By Frederic Geurts

This self-contained monograph is an built-in research of regularly occurring platforms outlined through iterated kinfolk utilizing the 2 paradigms of abstraction and composition. This incorporates the complexity of a few state-transition platforms and improves figuring out of complicated or chaotic phenomena rising in a few dynamical platforms. the most insights and result of this paintings crisis a structural kind of complexity bought via composition of straightforward interacting platforms representing hostile attracting behaviors. This complexity is expressed within the evolution of composed platforms (their dynamics) and within the relatives among their preliminary and ultimate states (the computation they realize). The theoretical effects are established via studying dynamical and computational homes of low-dimensional prototypes of chaotic structures, high-dimensional spatiotemporally complicated platforms, and formal structures.

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

**Example text**

As f is ﬁnite, there is only a ﬁnite number of such yi . Since (Xi )i is decreasing, one of these yi belongs to all Xi . Thus, ∃yj ∈ ∩i Xi , (yj , x) ∈ f −1 ≡ x ∈ f −1 (∩i Xi ). ⇒Let us now suppose that f is not ﬁnite. Let x be such that ∃(yi )i an inﬁnite sequence of distinct states of f −1 (x). For each i, we deﬁne Pi = {yi , yi+1 , yi+2 , · · ·}. By construction, (Pi )i is a decreasing sequence, whose intersection is empty. Thus f −1 (∩i Pi ) = f −1 (∅) = ∅. On the other hand, ∀i, x ∈ f −1 (Pi ), and x ∈ ∩i f −1 (Pi ), which entails a contradiction.

Finally, we have to discuss how to extend the notion of nondeterministic dynamics (Def. 9) to transﬁnite sequences. We use accumulation points again. 79 (Transﬁnite nondeterministic forward dynamics). The transﬁnite nondeterministic forward dynamics of a RDS (X, f ) from a set A ⊆ X of initial conditions is θ(A, f ) = {s ∈ X O | (s ∈ A) 0 ∧(∀n = 0 ∈ O s , (sn−1 , sn ) ∈ f ) ∧(∀n ∈ O l , sn ∈ i

A ⊆ B ⇒ ∃u ∈ B, x ∈ f (u) ≡ x ∈ f (B). The following trivial result involves monotonic relations. 36. Let f be a relation, and (Xi )i be any sequence of subsets of X. Then f (∩i Xi ) ⊆ ∩i f (Xi ) f (∪i Xi ) ⊇ ∪i f (Xi ). 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.