Abstract Compositional Analysis of Iterated Relations

Name: Abstract Compositional Analysis of Iterated Relations
Author: Frederic Geurts
ISBN: 9783540655060

Frederic Geurts

Language: English

Published by Springer Berlin Heidelberg, 1998

3540655069 / 9783540655060

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Prologue: Aims, Themes, and Motivations.- Prologue: Aims, Themes, and Motivations.- Mathematical Framework: Iterated Relations and Composition.- Dynamics of Relations.- Dynamics of Composed Relations.- Abstract Complexity: Abstraction, Invariance, Attractio.

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Title: Abstract Compositional Analysis of Iterated Relations
Author: Frederic Geurts
Publisher: Springer Berlin Heidelberg
Publication year: 1998
Condition: New
Binding: Kartoniert / Broschiert
Language: English
ISBN 10: 3540655069
ISBN 13: 9783540655060
Seller catalogs: Mathematik/Naturwissenschaften/Technik/Medizin

State-transition systems model machines, programs, and speci?cations [20, 23,284,329],butalsothegrowthanddeclineofantpopulations,?nancial markets, diseases and crystals [22, 35, 178, 209, 279]. In the last decade, thegrowinguseofdigitalcontrollersinvariousenvironmentshasentailed theconvergenceofcontroltheoryandreal-timesystemstowardhybrids- tems [16] by combining both discrete-event facets of reality with Nature's continuous-time aspects. The computing scientist and the mathematician have re-discovered each other. Indeed, in the late sixties, the programming language Simula, "father" of modern object-oriented languages, had already been speci?cally designed to model dynamical systems [76]. Today,theimportanceofcomputer-basedsystemsinbanks,telecom- nication systems, TVs, planes and cars results in larger and increasingly complex models. Two techniques had to be developed and are now fruitfully used to keep analytic and synthetic processes feasible: composition and - straction.Acompositionalapproachbuildssystemsbycomposingsubsystems that are smaller and more easily understood or built. Abstraction simpli?es unimportantmattersandputstheemphasisoncrucialparametersofsystems. Inordertodealwiththecomplexityofsomestate-transitionsystemsand tobetterunderstandcomplexorchaoticphenomenaemergingoutofthe behaviorofsomedynamicalsystems,theaimofthismonographistopresent ?rststepstowardtheintegratedstudyofcompositionandabstractionin dynamical systems de?ned by iterated relations. Themaininsightsandresultsofthisworkconcernastructuralorm f of complexityobtainedbycompositionofsimpleinteractingsystemspresenting opposedattractingbehaviors.Thiscomplexityexpressesitselfintheevo- tionofcomposedsystems,i.e.,theirdynamics,andintherelationsbetween their initial and ?nal states, i.e., the computations they realize. The theor- ical results presented in the monograph are then validated by the analysis ofdynamicalandcomputationalpropertiesoflow-dimensionalprototypesof chaotic systems (e.g. Smale horseshoe map, Cantor relation, logistic map), high-dimensional spatiotemporally complex systems (e.g. cellular automata), and formal systems (e.g. paperfoldings, Turing machines). Acknowledgements. ThismonographisarevisionofmyPhDthesiswhichwas completed at the Universit' e catholique de Louvain (Belgium) in March 96. VIII Preface The results presented here have been in?uenced by many people and I would like to take this opportunity to thank them all.

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