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View CartAlgebraic Theory of Automata Networks
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Description
Algebraic Theory of Automata Networks investigates automata networks as algebraic structures and develops their theory in line with other algebraic theories, such as those of semigroups, groups, rings, and fields. The authors also investigate automata networks as products of automata, that is, as compositions of automata obtained by cascading without feedback or with feedback of various restricted types or, most generally, with the feedback dependencies controlled by an arbitrary directed graph. This self-contained book surveys and extends the fundamental results in regard to automata networks, including the main decomposition theorems of Letichevsky, of Krohn and Rhodes, and of others.
Algebraic Theory of Automata Networks summarizes the most important results of the past four decades regarding automata networks and presents many new results discovered since the last book on this subject was published. It contains several new methods and special techniques not discussed in other books, including characterization of homomorphically complete classes of automata under the cascade product; products of automata with semi-Letichevsky criterion and without any Letichevsky criteria; automata with control words; primitive products and temporal products; network completeness for digraphs having all loop edges; complete finite automata network graphs with minimal number of edges; and emulation of automata networks by corresponding asynchronous ones.
Keywords: Finite Automata, Automata Networks, Products of Automata
Table of Contents
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Front Matter
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1. Preliminaries
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5. Letichevsky's Criterion
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Back Matter
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Excerpt
An automata network is a collection of automata connected together according to a directed graph D. The vertices of D are considered as automata and the edges indicate the existence of communication links. Thus D has no parallel edges. Each automaton can change its state at discrete time steps as a local transition function of the states and a global input, and synchronous action of the local state transitions defines a global transition on the entire network. We investigate automata networks as algebraic structures and develop their theory in line with other algebraic theories, such as those of semigroups, groups, rings, and fields.
In this monograph we restrict ourselves almost exclusively to finite automata networks (with notable exceptions in the study of asynchronous networks) for two reasons. This introductory monograph is devoted to the most fundamental cases. These occur when the network is finite: there are only finitely many component automata in the network (i.e., the interconnection digraph D is finite) and all component automata are also finite, having only finitely many states. On the other hand, finiteness is a natural constraint arising for real-world networks, including those in computational and technical applications.
Algebraic interpretations arise from consideration of the semigroup of transformations induced on the set of states by all possible finite sequences of inputs, but they also enter the subject in other ways when we study division relations of automata and the completeness of networks.
©2005 SIAM
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