CONTEMPORARY MATHEMATICS 257 Computability Theory and Its Applications Current Trends and Open Problems Proceedings of a 1999 AMS-IMS-SIAM Joint Summer Research Conference Computability Theory and Applications June 13-17 I 1999 University of Colorado I Boulder Peter A. Cholak Steffen Lempp Manuel Lerman Richard A. Shore Editors http://dx.doi.org/10.1090/conm/257 Selected Titles in This Series 257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000 256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfms and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000 255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear POE's, dynamics and continuum physics, 2000 254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000 253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000 252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999 251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000 250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999 248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum affine algebras and related topics, 1999 247 Lawrence Wasson Baggett and David Royal Larson, Editors, The functional and harmonic analysis of wavelets and frames, 1999 246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in dynamics, 1999 245 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrio, Ana Jeremias Lopez, and Joseph Lipman, Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999 243 Tsit Yuan Lam and Andy R. Magid, Editors, Algebra, K-theory, groups, and education, 1999 242 Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Editors, Geometric aspects of partial differential equations, 1999 241 Piotr Pragacz, Michal Szurek, and Jaroslaw Wisniewski, Editors, Algebraic geometry: Hirzebruch 70, 1999 240 Angel Carocca, Victor Gonzalez-Aguilera, and Rubi E. Rodriguez, Editors, Complex geometry of groups, 1999 239 Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson, Editors, Homotopy invariant alg~braic structures, 1999 238 Gui-Qiang 'ch.en and Emmanuele DiBenedetto, Editors, Nonlinear partial differential equations, 1999 237 Thomas Branson, Editor, Spectral problems in geometry and arithmetic, 1999 236 Bruce C. Berndt and Fritz Gesztesy, Editors, Continued fractions: From analytic number theory to constructive approximation, 1999 235 Walter A. Carnielli and !tala M. L. D'Ottaviano, Editors, Advances in contemporary logic and computer science, 1999 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Computability Theory and Its Applications Current Trends and Open Problems CoNTEMPORARY MATHEMATICS 257 Computability Theory and Its Applications Current Trends and Open Problems Proceedings of a 1999 AMS-IMS-SIAM Joint Summer Research Conference Computability Theory and Applications June 13-17 I 1999 University of Colorado I Boulder Peter A. Cholak Steffen Lempp Manuel Lerman Richard A. Shore Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius The AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Computability Theory and Applications was held at the University of Colorado, Boul- der, CO, June 13-17, 1999, with support from the National Science Foundation, Grant DMS-9618514. 2000 Mathematics Subject Classification. Primary 03C57, 03D25, 03D28, 03D30, 03D45, 03D80, 03E15, 03E35, 03F35, 03H15. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Library of Congress Cataloging-in-Publication Data Computability theory and its applications : current trends and open problems : proceedings of a 1999 AMS-IMS-SIAM, joint summer research conference, computability theory and applications, June 13-17, 1999, University of Colorado, Boulder / Peter A. Cholak ... [et a!.], editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 257) Includes bibliographical references. ISBN 0-8218-1922-4 (alk. paper) 1. Computable functions-Congresses. I. Cholak, Peter, 1962- II. Contemporary mathe- matics (American Mathematical Society) ; v. 257. QA9.59. C66 2000 511.3-dc21 00-036278 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 0:2940-6248. 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Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00 Contents Preface IX Program of invited talks XI Conference participants xii Randomness in computability theory KLAUS AMBOS-SPIES AND ANTONIN KUCERA 1 Open questions about the n-c.e. degrees MARAT ARSLANOV 15 The theory of numberings: Open problems 8ERIKZHAN BADAEV AND 8ERGEY GONCHAROV 23 n~ classes - structure and applications DOUGLAS CENZER AND CARL G. JOCKUSCH, JR. 39 The global structure of computably enumerable sets PETER A. CHOLAK 61 Computability theory in arithmetic: Provability, structure and techniques C. T. CHONG AND YUE YANG 73 How many Turing degrees are there? RANDALL DOUGHERTY AND ALEXANDER 8. KECHRIS 83 Questions in computable algebra and combinatorics ROD DOWNEY AND J. B. REMMEL 95 Issues and problems in reverse mathematics HARVEY FRIEDMAN AND STEPHEN G. SIMPSON 127 Open problems in the theory of constructive algebraic systems SERGEY GONCHAROV AND BAKHADYR KHOUSSAINOV 145 Independence results from ZFC in computability theory: Some open problems MARCIA G ROSZEK 171 Problems related to arithmetic JULIA F. KNIGHT 181 vii viii CONTENTS Embeddings into the computably enumerable degrees MANUEL LERMAN 191 Definability in the c.e. degrees: Questions and results ANDRE NIES 207 Strong reducibilities, again PIERGIORGIO 0DIFREDDI 215 Finitely axiomatizable theories and Lindenbaum algebras of semantic classes MIKHAIL PERETYAT1KIN 221 Towards an analog of Hilbert's tenth problem for a number field ALEXANDRA SHLAPENTOKH 241 Natural definability in degree structures RICHARD A. SHORE 255 Recursion theory in set theory THEODORE A. SLAMAN 273 Extensions, automorphisms, and definability ROBERT I. SOARE 279 Open problems in the enumeration degrees ANDREA SORB! 309 A sunny summer day in July, 1997. A walk through the woods outside Kazan, Along the Volga river. Ideas for a conference emerge. Focus only on open problems. Provide sufficient background for Fruitful discussions about the problems. Allow enough free time to Immerse ourselves in the problems. Assemble a mix of people with diverse, but con_nected interests. Provide new ideas and insights. A consensus emerges. A list of focus areas is proposed, discussed, modified. Potential speakers are matched with topics. A successful proposal to the AMS. A conference of a different nature sees the light of day. Preface This volume is a faithful and expanded reflection of most of the talks which were presented at the Conference on Computability Theory and Applications, as part of the Joint Summer Research Conferences in the Mathematical Sciences, Boulder, Colorado, June 13-17, 1999. The meeting focused on open problems in Computability Theory and some related areas in which the ideas, methods and/ or results of Computability Theory play a role. Some talks delved in depth into a narrowly focused group of problems, providing a description of what had been done and delineating the obstacles to solution. Others covered a wider area, providing the rationale for interest in the area and the directions pursued, and a broad cross- section of central open problems. Discussions ensued. Some problems were solved quickly at the meeting, and others since that time. The result is, we hope, a snapshot of the status of Computability Theory at the