By Frederic Geurts
This self-contained monograph is an built-in research of prevalent structures outlined by way of iterated family utilizing the 2 paradigms of abstraction and composition. This incorporates the complexity of a few state-transition structures and improves knowing of advanced or chaotic phenomena rising in a few dynamical structures. the most insights and result of this paintings trouble a structural kind of complexity got by means of composition of straightforward interacting platforms representing adversarial 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 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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Additional info for Abstract Compositional Analysis of Iterated Relations: A Structural Approach to Complex State Transition Systems
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 . If s represents a sequence, |s| denotes its length. F. Geurts: Abstract Compositional Analysis of Iterated Relations, LNCS 1426, pp. 21-52, 1998. Springer-Verlag Berlin Heidelberg 1998 22 2. Dynamics of Relations Juxtaposition of symbols stands for concatenation; exponentiation stands for multiple concatenation.
To illustrate chaos in ﬁnite (but large) spaces. 2 Set-Level Dynamics and Predicate-Transformers Set-transformers are not new. They are used in general topology  and fractal theory [328, 159, 140, 28, 325]. When sets are speciﬁed by predicates, set-transformers are expressed as predicate-transformers. g. g. ). 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− ≡  wp · R · A ≡ WR · R · A ≡ pre[R](A) ∧ ¬pre[R](¬A) ≡ R−1 (A) ∩ X\R−1 (X\A) post[R] pre[R]     [this monograph] assuming that a unique sink can be reached by R when non-termination is possible, according to [150, 130].
21. Otherwise, an “undeﬁned” relation based on an “undeﬁned” element could do (see also Rem. 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.
Abstract Compositional Analysis of Iterated Relations: A Structural Approach to Complex State Transition Systems by Frederic Geurts