The Collected Works of J. Richard Biichi

The Collected Works of J. Richard Biichi

J. Richard Biichi, 1983

Saunders Mac Lane Editors Dirk Siefkes The Collected Works of J. Richard Biichi With 60 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong

Saunders Mac Lane Department of Mathematics University of Chicago Chicago, Illinois 60637 USA Dirk Siefkes Technische Universitat Berlin Fachberich Informatik D-lOOO Berlin 10 Federal Republic of Germany Library of Congress Cataloging-in-Publication Data Biichi, J. Richard. [Works. 1990) The collected works of J. Richard BiichiJ Saunders Mac Lane, Dirk Siefkes, editors. p. em. ISBN-13: 978-1-4613-8930-9 e-isbn-13: 978-1-4613-8928-6 DOl: 10.1007/978-1-4613-8928-6 1. Machine theory. 2. Logic, symbolic and mathematical. 3. Biichi, J. Richard. I. Mac Lane, Saunders. II. Siefkes, Dirk. III. Title. QA267.B79 1990 511.3-dc20 89-21769 Printed on acid-free paper. 1990 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1 st edition 1990 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. 9 8 7 6 5 4 3 2 1

Preface J. Richard Biichi is well known for his work in mathematical logic and theoretical computer science. (He himself would have sharply objected to the qualifier "theoretical," because he more or less identified science and theory, using "theory" in a broader sense and "science" in a narrower sense than usual.) We are happy to present here this collection of his papers. I (DS)1 worked with Biichi for many years, on and off, ever since I did my Ph.D. thesis on his Sequential Calculus. His way was to travel locally, not globally: When we met we would try some specific problem, but rarely discussed research we had done or might do. After he died in April 1984 I sifted through the manuscripts and notes left behind and was dumbfounded to see what areas he had been in. Essentially I knew about his work in finite automata, monadic second-order theories, and computability. But here were at least four layers on his writing desk, and evidently he had been working on them all in parallel. I am sure that many people who knew Biichi would tell an analogous story. Therefore when Saunders Mac Lane asked me to help him edit a volume of collected papers of Biichi, I gladly accepted, although I was afraid too. Walter Kaufmann-Biihler from Springer-Verlag, who had struggled with Biichi for many years to get him to finish his book on a theory of automata, grammars, and terms, encouraged us. When Walter sadly died, Lynn Montz took over, so both books are now Springer volumes. We are grateful to the Springer people for that, and thank them for their help. I (SM)1 first met Richard in 1947. I was then spending six months in study at the Eidgenossische Technische Hochschule in Ziirich. Richard and I found and discussed common interests in Boolean algebras and in the foundations of mathematics. One week early in the spring of 1948 we went on a skiing trip together to St. Moritz, where Richard knew all the ropes and found a good pension for our stay. We skied up and down the slopes and down the glacier, talking the while about logic. Since then it has been my privilege to 1 The preface is written by both editors. Where necessary we add initials to indicate who is writing. v

PREFACE follow his career and now to share in this presentation of his varied and decisive work. At our request, several people wrote comments on different parts of Biichi's work, to introduce and evaluate what he did in an area. All of them knew Biichi personally, be it through a single visit or as a year-long colleague. We made it a policy, however, not to have coworkers of Biichi write commentsexcept for some help from doctoral students. Thus nine comments appear herein, each with the corresponding section of Biichi's work. These comments turned out quite differently, some involving a personal attitude, some expositing a whole area. We feel this is good, since it reflects the different reactions Biichi could draw from people. We thank all of them for the work they did. Many people helped by contributing to the volume in other waysyou will see their names while reading-or from the background. We are personally most indebted to Sylvia Biichi and Leonard Lipshitz, who gave time and space, advice, memories, material support, and a cheerful spirit when necessary. Walter Schnyder helped as both a colleague and a friend of Biichi. Beat Glaus from the ETH library in Zurich indexed and stored the many boxes of Biichi's notes and papers; through his immense effort all material is now publicly available. 2 Also I (DS) thank the people in Berlin who would sometimes rather have seen me doing my regular work at home or in my office than "Biichi again." I thank especially Helga Barnewitz for her excellent typing and my wife for loving patience. At the same time I (SM) thank Ann Kauth and Lynette Whalum for quickly typing the numerous letters needed to keep this project on track. SAUNDERS MAC LANE DIRK SIEFKES 2 ETH-Bibliothek, Riimistr. 101, CH-8092 Ziirich, Schweiz. vi

Contents Preface................................. v Publications................................................. xi Permissions... xv Part 1. The Person and His Work J. Richard Biichi (1924-1984).................................. 2 Dirk Siefkes, Paul Young, and Leonard Lipshitz The Life of J. Richard Biichi................................... 4 Sylvia Biichi The Work of J. Richard Biichi... 7 Dirk Siefkes The Role of Biichi's Automata in Computer Science......... 18 E. Allen Emerson J. Richard Biichi's Doctoral Students............................ 23 Abstracts Published by J. Richard Biichi......................... 24 Part 2. The Publications, with Comments Section 1. Boolean Algebras................................... 28 With comments by Saunders Mac Lane, The University of Chicago [1]* Die Boole'sche Partialordnung und die Paarung von Gefiigen... 33 [2] Representation of Complete Lattices by Sets.................. 103 [3] Investigation of the Equivalence of the Axiom of Choice and Zorn's Lemma from the Viewpoint of the Hierarchy of Types... 121 * Numbers in brackets refer to the Publications section on pp. xi-xiii. vii

CONTENTS Section 2. Discrete Spaces..................................... 132 With comments by Don Pigozzi, Iowa State University, Ames [4] On the Existence of Totally Heterogeneous Spaces............. 141 [19] Jordan Circuits ofa Graph (with Gary Haggard)... 147 [37] Skolem Rings and Their Varieties (with T. Michael Owens)..... 161 Section 3. Towards a Theory of Definability..................... 222 With comments by Ernst Specker, ETH Zurich, with assistance from Kenneth Danhof, Southern Illinois University at Carbondale [5] The Theory of Proportionality as an Abstraction of Group Theory (with Jesse B. Wright)..................................... 226 [6] Invariants of the Anti-Automorphisms of a Group (with Jesse B. Wright)................................................. 234 [21] Model Theoretic Approaches to Definability (with Kenneth J. Danhof)................................................ 241 [22] Definability in Normal Theories (with Kenneth J. Danhof)..... 252 [23] Variations on a Theme of Cantor in the Theory of Relational Structures (with Kenneth J. Danhof)........................ 261 [40] Relatively Categorical and Normal Theories.................. 277 Abstract 11. Relatively Categorical and Normal Theories.......... 288 Section 4. Automata and Grammars............................ 291 With comments by Peter Deussen, Universitiit Karlsruhe [10] Mathematische Theorie des Verhaltens endlicher Automaten.... 295 Mathematical Theory of the Behavior of Finite Automata (English translation by Sylvia Biichi, Peter Deussen, and Dirk Siefkes)... 303 [11] Regular Canonical Systems................................ 317 [12] Algebraic Theory of Feedback in Discrete Systems............ 338 [17] Canonical Systems which Produce Periodic Sets (with William H. Hosken)................................................ 371 Section 5. Automata and Monadic Theories... 381 With comments by Robert McNaughton, Rensselaer Polytechnic Institute [7] Weak Second-Order Arithmetic and Finite Automata.......... 398 [8] On a Decision Method in Restricted Second Order Arithmetic.. 425 [13] Transfinite Automata Recursions and Weak Second Order Theory of Ordinals.............................................. 437 [14] Decision Methods in the Theory of Ordinals.................. 459 [16] Definability in the Monadic Second-Order Theory of Successor (with Lawrence H. Landweber)............................. 464 viii

CONTENTS [26] The Complete Extensions of the Monadic Second Order Theory of Countable Ordinals (with Dirk Siefkes)...................... 469 [28] Deterministic Automata and the Monadic Theory of Ordinals < W 2 (with Charles Zaiontz)............................... 493 Section 6. Games and Determinacy............................. 517 With comments by Yuri Gurevich, University of Michigan, Ann Arbor [15] Solving Sequential Conditions by Finite-State Strategies (with Lawrence H. Landweber).................................. 525 [18] Algorithmisches Konstruieren von Automaten und die Herstellung von Gewinnstrategien nach Cantor-Bendixson................ 543 Algorithmic Construction of Automata and the Production of Winning Strategies According to Cantor-Bendixson (English translation by Sylvia Biichi, Saunders Mac Lane, Walter Schnyder, and Dirk Siefkes)......................................... 557 [20] On the Presentation of Winning Strategies via the Cantor Bendixson Method (with Stephen Klein)..................... 567 [27] Using Determinancy of Games to Eliminate Quantifiers........ 581 [29] State-Strategies for Games in Fad n Gdt1...................... 593 Section 7. Computability... 621 With comments by Martin Davis, The Courant Institute [9] Turing Machines and the Entscheidungsproblem.............. 627 [31] Recursive Definition and Complexity of Functions Over Arbitrary Data Structures (with Bernd Mahr and Dirk Siefkes).......... 641 [32] Coding in the Existential Theory of Concatenation (with Steven Senger)................................................. 665 [33] Definability in the Existential Theory of Concatenation and Undecidable Extensions of This Theory (with Steven Senger)...... 671 Section 8. Quadratic Forms, the Five Squares Problem, and Diophantine Equations...................................... 677 With comments by Leonard Lipshitz, Purdue University Section 9. Graphs and Matroids................................ 681 With comments by William E. Fenton, Bellarmine College and Saunders Mac Lane, The University of Chicago [34] Large Convex Sets in Oriented Matroids (with William E. Fenton) 685 IX

Publications [1] Die Boole'sche Partialordnung und die Paarung von Gefiigen. Port. Math. 7 (1950), pp. 119-178. Doctoral dissertation. [2] Representation of complete lattices by sets. Port. Math. 11 (1952), pp. 151-167. [3] Investigation of the equivalence of the axiom of choice and Zorn's lemma from the viewpoint of the hierarchy of types. J. Symbolic Logic 18 (1953), pp. 125-135. [4] On the existence of totally heterogeneous spaces. Fund. Math. 41 (1954), pp. 97-102. [5] (with Jesse B. Wright) The theory of proportionality as an abstraction of group theory. Math. Annalen 130 (1955), pp. 102-108. [6] (with Jesse B. Wright) Invariants of the anti-automorphisms of a group. Proc. AMS 8 (1957), pp. 1134-1140. [7] Weak second-order arithmetic and finite automata. Zeitschrift Math. Logik und Grundlagen der Mathematik 6 (1960), pp. 66-92. [8] On a decision method in restricted second order arithmetic. Proc. Int. Congress Logic, Methodology, and Philosophy of Science, Berkeley 1960. Stanford University Press, Stanford, Calif. (1962), pp. 1-11. Invited address. [9] Turing machines and the Entscheidungsproblem. Math. Annalen 148 (1962), pp.201-213. [10] Mathematische Theorie des Verhaltens endlicher Automaten. Zeitschrift Angew. Mathematik und Mechanik 42 (1962), T9-Tl6. Invited address, yearly meeting of the German Soc. Appl. Math. and Mech. [11] Regular canonical systems. Archiv Math. Logik und Grundlagenforschung 6 (1964), pp. 91-111. [12] Algebraic theory offeedback in discrete systems. In E. Caianello (Ed.), Automata Theory. First Course on Automata Theory, Ravello, Italy, 1964. Academic Press, New York (1966), pp. 70-101. [13] Transfinite automata recursions and weak second order theory of ordinals. Proc. Int. Congress Logic, Methodology, and Philosophy of Science, Jerusalem 1964. North-Holland Publ. Co., Amsterdam (1965), pp. 2-23. Invited address. [14] Decision methods in the theory of ordinals. Bull. AMS 71 (1965), pp. 767-770. [15] (with Lawrence H. Landweber) Solving sequential conditions by finite-state strategies. Trans. AMS 138 (1969), pp. 295-311. [16] (with Lawrence H. Landweber) Definability in the monadic second-order theory of successor. J. Symbolic Logic 34 (1969), pp. 166-170. Xl

PUBLICATIONS [17] (with William H. Hosken) Canonical systems which produce periodic sets. Math. System Theory 4 (1970), pp. 81-90. [18] Algorithmisches Konstruieren von Automaten und die Herstellung von Gewinnstrategien nach Cantor-Bendixson. In J. Dorr and G. Hotz (Eds.), Automatentheorie und Formale Sprachen, Tagung Math. Forschungsinst. Oberwolfach, 1969. Mannheim, F.R. Germany (1970), pp. 385-398. Invited address. [19] (with Gary Haggard) Jordan circuits of a graph. J. Combinatorial Theory 10 (1971), pp. 185-197. [20] (with Stephen Klein) On the presentation of winning strategies via the Cantor Bendixson Method. Report Purdue University CSD TR-81 (1972), 14 pp. [21] (with Kenneth J. Danhof) Model theoretic approaches to definability. Zeitschrift Math. Logik und Grundlagen der Mathematik 18 (1972), pp. 61-70. [22] (with Kenneth J. Danhof) Definability in normal theories. Israel Journal of Mathematics 14 (1973), pp. 248-256. [23] (with Kenneth J. Danhof) Variations on a theme of Cantor in the theory of relational structures. Zeitschrift Math. Logik und Grundlagen der Mathematik 19 (1973), pp. 411-426. *[24] The monadic second order theory of w 1 In J.R. Buchi and D. Siefkes, The Monadic Second Order Theory of all Countable Ordinals. Lect. Notes Math. vol. 328 (1973), Springer-Verlag, pp. 1-127. *[25] (with Dirk Siefkes) Axiomatization of the monadic second order theory of w 1 The Monadic Second Order Theory of All Countable Ordinals. Lect. Notes Math. vol. 328 (1973), Springer-Verlag, pp. 129-217. [26] (with Dirk Siefkes) The complete extensions ofthe monadic second order theory of countable ordinals. Report Forschungsinstitut Mathematik, ETH Zurich (1974). Zeitschrift Math. Logik und Grundlagen der Mathematik 29 (1983), pp. 289-312. [27] Using determinancy of games to eliminate quantifiers. In M. Karpinski (Ed.), Fundamentals of Computation Theory, 1977. Lecture Notes Compo Science vol. 56, Springer-Verlag, pp. 367-378. Invited address. [28] (with Charles Zaiontz) Deterministic automata and the monadic theory of ordinals <w 2 Zeitschrift Math. Logik und Grundlagen der Mathematik 29 (1983), pp. 313-336. [29] State-strategies for games in F"6 n G6,.. J. Symbolic Logic 48 (1983), pp. 1171-1198. *[30] (with Bernd Mahr, Dirk Siefkes) Manual on REC-A language for use and cost analysis of recursion over arbitrary data structures. Techn. Univ. Berlin, FB Informatik (20), Bericht Nr. 84-06 (1984), 79 pp. [31] (with Bernd Mahr and Dirk Siefkes) Recursive definition and complexity of functions over arbitrary data structures. In G. Wechsung (Ed.), Proc. 2. Frege Conference. Akademie-Verlag Berlin (1984), pp. 303-308. [32] (with Steven Senger) Coding in the existential theory of concatenation. Archiv Math. Logik und Grundlagenforschung 26 (1986/87), pp. 101-106. [33] (with Steven Senger) Definability in the existential theory of concatenation and undecidable extensions of this theory. Zeitschrift Math. Logik und Grundlagen der Mathematik 34 (1988), pp. 337-342. [34] (with William E. Fenton) Large convex sets in oriented matroids. Journal Combinatorial Theory, series B 45 (1988), pp. 293-304. * Does not appear in this book. xii

PUBLICATIONS *[35] (with William E. Fenton) Directed Circuits of a Graph with an Application to Series-Parallel Graphs. Manuscript. *[36] Implicative Boolean algebras, Copeland's theory of conditional probability. Manuscript. [37] (with T. Michael Owens) Skolem rings and their varieties. Report Purdue University CSD TR-140 (1975). *[38] (with T. Michael Owens) Complemented Monoids and Hoops. To be submitted. *[39] Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. D. Siefkes (Ed.). Springer-Verlag, New York (1989). [40] Relatively Categorical and Normal Theories. Manuscript (1963-1965). * Does not appear in this book. xiii

Permissions [1]* Reprinted with permission. [2] Reprinted with permission. [3] Reprinted with permission. [4] Reprinted with permission. [5] Reprinted with permission from Math. Annal. Volume 130, 102-108. 1955 Springer-Verlag. [6] Reprinted from Proceedings of the American Mathematical Society (1957), "Invariants of the Anti-Automorphism of a Group," J. Richard Biichi and Jesse B. Wright, Volume 8, pages 1134-1140, with permission of the American Mathematical Society and Jesse B. Wright. [7] Reprinted with permission. [8] Reprinted from Logic, Methodology, and Philosophy of Science, Proceedings of the 1960 International Congress, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, with the permission ofthe publishers, Stanford University Press. 1962 by the Board of Trustees of the Leland Stanford Junior University. [9] Reprinted with permission from Math. Annal. Bd. Volume 148,201-213. 1962 Springer-Verlag. [10] Reprinted with permission. [11] Reprinted with permission. [12] Reprinted from Automata Theory. 1966 Academic Press, New York. [13] Reprinted with permission from Proc.lnt. Congress Logic, Method., Philos. of Sci. 1964, Jerusalem. 1965 North-Holland Publishing Company. [14] Reprinted from Bulletin of the American Mathematical Society Volume 71 (1965) 767-770. Reprinted with permission of the American Mathematical Society. [15] Reprinted from Trans. of the American Mathematical Society Volume 138 (1969) 295-311. Reprinted with permission of the American Mathematical Society. [16] Reprinted with permission from the Association for Symbolic Logic and Lawrence H. Landweber. [17] Reprinted with permission from Mathematical Systems Theory Volume 4,81-90. 1970 Springer-Verlag. [18] Reprinted with permission. [19] Reprinted from Journal of Combinatorial Theory Volume 10, 185-197. 1971 Academic Press, New York. * Numbers in brackets refer to the Publications section on pp. xi-xiii. xv

PERMISSIONS [20] Reprinted with permission of Stephen J. Klein. [21] Reprinted with permission. [22] Reprinted with permission. [23] Reprinted with permission. [26] Reprinted with permission. [27] Reprinted with permission from Fundamentals of Computation Theory Lecture Notes in Computer Science Volume 56,367-378. Editor: Marek Karpinski. 1977 Springer-Verlag. [28] Reprinted with penilission from V.E.B. Deutscher Verlag der Wissenshaften and Charles Zaiontz. [29] Reprinted with permission. [31] Reprinted with pertnission. [32] Reprinted with permission. [33] Reprinted with permission. [34] Reprinted with permission from Journal of Combinatorial Theory Series B, Volume 45, Number 3, December 1988, pp. 293-304. 1988 Academic Press. [37] Reprinted with permission. Abstract 11 Reprinted with permission from Proceedings of the Symposium on the Theory of Models. 1965 North-Holland Publishing Company, Amsterdam. xvi