EATCS Monographs on Theoretical Computer Science

Transcription

1 EATCS Monographs on Theoretical Computer Science Volume 5 Editors: W. Brauer G. Rozenberg A. Salomaa Advisory Board: G. Ausiello S. Even M. Nivat Chr. Papadimitriou A.L. Rosenberg D. Scott

2 EATCS Monographs on Theoretical Computer Science Vol. 1: K. Mehlhorn: Data Structures and Algorithms 1: Sorting and Searching. XIV, 336 pages, 87 figs Vol. 2: K. Mehlhorn: Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness. XII, 260 pages, 54 figs Vol. 3: K. Mehlhorn: Data Structures and Algorithms 3: M ultidimensional Searching and Computational Geometry. XII, 284 pages, 134 figs Vol. 4: W. Reisig: Petri Nets. An Introduction. X, 161 pages, 111 figs Vol. 5: W. Kuich, A. Salomaa: Semirings, Automata, Languages. IX, 374 pages, 23 figs Vol. 6: H. Ehrig, B. Mahr: Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. XI, 321 pages

3 Werner Kuich Arta Salamaa Semirings, Automata, Languages With 23 Figures Springer-Verlag Berlin Heidelberg New York Tokyo

4 Editors Prof. Dr. Wilfried Brauer Institut fur Informatik der Technischen Universitat Arcisstr. 21, 8000 Munchen 2, Germany Prof. Dr. Grzegorz Rozenberg Institute of Applied Mathematics and Computer Science University of Leiden, Wassenaarseweg 80, P. O. Box RA Leiden, The Netherlands Prof. Dr. Arto Salomaa Department of Mathematics, University of Turku Turku 50, Finland Authors Prof. Dr. Werner Kuich Institut fur Algebra und Diskrete Mathematik Abteilung Theoretische Informatik Technische Universitat Wi en Wiedner Hauptstraf3e Wien, Austria Prof. Dr. Arto Salomaa Department of Mathematics University of Turku Turku, Finland ISBN-13: e-isbn-13: DOl: / Library of Congress Cataloging in Publication Data. Kuich, Werner, Semirings, automata, languages. (EATCS monographs on theoretical computer science; vol. 5) Bibliography: p. Includes indexes. 1. Machine theory. 2. Semirings (Mathematics). 3. Formal languages. I. Salomaa, Arto. II. Title. III. Series: EATCS monographs on theoretical computer science; v. 5. QA267.K This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "Verwertungsgesellschaft Wort", Munich. Springer-Verlag Berlin Heidelberg 1986 Sof'tcover reprint of the hardcover 1 st edition 1986 Typesetting: Hagedorn, Berlin. Printing: Color-Druck, Berlin. Bookbinding: Schiineberger Buchbinderei, Berlin. 2145/

5 To Beate, Kaarina and Dancwart

6 Acknowledgments This book would probably not exist without Hermann Maurer. Being a close friend of both authors, he knew that the first author already had many results and a smaller book in this research area, which for years had belonged to the main interests of the second author. With efficiency typical for him, Hermann Maurer got the authors to work together. Juha Honkala has read through most of the manuscript; his unusually deep mathematical insight has given rise to many really invaluable comments that have essentially improved the presentation in many respects. Otherwise, it is rather difficult to list all persons who have in some way or other contributed to this book. Martin Goldstern, Georg Karner, Gerhard Kuster, Valtteri Niemi, Helmut Prodinger, Friedrich J. Urbanek, Wolfgang Wechler and Emo Welzl have also read through substantial parts of the manuscript and given very helpful comments. Special thanks are due to Franz Halter-Koch, Ludwig Reich and Kalevi Suo min en for many helpful discussions about the tools required in Section 16. We have benefited from discussions with or comments from Gerd Baron, Juhani Karhumaki, Grzegorz Rozenberg, Matti Soittola and Derick Wood. The difficult task of typing the manuscript was performed in an excellent fashion by Li Urban. The figures were drawn by Michael Kunerth. Finally we want to thank Suomen Akatemia and Osterreichische Forschungsgemeinschaft for financial support required for our trips between Turku and Wien. WienjTurku, September 1985 Werner Kuich Arto Salomaa

7 Contents Introduction Chapter I. Linear Algebra 1. Semirings and Power Series. 2. Convergence, Equations and Identities 3. Strong Convergence and Cycle-Free Power Series 4. Matrices, Linear Systems and Identities 5. Semirings with Particular Properties 6. Morphisms and Representations Chapter II. Automata Automata in Terms of Matrices. 8. Rational Power Series and Decidability. 9. Rational Transductions Pushdown Automata 11. Abstract Families of Power Series 12. Substitutions Reset Pushdown Automata and Counter Automata Chapter III. Algebraic Systems Algebraic Series and Context-Free Languages 15. The Super Normal Form Commuting Variables: Decidability and Parikh's Theorem Historical and Bibliographical Remarks References. Subject Index Symbol Index