Products of Automata
Published on 23. July 1986
Book
Hardback
VIII, 108 pages
978-3-540-13719-1 (ISBN)
Description
Both theoretical and practical considerations motivate the repre sentation of objects as certain compositions of simpler ones. In the theory of automata this observation has led to the concepts of pro ducts and complete systems of automata. In the general form of the products of automata all the component automata are fed back to one another. With this very broad notion of products, the realization of automata with large numbers of states by means of compositions of basic components is a highly involved process; this increases the possibility of errors. In order to decrease the complexity of feedbacks, a hierarchy of products called lXi-pro ducts was introduced some 10 years ago, where i runs over the set of all non-negative integers. In an IXcproduct the index set of the component automata is linearly ordered. The input of each automaton in the product may depend on the states of all automata preceding it, i. e. , all component automata steer all those automata which follow them in the product. Furthermore, at most the next i-I automata (including itself) may be fed back to the input of a given component automaton. Thus for iXcproducts the lengths of feedbacks are at most i. The aim of this monograph is to give a systematic account of iXi-Products. It consists of five chapters, a reference section, and an index. The first chapter contains the necessary concepts and results from universal algebra, automata, and sequential machines.
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Content
1. Basic Concepts and Preliminaries.- 1.1 Sets and Relations.- 1.2 Algebraic Structures.- 1.3 Automata and Sequential Machines.- 1.4 Products and Complete Systems of Sequential Machines and Automata.- 2. Homomorphic Representations.- 2.1 A Homomorphically Complete System for the ?0-Product.- 2.2 A Minimal Homomorphically Complete System with Respect to the ?0-Product.- 2.3 Homomorphic Representations of Automata by ?0- and ?1-Products of Smaller Automata.- 2.4 Homomorphically Complete Systems for ?i-Products withu i> 1.- 2.5 Comparison of the Homomorphic Representation Powers of ?i-Products.- 2.6 Homomorphically ?i-Simple Automata.- 2.7 A Decidability Result.- 3. Isomorphic Representations.- 3.1 Embedding into ?i-Products of Automata with Fewer States Than a Given Integer.- 3.2 Isomorphically Complete Systems for the ?0-Product.- 3.3 Isomorphically Complete Systems for ?i-Products with i? 1.- 3.4 Comparison of the Isomorphic Representation Powers of ?i-Products.- 3.5 Isomorphically Complete Classes for Nilpotent Automata.- 4. Generalized Products and Simulations.- 4.1 Basic Concepts.- 4.2 Simulations by Generalized ?0-Products.- 4.3 Simulations by Generalized ?1-Products.- 4.4 Simulations by Generalized Products and Generalized ?i-Products with i> 1.- 4.5 Homomorphic Representations by a Restricted Form of Generalized Products.- 5. Representation of Automaton Mappings in Finite Length. Infinite Products.- 5.1 Metric Completeness.- 5.2 Equational Classes of Automata.- 5.3 Metric Equivalence of Products.- Bibliographical Remarks.- References.