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 mathematics (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. Requests can also be made by e-mail to reprint-permissioniDams. erg. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 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 end of the millennium, and a list of fruitful directions for research early in the next millennium. All papers in this volume reflect invited talks, are written by the invited speakers, sometimes with a co-author, and are refereed. Alekos Kechris and George Odifreddi were unable to speak, but submitted papers. Anil Nerode spoke on Computable Analysis and Topology and Gerald Sacks on Higher Recursion Theory, but there are no follow-up papers for these talks. Yiannis Moschovakis was, un- fortunately, not able to attend and talk about Recursion in Computer Science. In addition, two three-hour blocks of time were allocated for Barry Cooper to describe, in detail, his work on constructing automorphisms of the degrees. Of particular note was the intersection of this meeting with one on Topology which was being held simultaneously. In particular, schedules were arranged so that ix X PREFACE both groups could meet to hear the talk by Shmuel Weinberger on joint work1 with Alex Nabutovsky which uses results about computably enumerable sets and degrees to obtain results on manifolds with Riemannian metrics in differential topology. This application of Computability Theory was brought to our attention by Robert Soare who had a part in developing some of the computability-theoretic results used in this work. It added yet another direction of application beyond those which were represented by our original list of topics. We thank the AMS and NSF for funding the conference. Special thanks go to Donna Salter who handled the organization and logistics superbly, and to Christine Thivierge who assisted with the preparation of this volume. Of course, the success of the Conference can be mainly attributed to the speakers and participants, whose active involvement brought many of the initial goals to fruition.

Peter A. Cholak Steffen Lempp Manuel Lerman Richard A. Shore

Editors and Conference Co-organizers December 15, 1999

1 These results are presented in A. Nabutovsky and S. Weinberger: Variational problems for Riemannian functionals and arithmetic groups, and The fractal nature of Riem/Diff, both to appear. For background on this material, see also A. Nabutovsky: Disconnectedness of sublevel sets of some Riemannian functionals, Geometric and Functional Analysis, 6 (1996), 703-725, and Geometry of the space of triangulations of a compact manifold, Communications in Mathematical Physics, 181 (1996), 303-330. Program of Invited Talks, AMS Summer Research Conference on Computability Theory and Applications

Sunday, June 13, 1999 • Sergey Goncharov, Novosibirsk, Computable model theory • Julia Knight, Notre Dame, Models of arithmetic • Serikzhan Badaev, Almaty, Numeration theory • Jeffrey Remmel, San Diego, Computable algebra • Bakhadyr Khoussainov, Auckland, Issues in computable presentations of models • Mikhail Peretyat 'kin, Almaty, Finitely axiomatizable theories and Lindenbaum algebras Monday, June 14, 1999 • Peter Cholak, Notre Dame, The lattice of computably enumerable sets • Robert Soare, Chicago, The lattice of computably enumerable sets • Klaus Ambos-Spies, Heidelberg, Genericity and randomness • Barry Cooper, Leeds, Proof of the automorphism theorem, parts 1,2 Tuesday, June 15, 1999 • Andre Nies, Chicago, Definability and coding • Richard Shore, Cornell, Natural definability in degree structures • Gerald Sacks, Harvard & MIT, Higher recursion theory Wednesday, June 16, 1999 • Stephen Simpson, Penn State, Reverse mathematics • Carl Jockusch, Urbana, IT~ classes and computable combinatorics • C. T. Chong, Singapore, Reverse computability theory • Harvey Friedman, Ohio State, Reverse mathematics • Alexandra Shlapentokh, East Carolina, Issues related to Hilbert's Tenth Problem • Anil Nerode, Cornell, Computable analysis and topology Thursday, June 17, 1999 • Theodore Slaman, Berkeley, Applications of recursion theoretic methods in set theory • Marat Arslanov, Kazan, D.c.e. and n-c.e. degrees • Marcia Groszek, Dartmouth, Independence results (from ZFC) in recursion theory • Manuel Lerman, Connecticut, Lattice embeddings into the computably enumerable degrees • Andrea Sorbi, Siena, Enumeration degrees

xi Conference Participants

Ambos-Spies, Klaus Chong, Chi Tat Universitiit Heidelberg National University of Singapore Mathematisches Institut Department of Mathematics Im N euenheimer Feld 294 Lower Kent Ridge Road D-69120 Heidelberg, Germany Singapore 119260, Singapore ambos@math. uni-heidelberg.de [email protected]

Arana, Andrew Coles, Richard University of Notre Dame 30 Slade Rd Four Oaks Department of Mathematics Sutton Coldfield Mail Distribution Center West Midlands B75 5PG, England Notre Dame, IN 46556-5683, USA [email protected] E-mail: [email protected] Cooper, Barry University of Leeds Arslanov, Mar at School of Mathematics Kazan State University Leeds LS2 9JT, England Department of Mechanics and Math. [email protected]. uk ul. Kremlevskaya 18 420008 Kazan, Russia Davis, Martin [email protected] 3360 Dwight Way Berkeley, CA 94704-2523, USA Badaev, Serikzhan [email protected] Kazakh Academy of Sciences Institute of Mathematics Ealy, Jr. Clifton E. 125 Pushkin Street Western Michigan University Almaty 480100, Kazakhstan Department of Mathematics & Statistics [email protected] Kalamazoo, MI 49008-5152, USA [email protected] Calhoun, William C. Englert, Burkhard Bloomsburg University University of Connecticut Department of Math., CS & Stat. Department of Mathematics Bloomsburg, PA 17815, USA U-9 Rm 111, 196 Auditorium Rd. [email protected] Storrs, CT 06269-3009, USA Cenzer, Douglas englert@math. uconn.edu University of Florida Ershov, Yuri Department of Mathematics Academy of Sciences 310 Little Hall Siberian Branch Gainesville, FL 32611-8105, USA Mathematical Institute cenzer@math. uft.edu 630090 Novosibirsk, Russia root@ershov .nsu.ru Cholak, Peter University of Notre Dame Fejer, Peter Department of Mathematics University of Massachusetts Mail Distribution Center Department of Mathematics & CS Notre Dame, IN 46556-5683, USA Boston, MA 02125-3393, USA peter.cholak.1 @nd.edu fejer@cs. umb.edu xii CONFERENCE PARTICIPANTS xiii

Friedman, Harvey Herrmann, Eberhard The Ohio State University Institut fiir Mathematik Department of Mathematics Humboldt-Universitat zu Berlin 231 West 18th Avenue Math.-Naturwiss. Fakultat II Columbus, OH 43210-1101, USA U nter den Linden 6 [email protected] D-10099 Berlin, Germany [email protected] Galminas, Lisa Northwestern State Univ of LA Department of Mathematics Hirschfeldt, Denis Natchitoches, LA 71497, USA Cornell University [email protected] Department of Mathematics Malott Hall Giorgi, Matthew B. Ithaca, NY 14853-7901, USA School of Mathematics [email protected] University of Leeds Leeds LS2 9JT, United Kingdom [email protected]. uk Hirst, Jeff Appalachian State University Goncharov, Sergey Department of Mathematical Sciences Academy of Sciences Boone, NC 28608, USA Siberian Branch [email protected] Mathematical Institute 630090 Novosibirsk, Russia [email protected] Ho, Kejia (Joyce) University of Illinois Griffor, Edward R. Department of Mathematics ComAdvisors, Inc. 1409 West Green Street 19111 West Ten Mile Road Urbana, IL 61801-2975, USA Suite 167 [email protected] Southfield, MI 48075, USA [email protected] Hummel, Tamara Groszek, Marcia Allegheny College Dartmouth College Department of Mathematics Department of Mathematics Meadville, PA 16335, USA 6188 Bradley Hall [email protected] Hanover, NH 03755-3551, USA [email protected] Jockusch, Carl Harizanov, Valentina University of Illinois George Washington University Department of Mathematics Department of Mathematics 1409 W. Green St. Funger Hall Urbana, IL 61801-2975, USA 2201 G Street NW jockusch@math. uiuc.edu Washington, DC 22052 [email protected] Khoussainov, Bakh Harrington, Leo University of Auckland University of California Department of Computer Science Department of Mathematics Private Bag 92019 Berkeley, CA 94720-0001, USA Auckland, New Zealand leo@ math. berkeley.edu [email protected] xiv CONFERENCE PARTICIPANTS

Knight, Julia Lerman, Manuel University of Notre Dame University of Connecticut Department of Mathematics Department of Mathematics Mail Distribution Center U-9 Rm 111, 196 Auditorium Rd Notre Dame, IN 46556-5683, USA Storrs, CT 06269-3009, USA [email protected] mlerman@math. uconn.edu

Kucera, Antonfn Li, Angsheng Charles University Academica Sinica Department of Theoretical CS Institute of Software Faculty of Mathematics & Physics Post Office Box 8718 Malostranske nam. 25 Beijing 100080, PR of China CZ-110 00 Prague 1, Czech Republic [email protected] [email protected] Marcone, Alberto Kudinov, Oleg Universita di Udine Academy of Sciences Dip. di Matematica e Inform. Siberian Branch Via delle Scienze 206 Mathematical Institute 33100 Udine, Italy 630090 Novosibirsk, Russia marcone@dimi. uniud.it kud @math.nsc.ru

McAllister, Alex LaForte, Geoff Centre College University of West Florida Department of Mathematics Institute for Human & Machine Cog. 600 W. Walnut Street 11000 University Parkway Danville, KY 40422-1394, USA Pensacola, FL 32514, USA [email protected] glaforte@coginst. uwf.edu

Lawton, Linda McNicholl, Timothy H. University of Illinois University of Dallas Department of Mathematics Department of Mathematics 1409 W. Green Street Irving, TX 75062, USA Urbana, IL 61801-2975, USA tmcnicho@acad. udallas.edu lawton@math. uiuc.edu Miller, Russell Lempp, Steffen University of Chicago University of Wisconsin Department of Mathematics Department of Mathematics 5734 University Avenue 480 Lincoln Drive Chicago, IL 60637-1514, USA Madison, WI 53706-1388, USA russell@math. uchicago.edu lempp@math. wisc.edu Morozov, Andrei Leonhardi, Steven Academy of Sciences Winona State University Siberian Branch Department of Mathematics & Statistics Mathematical Institute Winona, MN 55987, USA 630090 Novosibirsk, Russia [email protected] [email protected] CONFERENCE PARTICIPANTS XV

Nerode, Anil Slaman, Theodore Cornell University University of California Department of Mathematics Department of Mathematics Malott Hall Berkeley, CA 94720-0001, USA Ithaca, NY 14853-4201, USA slaman@math. berkeley.edu [email protected] Smuga-Otto, Maciej Nies, Andre University of Wisconsin University of Chicago Department of Mathematics Department of Mathematics 480 Lincoln 'Drive 5734 University Avenue Madison, WI 53706-1388, USA Chicago IL 60637-1514, USA [email protected] nies@math. uchicago.edu Soare, Robert Peretyat 'kin, Mikhail University of Chicago Kazakh Academy of Sciences Department of Mathematics Institute of Mathematics 5734 University Avenue 125 Pushkin Street Chicago, IL 60637-1514, USA Almaty 480021, Kazakhstan soare@cs. uchicago.edu [email protected] Solomon, Reed Remmel, Jeffrey University of Wisconsin University of California Department of Mathematics Department of Mathematics 480 Lincoln Drive La Jolla, CA 92093-0112, USA Madison. WI 53706-1388, USA [email protected] [email protected] Sacks, Gerald Sorbi, Andrea Massachusetts Institute of Technology Dip. di Matematica Department of Mathematics Via del Capitano 15 Cambridge, MA 02139-4358, USA I-53100 Siena, Italy [email protected] [email protected] Shlapentokh, Alexandra Stephan, Frank East Carolina University Universitat Heidelberg Deptartment of Mathematics Mathematisches Institut Greenville, NC 27858-4353, USA Im Neuenheimer Feld 294 [email protected] D-69120 Heidelberg, Germany fstephan@math. uni-heidelberg.de Shore, Richard A. Cornell University Thurber, John Department of Mathematics Eastern Oregon University Malott Hall Department of Mathematics Ithaca, NY 14853-4201, USA La Grande, OR 97850, USA [email protected] [email protected] Simpson, Stephen Wald, Kevin Pennsylvania State University University of Chicago Department of Mathematics Department of Mathematics McAllister Building 5734 University Ave University Park, PA 16802-6401, USA Chicago, IL 60637-1514, USA [email protected] wald@math. uchicago.edu xvi CONFERENCE PARTICIPANTS

Walk, Steve White, Walker University of Notre Dame Cornell University Department of Mathematics Department of Mathematics Mail Distribution Center Malott Hall Notre Dame, IN 46556-5683, USA Ithaca, NY 14853-4201, USA [email protected] [email protected] Wang, Dejia University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1388, USA dwang@math. wisc.edu Computability Theory and Its Applications Current Trends and Open Problems Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors

This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers repre- sent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM Joint Summer Conference on Computability Theory and Applications held at the University of Colorado (Boulder). The conference focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P. Cholak, R. Soare), definability issues in the c.e. and Turing degrees (A Nies, R. Shore) and other degree structures (M. Arslanov, S. Badaev and S. Goncharov, P. Odifreddi, A Sorbi). The topics involving relations between computability and other areas of logic and mathematics are reverse mathematics and proof theory (D. Cenzer and C. Jockusch, C. Chong andY. Yang, H. Friedman and S. Simpson), set theory (R. Dougherty and A Kechris, M. Groszek, T. Slaman) and computable mathematics and model theory (K. Ambos-Spies and A Kucera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A Shlapentokh).

ISBN 0-8218-1922-4

9 780821 819227 CONM/257