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Searching for Chaos in Cellular Automata: New Tools for Classification
 Complex Systems, Mechanism of Adaptation
, 1994
"... this paper is twofold: to propose a new classification of CA, formally and precisely defined; to investigate the class of complex behaviors (particularly "aperiodic" behaviors). We propose new tools, i.e. transfinite attraction and shifted hamming distance, giving us a way of defining a ne ..."
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this paper is twofold: to propose a new classification of CA, formally and precisely defined; to investigate the class of complex behaviors (particularly "aperiodic" behaviors). We propose new tools, i.e. transfinite attraction and shifted hamming distance, giving us a way of defining a new classification of CA in which every class is formally defined. We also analyze three different ways of grouping these classes, which bring new insights in understanding chaotic behaviors. Before concluding,
Searching for Chaos in Cellular Automata: Compositional Approach
 Complex Systems, Mechanism of Adaptation
, 1994
"... . We propose composition operators allowing to study simple cellular automata and to extend individual results to global ones. This compositional approach can be used to reach a better understanding of chaos in discretetime multidimensional systems. 1. Introduction The notion of chaos is still u ..."
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. We propose composition operators allowing to study simple cellular automata and to extend individual results to global ones. This compositional approach can be used to reach a better understanding of chaos in discretetime multidimensional systems. 1. Introduction The notion of chaos is still unprecise in discretetime discretespace multidimensional dynamical systems like, for example, cellular automata (CA). Recently, many authors have tried to formalize it [1, 2, 3, 4]. Classification of CA w.r.t. their asymptotic behavior is a central theme in the field, and should lead to a better understanding of chaotic and related behaviors [5]. However, many problems have to be tackled: the asymptotic behavior is easy to study theoretically for very few simple cases only; in general, simulation is used to get results for complex behaviors. Our purpose is to move the first steps towards a definition of "chaos" in cellular automata using a compositional approach. We consider elementary ce...
New Connections between Mathematics and Computer Science
, 1996
"... A workshop on "New Connections between Mathematics and Computer Science" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England from 2024 November 1995. The workshop was supported by the Engineering and Physical Science Research Council of the United Kingdo ..."
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A workshop on "New Connections between Mathematics and Computer Science" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England from 2024 November 1995. The workshop was supported by the Engineering and Physical Science Research Council of the United Kingdom, the London Mathematical Society and HewlettPackard's Basic Research Institute in the Mathematical Sciences. This document contains a report on the workshop, the abstracts of the talks and the accompanying bibliography.
Hierarchy of DiscreteTime Dynamical Systems
, 1994
"... This paper is an attempt to unify classical automata theory and dynamical systems theory. We present a notion of generalized dynamical systems, allowing us to compare properties of both types of systems. Then we establish a hierarchy of dynamical systems, including Turing machines, cellular automata ..."
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This paper is an attempt to unify classical automata theory and dynamical systems theory. We present a notion of generalized dynamical systems, allowing us to compare properties of both types of systems. Then we establish a hierarchy of dynamical systems, including Turing machines, cellular automata and classical dynamical systems. We finish with some conclusions and motivations for future work.
Compositional Complexity in Cellular Automata: a Case Study
, 1996
"... We relate two compositional approaches of dynamical systems showing the same emergence of dynamical complexity from the interaction of two simple and similar systems attracting their underlying space to different regions: two shifting cellular automata produce complexity, as well as classical dynami ..."
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We relate two compositional approaches of dynamical systems showing the same emergence of dynamical complexity from the interaction of two simple and similar systems attracting their underlying space to different regions: two shifting cellular automata produce complexity, as well as classical dynamical systems like Cantor's relation or Smale's horseshoe map. 1 Introduction Cellular automata (CA, for short) are discretetime, discretespace, massively parallel dynamical systems. The rich variety of their behaviors as well as their universal computation ability allow their use as models in many disciplines, ranging from parallel computing to ecology or physics. Indeed, they can exhibit very simple (destruction of information) to very complex (propagation of information following complex rules) dynamics, including spatiotemporal chaos. Dynamics. In [4], the classification of CA dynamics of [32] was refined, and it was then formalized and structured in [9, 13]. This led to a hierarchy of ...
Compositional Complexity in Dynamical Systems
 In Proc. International Symposium on Nonlinear Theory and its Applications, Las Vegas
, 1996
"... This paper presents applications of the composition principle in an informal way, emphasizing the qualitative aspects of the approach. Keywords: dynamical system, complexity, composition, invariance, attraction. ..."
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This paper presents applications of the composition principle in an informal way, emphasizing the qualitative aspects of the approach. Keywords: dynamical system, complexity, composition, invariance, attraction.
Compositional Experimental Analysis of Cellular Automata: Attraction Properties and Logic Disjunction
, 1996
"... In this paper, we analyze attraction properties of elementary (i.e. Boolean, onedimensional, biinfinite) cellular automata (for short, CA). To overcome the wellknown undecidability constraints met by these systems, in addition to the classical extensive use of computer simulations, we introduc ..."
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In this paper, we analyze attraction properties of elementary (i.e. Boolean, onedimensional, biinfinite) cellular automata (for short, CA). To overcome the wellknown undecidability constraints met by these systems, in addition to the classical extensive use of computer simulations, we introduce composition: we first characterize basis CA, which we then use as building blocks to understand a whole family of CAbased systems obtained by composing them using logic disjunction. The compositional approach allows deep structured investigations, and it permits to define a new notion of dynamical complexity. 1 Experimental analysis of cellular automata Cellular automata. Cellular automata (for short, CA) are massively parallel systems obtained by composition of myriads of simple agents interacting locally, i.e. with their closest neighbors. In spite of their simplicity, the dynamics of CA is potentially very rich, and ranges from attracting stable configurations to spatiotemporal ...
Lattice FixedPoint Theorem for Class F Relations
, 1993
"... this paper seems also very promising as a basis for composition of systems [SG93]: we could take the single functions as components, and union as a first type of composition. Another essential type of composition should be a cartesian product, also extended to linked products of systems (i.e. with i ..."
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this paper seems also very promising as a basis for composition of systems [SG93]: we could take the single functions as components, and union as a first type of composition. Another essential type of composition should be a cartesian product, also extended to linked products of systems (i.e. with interactions). There, it should be very interesting to propose compositions of dynamical properties, instead of trying to compute global properties. 6 Acknowledgements
Hybrid Systems' Properties  Classification and Relation to Computer Science
"... In order to promote a deeper understanding of hybrid, i.e. mixed discrete and continuous, systems, we introduce a set of important properties of such systems and classify them. For the properties of stability and attraction which are central for continuous systems we discuss their relationship t ..."
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In order to promote a deeper understanding of hybrid, i.e. mixed discrete and continuous, systems, we introduce a set of important properties of such systems and classify them. For the properties of stability and attraction which are central for continuous systems we discuss their relationship to discrete systems usually studied in computer science. An essential result is that the meaning of these properties for discrete systems vitally depends on the used topologies. Based on the classication we discuss the utility of a renement notion based on trace inclusion. Furthermore, for proofs of stability the role of Liapunov functions as abstractions is emphasized by identifying conditions under which they dene Galois connections. 1
Paper Foldings as Chaotic Dynamical Systems
, 1996
"... A paper folding sequence is the sequence of ridges and valleys obtained by unfolding a sheet of paper which has been folded infinitely many times. To study the complexity of such sequences, we consider foldings as a dynamical system obtained by composing very simple systems. This allows to prove the ..."
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A paper folding sequence is the sequence of ridges and valleys obtained by unfolding a sheet of paper which has been folded infinitely many times. To study the complexity of such sequences, we consider foldings as a dynamical system obtained by composing very simple systems. This allows to prove the existence of a Cantor invariant set in the space of infinite landscapes, and that folding systems are chaotic on this invariant. Keywords: paper folding sequence, dynamical system, composition, chaos. 1 Introduction Although it is well known that no reasonable sheet of paper can be folded more than 7 times, a paper folding sequence is the sequence of ridges and valleys obtained by unfolding a sheet of paper which has been folded infinitely many times. Paper folding sequences and their complexity have been studied by several authors, using formal power series, continued fractions, language theory and morphisms, measure theory, group theory, etc. [11, 23, 12, 22, 2, 5]. The folding process b...