contemporary mathematics mathematics 257 computability theory and its applications current trends...

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 13-1 7 I 1999

University of Colorado I Boulder

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

Editors

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/.

http://dx.doi.org/10.1090/conm/257

CoNTEMPORARY MATHEMATICS

257



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


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. 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 speak-ers, 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.


Editors and Conference Co-organizers December 15, 1999

1These 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 Universitiit Heidelberg Mathematisches Institut Im N euenheimer Feld 294 D-69120 Heidelberg, Germany ambos@math. uni-heidelberg.de

Arana, Andrew University of Notre Dame Department of Mathematics Mail Distribution Center Notre Dame, IN 46556-5683, USA E-mail: [email protected]

Arslanov, Mar at Kazan State University Department of Mechanics and Math. ul. Kremlevskaya 18 420008 Kazan, Russia [email protected]

Badaev, Serikzhan Kazakh Academy of Sciences Institute of Mathematics 125 Pushkin Street Almaty 480100, Kazakhstan [email protected]

Calhoun, William C. Bloomsburg University Department of Math., CS & Stat. Bloomsburg, PA 17815, USA [email protected]

Cenzer, Douglas University of Florida Department of Mathematics 310 Little Hall Gainesville, FL 32611-8105, USA cenzer@math. uft.edu

Cholak, Peter University of Notre Dame Department of Mathematics Mail Distribution Center Notre Dame, IN 46556-5683, USA peter.cholak.1 @nd.edu

xii

Chong, Chi Tat National University of Singapore Department of Mathematics Lower Kent Ridge Road Singapore 119260, Singapore [email protected]

Coles, Richard 30 Slade Rd Four Oaks Sutton Coldfield West Midlands B75 5PG, England [email protected]

Cooper, Barry University of Leeds School of Mathematics Leeds LS2 9JT, England [email protected]. uk

Davis, Martin 3360 Dwight Way Berkeley, CA 94704-2523, USA [email protected]

Ealy, Jr. Clifton E. Western Michigan University Department of Mathematics & Statistics Kalamazoo, MI 49008-5152, USA [email protected]

Englert, Burkhard University of Connecticut Department of Mathematics U-9 Rm 111, 196 Auditorium Rd. Storrs, CT 06269-3009, USA englert@math. uconn.edu

Ershov, Yuri Academy of Sciences Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia root@ershov .nsu.ru

Fejer, Peter University of Massachusetts Department of Mathematics & CS Boston, MA 02125-3393, USA fejer@cs. umb.edu

CONFERENCE PARTICIPANTS xiii

Friedman, Harvey The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus, OH 43210-1101, USA [email protected]

Galminas, Lisa Northwestern State Univ of LA Department of Mathematics Natchitoches, LA 71497, USA [email protected]

Giorgi, Matthew B. School of Mathematics University of Leeds Leeds LS2 9JT, United Kingdom [email protected]. uk

Goncharov, Sergey Academy of Sciences Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia [email protected]

Griffor, Edward R. ComAdvisors, Inc. 19111 West Ten Mile Road Suite 167 Southfield, MI 48075, USA [email protected]

Groszek, Marcia Dartmouth College Department of Mathematics 6188 Bradley Hall Hanover, NH 03755-3551, USA [email protected]

Harizanov, Valentina George Washington University Department of Mathematics Funger Hall 2201 G Street NW Washington, DC 22052 [email protected]

Harrington, Leo University of California Department of Mathematics Berkeley, CA 94720-0001, USA leo@ math. berkeley.edu

Herrmann, Eberhard Institut fiir Mathematik Humboldt-Universitat zu Berlin Math.-Naturwiss. Fakultat II U nter den Linden 6 D-10099 Berlin, Germany [email protected]

Hirschfeldt, Denis Cornell University Department of Mathematics Malott Hall Ithaca, NY 14853-7901, USA [email protected]

Hirst, Jeff Appalachian State University Department of Mathematical Sciences Boone, NC 28608, USA [email protected]

Ho, Kejia (Joyce) University of Illinois Department of Mathematics 1409 West Green Street Urbana, IL 61801-2975, USA [email protected]

Hummel, Tamara Allegheny College Department of Mathematics Meadville, PA 16335, USA [email protected]

Jockusch, Carl University of Illinois Department of Mathematics 1409 W. Green St. Urbana, IL 61801-2975, USA jockusch@math. uiuc.edu

Khoussainov, Bakh University of Auckland Department of Computer Science Private Bag 92019 Auckland, New Zealand [email protected]

xiv CONFERENCE PARTICIPANTS

Knight, Julia University of Notre Dame Department of Mathematics Mail Distribution Center Notre Dame, IN 46556-5683, USA [email protected]

Kucera, Antonfn Charles University Department of Theoretical CS Faculty of Mathematics & Physics Malostranske nam. 25 CZ-110 00 Prague 1, Czech Republic [email protected]

Kudinov, Oleg Academy of Sciences Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia kud @math.nsc.ru

LaForte, Geoff University of West Florida Institute for Human & Machine Cog. 11000 University Parkway Pensacola, FL 32514, USA glaforte@coginst. uwf.edu

Lawton, Linda University of Illinois Department of Mathematics 1409 W. Green Street Urbana, IL 61801-2975, USA lawton@math. uiuc.edu

Lempp, Steffen University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1388, USA lempp@math. wisc.edu

Leonhardi, Steven Winona State University Department of Mathematics & Statistics Winona, MN 55987, USA [email protected]

Lerman, Manuel University of Connecticut Department of Mathematics U-9 Rm 111, 196 Auditorium Rd Storrs, CT 06269-3009, USA mlerman@math. uconn.edu

Li, Angsheng Academica Sinica Institute of Software Post Office Box 8718 Beijing 100080, PR of China [email protected]

Marcone, Alberto Universita di Udine Dip. di Matematica e Inform. Via delle Scienze 206 33100 Udine, Italy marcone@dimi. uniud.it

McAllister, Alex Centre College Department of Mathematics 600 W. Walnut Street Danville, KY 40422-1394, USA [email protected]

McNicholl, Timothy H. University of Dallas Department of Mathematics Irving, TX 75062, USA tmcnicho@acad. udallas.edu

Miller, Russell University of Chicago Department of Mathematics 5734 University Avenue Chicago, IL 60637-1514, USA russell@math. uchicago.edu

Morozov, Andrei Academy of Sciences Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia [email protected]

CONFERENCE PARTICIPANTS

Nerode, Anil Cornell University Department of Mathematics Malott Hall Ithaca, NY 14853-4201, USA [email protected]

Nies, Andre University of Chicago Department of Mathematics 5734 University Avenue Chicago IL 60637-1514, USA nies@math. uchicago.edu

Peretyat 'kin, Mikhail Kazakh Academy of Sciences Institute of Mathematics 125 Pushkin Street Almaty 480021, Kazakhstan [email protected]

Remmel, Jeffrey University of California Department of Mathematics La Jolla, CA 92093-0112, USA [email protected]

Sacks, Gerald Massachusetts Institute of Technology Department of Mathematics Cambridge, MA 02139-4358, USA [email protected]

Shlapentokh, Alexandra East Carolina University Deptartment of Mathematics Greenville, NC 27858-4353, USA [email protected]

Shore, Richard A. Cornell University Department of Mathematics Malott Hall Ithaca, NY 14853-4201, USA [email protected]

Simpson, Stephen Pennsylvania State University Department of Mathematics McAllister Building University Park, PA 16802-6401, USA [email protected]

Slaman, Theodore University of California Department of Mathematics Berkeley, CA 94720-0001, USA slaman@math. berkeley.edu

Smuga-Otto, Maciej University of Wisconsin Department of Mathematics 480 Lincoln 'Drive Madison, WI 53706-1388, USA [email protected]

Soare, Robert University of Chicago Department of Mathematics 5734 University Avenue Chicago, IL 60637-1514, USA soare@cs. uchicago.edu

Solomon, Reed University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison. WI 53706-1388, USA [email protected]

Sorbi, Andrea Dip. di Matematica Via del Capitano 15 I-53100 Siena, Italy [email protected]

Stephan, Frank Universitat Heidelberg Mathematisches Institut Im Neuenheimer Feld 294 D-69120 Heidelberg, Germany fstephan@math. uni-heidelberg.de

Thurber, John Eastern Oregon University Department of Mathematics La Grande, OR 97850, USA [email protected]

Wald, Kevin University of Chicago Department of Mathematics 5734 University Ave Chicago, IL 60637-1514, USA wald@math. uchicago.edu

XV

xvi CONFERENCE PARTICIPANTS

Walk, Steve University of Notre Dame Department of Mathematics Mail Distribution Center Notre Dame, IN 46556-5683, USA [email protected] Wang, Dejia University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1388, USA dwang@math. wisc.edu

White, Walker Cornell University Department of Mathematics Malott Hall Ithaca, NY 14853-4201, USA [email protected]

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 math-ematics 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

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contemporary mathematics mathematics 257 computability theory and its applications current trends...

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