By Frederic Geurts

State-transition platforms version machines, courses, and speci?cations [20, 23,284,329], butalsothegrowthanddeclineofantpopulations, ?nancial markets, ailments and crystals [22, 35, 178, 209, 279]. within the final decade, thegrowinguseofdigitalcontrollersinvariousenvironmentshasentailed theconvergenceofcontroltheoryandreal-timesystemstowardhybrids- tems [16] by way of combining either discrete-event features of truth with Nature's continuous-time points. The computing scientist and the mathematician have re-discovered one another. certainly, within the past due sixties, the programming language Simula, "father" of recent object-oriented languages, had already been speci?cally designed to version dynamical platforms [76]. at the present time, theimportanceofcomputer-basedsystemsinbanks, telecom- nication structures, TVs, planes and vehicles ends up in greater and more and more advanced types. options needed to be built and are actually fruitfully used to maintain analytic and artificial methods possible: composition and - straction.Acompositionalapproachbuildssystemsbycomposingsubsystems which are smaller and extra simply understood or outfitted. Abstraction simpli?es unimportantmattersandputstheemphasisoncrucialparametersofsystems. Inordertodealwiththecomplexityofsomestate-transitionsystemsand tobetterunderstandcomplexorchaoticphenomenaemergingoutofthe behaviorofsomedynamicalsystems, theaimofthismonographistopresent ?rststepstowardtheintegratedstudyofcompositionandabstractionin dynamical platforms de?ned through iterated relatives. Themaininsightsandresultsofthisworkconcernastructuralorm f of complexityobtainedbycompositionofsimpleinteractingsystemspresenting opposedattractingbehaviors.Thiscomplexityexpressesitselfintheevo- tionofcomposedsystems, i.e., theirdynamics, andintherelationsbetween their preliminary and ?nal states, i.e., the computations they detect. The theor- ical effects offered within the monograph are then proven by means of the research ofdynamicalandcomputationalpropertiesoflow-dimensionalprototypesof chaotic platforms (e.g. Smale horseshoe map, Cantor relation, logistic map), high-dimensional spatiotemporally complicated structures (e.g. mobile automata), and formal structures (e.g. paperfoldings, Turing machines). Acknowledgements. ThismonographisarevisionofmyPhDthesiswhichwas accomplished on the Universit e catholique de Louvain (Belgium) in March ninety six. VIII Preface the consequences offered right here were in?uenced by means of many of us and that i want to take this chance to thank all of them.

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**Additional info for Abstract Compositional Analysis of Iterated Relations: A Structural Approach to Complex State Transition Systems (Lecture Notes in Computer Science) (v. 1426)**

**Sample text**

We have ∀i, ∩i Xi ⊆ Xi , and monotonicity of f (Prop. 35) gives ∀i, f (∩i Xi ) ⊆ f (Xi ). This entails f (∩i Xi ) ⊆ ∩i f (Xi ). By monotonicity, ∀i, Xi ⊆ ∪i Xi ⇒ f (Xi ) ⊆ f (∪i Xi ). Hence, ∪i f (Xi ) ⊆ f (∪i Xi ). Stronger properties are interesting, where inclusions are replaced by equalities. Intersection is equivalent to conjunction, and union is equivalent to disjunction, whence the generic term “junctivity”. Such junctivity properties are useful when using various ﬁxpoint theorems. All of them are further discussed in [284, 93].

Finally, nondeterminism and probabilities can be related, like in the semantics of programs.

Therefore, we introduce an iteration scheme based on transﬁnite ordinal numbers, which generalizes classical iteration schemes. 5. Before introducing this chapter, some useful notational conventions are ﬁrst mentioned. They concern sequences of arbitrary spaces, or words of formal languages. 1 (Sequences, words). Let X is be an arbitrary space (resp. an alphabet). Then X ≤n is the set of sequences (resp. words) of length smaller than n, X n is the set of sequences of length n, X ∗ is the set of ﬁnite sequences of X (including the empty sequence ε), X ω is the set of inﬁnite sequences, X ∞ = Σ ∗ ∪ Σ ω , X n>ω is the set of sequences longer than ω, and X O = X ∗ ∪ X ω ∪ X n>ω .