Download Cellular Automata and Discrete Complex Systems: 21st IFIP WG by Jarkko Kari PDF

By Jarkko Kari

This quantity constitutes the completely refereed court cases of the twenty first foreign Workshop on mobile Automata and Discrete complicated structures, AUTOMATA 2015, held in Turku, Finland, in June 2015. This quantity comprises four invited talks in full-paper size and 15 usual papers, that have been rigorously reviewed and chosen from a complete of 33 submissions. themes of curiosity contain, the next points and lines of such platforms: dynamical, topological, ergodic and algebraic facets; algorithmic and complexity concerns; emergent houses; formal language processing features; symbolic dynamics; types of parallelism and allotted structures; timing schemes; phenomenological descriptions; clinical modeling; and useful applications.

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Extra resources for Cellular Automata and Discrete Complex Systems: 21st IFIP WG 1.5 International Workshop, AUTOMATA 2015, Turku, Finland, June 8-10, 2015. Proceedings

Example text

Proof. Again, we only need to prove the claim on full shifts. Let f ∈ Aut(S Z ) be arbitrary. We again show that an algorithm for the problem in the statement also decides whether f is finite. Without loss of generality, we may assume G is a subgroup of a symmetric group Sk , so that G acts nontrivially on [1, k]. Let F = {fg | g ∈ G} be the subgroup of Aut((S k )Z ) permuting the tracks according to this embedding: fg (x1 , . . , xk ) = (xg−1 (1) , xg−1 (2) , . . , xg−1 (k) ). 21 In fact, the CA constructed in [KO08] are already time-symmetric, but the idea on permuting tracks illustrates Theorem 2 better.

The paper [KR90] contains many more examples of subgroups that can be embedded (for example, fundamental groups of 2-manifolds), and proves that the set of subgroups of Aut(X) is closed under finite extensions when X is a full shift. That is, if H ≤ Aut(X) where X is a full shift, and [G : H] < ∞, then G ≤ Aut(X). Note that by Lemma 3, the groups Aut({0, 1}Z ) and Aut({0, 1, 2}Z ) embed into each other, and thus have the same subgroups. 19 This is one of the open problems in symbolic dynamics listed in [Boy08].

Again, it is enough to prove the result for full shifts by Lemma 3. We show that an algorithm for this problem would give an algorithm for the torsion problem as well. Let f ∈ Aut(S Z ) be given. Consider the full shift (S × S)Z , and define g, h ∈ Aut((S × S)Z ) by g(x, y) = (f (y), f −1 (x)) and h(x, y) = (y, x). Then (g ◦ h)(x, y) = (f (x), f −1 (y)). If f is infinite, then the orbit of some point x is infinite by Theorem 2, and thus the orbit of (x, y) is of infinite order by Theorem 2 for any y ∈ S Z .

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