By Henning Mortveit, Christian Reidys

This introductory textual content to the category of Sequential Dynamical platforms (SDS) is the 1st textbook in this well timed topic. pushed by means of a number of examples and thought-provoking difficulties all through, the presentation deals solid foundational fabric on finite discrete dynamical structures, which then leads systematically to an advent of SDS. From a huge diversity of issues on constitution thought - equivalence, mounted issues, invertibility and different section area houses - thereafter SDS kinfolk to graph conception, classical dynamical platforms in addition to SDS functions in laptop technological know-how are explored. this can be a flexible interdisciplinary textbook.

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**Example text**

The automorphisms of Y form a group under function composition. This is the automorphism group of Y , and it is denoted Aut(Y ). Let Y and Z be undirected graphs and let ϕ : Y −→ Z be a graph morphism. 5) are all surjective or all injective, respectively. A graph morphism that is both locally surjective and locally injective is called a local isomorphism or a covering. 1. 1 is surjective but not locally surjective. −→ Y = =Z Fig. 1. 1. 1 Simple Graphs and Combinatorial Graphs An undirected graph Y is a simple graph if the mapping {e, e} → {ω(e), τ (e)} is injective.

Closely related to this is the #predecessor problem, which asks for the number of predecessors of a system state x. 1) in, for example, [17]. Exactly as for the previous problems the predecessor existence problem is NP-complete in the general case, but can be solved eﬃciently for restricted classes of vertex functions and/or graphs. Examples include SDS where the vertex functions are given by logical And 4 The problem is PSPACE-complete; see, for example, [15]. 20 1 What is a Sequential Dynamical System?

7. Let Y be a graph with adjacency matrix A. The number of walks of length k in Y that start at vertex vi and end at vertex vj is [Ak ]i,j , the (i, j) entry of the kth power of A. The result is proved by induction. Obviously, the assertion holds for k = 1. Assume it is true for k = m. We can show that it holds for k = m + 1 by decomposing a walk of length m + 1 from vertex vi to vertex vj into a walk of length m from the initial vertex vi to an intermediate vertex vk followed by a walk of length 1 from the intermediate vertex vk to the ﬁnal vertex vj .