Download Abstract Compositional Analysis of Iterated Relations: A by Frederic Geurts PDF

By Frederic Geurts

This self-contained monograph is an built-in research of normal platforms outlined via iterated family members utilizing the 2 paradigms of abstraction and composition. This contains the complexity of a few state-transition structures and improves realizing of complicated or chaotic phenomena rising in a few dynamical platforms. the most insights and result of this paintings main issue a structural type of complexity received by way of composition of straightforward interacting structures representing hostile attracting behaviors. This complexity is expressed within the evolution of composed structures (their dynamics) and within the family members among their preliminary and ultimate states (the computation they realize). The theoretical effects are demonstrated by means of examining dynamical and computational houses of low-dimensional prototypes of chaotic structures, high-dimensional spatiotemporally complicated platforms, and formal platforms.

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

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Graph of f (x) (and y = x, dotted line) f (x) = 1 12 x 1 12 x + + 1 3 7 11 4 on [0, 11 ) 4 on [ 11 , 1]. 4 4 The part of f defined on [0, 11 ) seems to have a fixed point in 11 but there it is not defined. Actually, at this point, the right branch is defined, for which 84 . e. the there is a true fixed point in 121 1 contractivity factor is strictly smaller than 1 (here, it is equal to 12 ). 4 Thus, starting from any point in [0, 11 ), ω iterations are needed to reach the 4 “virtual” attracting fixed point 11 .

They are used in general topology [9] and fractal theory [328, 159, 140, 28, 325]. When sets are specified by predicates, set-transformers are expressed as predicate-transformers. g. g. [284]). The interesting relationships between relations, predicate-transformers, multi-valued functions, and their algebraic construction have been investigated in [39, 112]. We summarize below the equivalences between set-transformers and existing operators: R ≡ R−1 ≡ [this monograph] and R+ ≡ R− ≡ [286] wp · R · A ≡ WR · R · A ≡ pre[R](A) ∧ ¬pre[R](¬A) ≡ R−1 (A) ∩ X\R−1 (X\A) post[R] pre[R] [284] [91] [246] [284] [this monograph] assuming that a unique sink can be reached by R when non-termination is possible, according to [150, 130].

Let (X, d) be a metric space. Then, for any subset A ⊆ X, its diameter is diam(A) = sup d(x, y). x,y∈A In general, we use a RDS, which involves a compact metric space (X, d), where d is the metric defined on X. Relations are regarded as multi-valued functions from X to P(X). This requires a metric on P(X). We consider the standard Hausdorff metric. 56 (Hausdorff metric). Let (X, d) be a metric space. The Hausdorff metric h on P(X) is given as follows: ∀A, B ∈ P(X), h(A, B) = max{h (A, B), h (B, A)} where h (A, B) = sup h (x, B) x∈A h (x, B) = inf d(x, y).

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