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Abstract Compositional Analysis of Iterated Relations: A Structural Approach to Complex State Transition Systems

Abstract Compositional Analysis of Iterated Relations: A Structural Approach to Complex State Transition Systems

Geurts, Frederic

Paperback

Series: Lecture Notes in Computer Science, Book 1426

Medical ReferenceGeneral Computers

ISBN10: 3540655069
ISBN13: 9783540655060
Publisher: Springer Nature
Published: Dec 18 1998
Pages: 280
Weight: 0.91
Height: 0.61 Width: 6.14 Depth: 9.21
Language: English
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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