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

By Frederic Geurts

This self-contained monograph is an built-in learn of familiar platforms outlined via iterated family members utilizing the 2 paradigms of abstraction and composition. This contains the complexity of a few state-transition platforms and improves figuring out of advanced or chaotic phenomena rising in a few dynamical structures. the most insights and result of this paintings challenge a structural kind of complexity bought via composition of straightforward interacting structures representing hostile attracting behaviors. This complexity is expressed within the evolution of composed platforms (their dynamics) and within the kin among their preliminary and ultimate states (the computation they realize). The theoretical effects are proven via reading dynamical and computational houses of low-dimensional prototypes of chaotic structures, high-dimensional spatiotemporally complicated platforms, and formal platforms.

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

Example text

55 (Diameter). 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).

The limit of this sequence is Xn . Based on this we can propose an evolution scheme: the basic case is f 0 (A) = A; for all successor ordinals n = 0 ∈ O s , f n (A) = f (f n−1 (A)); and for all limit ordinals n ∈ O l , f n (A) = m ω, the sequence of successive iterates (Xi )i<ηL is stable in at most ω steps iff f is andcontinuous.

Proof. 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.

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