Download e-book for kindle: Algebraic Theory of Automata Networks (SIAM Monographs on by Pal Domosi, Chrystopher L. Nehaniv

By Pal Domosi, Chrystopher L. Nehaniv

ISBN-10: 0898715695

ISBN-13: 9780898715699

Algebraic conception of Automata Networks investigates automata networks as algebraic buildings and develops their idea based on different algebraic theories, comparable to these of semigroups, teams, earrings, and fields. The authors additionally examine automata networks as items of automata, that's, as compositions of automata bought by way of cascading with no suggestions or with suggestions of assorted limited varieties or, most widely, with the suggestions dependencies managed by way of an arbitrary directed graph. This self-contained e-book surveys and extends the elemental leads to regard to automata networks, together with the most decomposition theorems of Letichevsky, of Krohn and Rhodes, and of others.

Algebraic conception of Automata Networks summarizes crucial result of the prior 4 a long time concerning automata networks and provides many new effects stumbled on because the final booklet in this topic used to be released. It includes a number of new tools and unique suggestions now not mentioned in different books, together with characterization of homomorphically whole periods of automata lower than the cascade product; items of automata with semi-Letichevsky criterion and with none Letichevsky standards; automata with keep an eye on phrases; primitive items and temporal items; community completeness for digraphs having all loop edges; entire finite automata community graphs with minimum variety of edges; and emulation of automata networks by way of corresponding asynchronous ones.

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Extra info for Algebraic Theory of Automata Networks (SIAM Monographs on Discrete Mathematics and Applications, 11)

Example text

N}, we shall write F ( l , . . , n) = ( f ( 1 ) , . . , c f(n) ). Take a digraph D = (V, E) with an ordered set V = { v 1 , . . , vn}, n > 1 of vertices. Place a coin ci onto vi for every i = 1,... ,n such that ci Cj whenever i j for some 1 i,j n. Let us say that a vertex is free if either it is covered by a coin cn or there exists another vertex that is covered by the same type of coin. ) Suppose that we are allowed to change the coins according to the following conditions: (1) For every i, j = 1,...

Then duplicate the coin C2 and put one of its copies to the vertex 3 (leaving the other one on the vertex 2). Then we get the configuration (c 1 , c2, c 3 , . . , c n - 1 ). Shift the coins right cyclically n — 1 times reaching (c 2 , c 2 , c 3 , . . , c n - 1 , c 1 ). Now we can shift the first m coins right cyclically m — 1 times, which results in ( c 2 , . . , c n - 1 , c 1 ). We exchange the coin c2 of the first vertex for a copy of c1 covering the last vertex. Finally, shift again the first m coins right cyclically, obtaining the configuration (c2, c1, c 3 , .

Formally, define the mappings [shifting first m right cyclically], [collapsing [collapsing n to 1], [col- lapsing By an elementary computation we may check that is allowed and This shows that D penultimately realizes y with respect to 1 , . . , n. Assuming again that every vertex iis covered by a coin ci, i = 1 , . . , n, we distinguish two cases. Case 1. m — n — 1. Repeat u — 1 times the above procedure of our proof resulting in F1. , c n - 1 , c 1 , . . , c n - u , c n - u + 1 ) such that u is covered by c\.

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Algebraic Theory of Automata Networks (SIAM Monographs on Discrete Mathematics and Applications, 11) by Pal Domosi, Chrystopher L. Nehaniv

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