Abstract Compositional Analysis of Iterated Relations: A by Frederic Geurts

By Frederic Geurts

This self-contained monograph is an built-in research of standard platforms outlined by means of iterated kin utilizing the 2 paradigms of abstraction and composition. This comprises the complexity of a few state-transition structures and improves knowing of advanced or chaotic phenomena rising in a few dynamical platforms. the most insights and result of this paintings quandary a structural type of complexity bought via composition of straightforward interacting platforms representing antagonistic attracting behaviors. This complexity is expressed within the evolution of composed structures (their dynamics) and within the relatives among their preliminary and ultimate states (the computation they realize). The theoretical effects are confirmed through examining dynamical and computational houses of low-dimensional prototypes of chaotic platforms, high-dimensional spatiotemporally complicated structures, and formal platforms.

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41. This slightly differs from [92], where the or-continuity of the predicate-transformer wp is also restricted to bounded nondeterminism. This difference is due to the fact that, stated in our framework, R being a relation on X and P ⊆ X, wp · R · P is not equivalent to R−1 (P ) but wp · R · P = R−1 (P ) ∩ (X\R−1 (X\P )). The set-difference modifies the result because it is no more or-continuous, since it is even not monotonic but anti-monotonic. Of course, and-continuity suffers from the same drawback.

G. K (X) the nonempty compact subsets of X. 72 is based, is too restrictive because it happens that successive iterations from a set A are neither increasing nor decreasing. In this case, one would like to have notion of limit that remains compatible with these particular cases. In terms of RDS, it is easy to define such a notion since every sequence in a compact set has accumulations points in this set. These accumulation points can serve as limit elements. 74 (Limit set). Let A be a subset of X.

X ∈ f (A) ∃u ∈ A, x ∈ f (u) ≡ ≡ ∃u ∈ A, x ∈ {v | (u, v) ∈ f } * Hyp. f ⊆ g ⇒ ∃u ∈ A, x ∈ {v | (u, v) ∈ g} ≡ x ∈ g(A). 35 (Monotonicity). Let f be a relation on X, and A, B ⊆ X, then A ⊆ B ⇒ f (A) ⊆ f (B). Proof. x ∈ f (A) ≡ ∃u ∈ A, x ∈ f (u) * Hyp. 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.

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