
and is therefore a good candidate for a complexity measure and a classification
method. Finally we argue that the large scale dynamics of CA can be very
simple, at least when measured by the Kolmogorov complexity of the large scale
update rule, and moreover exhibits a novel scaling law. We show that because of
this large-scale simplicity, the probability of finding a coarse-grained
description of CA approaches unity as one goes to increasingly coarser scales.
We interpret this large scale simplicity as a pattern formation mechanism in
which large scale patterns are forced upon the system by the simplicity of the
rules that govern the large scale dynamics.Comment: 18 pages, 9 figure
On the necessity of complexity
Wolfram's Principle of Computational Equivalence (PCE) implies that universal
complexity abounds in nature. This paper comprises three sections. In the first
section we consider the question why there are so many universal phenomena
around. So, in a sense, we week a driving force behind the PCE if any. We
postulate a principle GNS that we call the Generalized Natural Selection
Principle that together with the Church-Turing Thesis is seen to be equivalent
to a weak version of PCE. In the second section we ask the question why we do
not observe any phenomena that are complex but not-universal. We choose a
cognitive setting to embark on this question and make some analogies with
formal logic. In the third and final section we report on a case study where we
see rich structures arise everywhere.Comment: 17 pages, 3 figure
Computing with Coloured Tangles
We suggest a diagrammatic model of computation based on an axiom of
distributivity. A diagram of a decorated coloured tangle, similar to those that
appear in low dimensional topology, plays the role of a circuit diagram.
Equivalent diagrams represent bisimilar computations. We prove that our model
of computation is Turing complete, and that with bounded resources it can
moreover decide any language in complexity class IP, sometimes with better
performance parameters than corresponding classical protocols.Comment: 36 pages,; Introduction entirely rewritten, Section 4.3 adde
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Universalities in cellular automata; a (short) survey
This reading guide aims to provide the reader with an easy access to the study of universality in the field of cellular automata. To fulfill this goal, the approach taken here is organized in three parts: a detailled chronology of seminal papers, a discussion of the definition and main properties of universal cellular automata, and a broad bibliography
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