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

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  • Format: Paperback
  • Copyright: 2000-06-01
  • Publisher: Amer Mathematical Society

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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 represent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM 1999 Summer Conference on Computability Theory and Applications, which 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 and Y. 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).

Table of Contents

Preface ix
Program of invited talks xi
Conference participants xii
Randomness in computability theory
Klaus Ambos-Spies
Antonin Kucera
Open questions about the n-c.e. degrees
Marat Arslanov
The theory of numberings: Open problems
Serikzhan Badaev
Sergey Goncharov
Π01 classes---structure and applications
Douglas Cenzer
Carl G. Jockusch, Jr.
The global structure of computably enumerable sets
Peter A. Cholak
Computability theory in arithmetic: Provability, structure and techniques
C. T. Chong
Yue Yang
How many Turing degrees are there?
Randall Dougherty
Alexander S. Kechris
Questions in computable algebra and combinatorics
Rod Downey
J. B. Remmel
Issues and problems in reverse mathematics
Harvey Friedman
Stephen G. Simpson
Open problems in the theory of constructive algebraic systems
Sergey Goncharov
Bakhadyr Khoussainov
Independence results from ZFC in computability theory: Some open problems
Marcia Groszek
Problems related to arithmetic
Julia F. Knight
Embeddings into the computably enumerable degrees
Manuel Lerman
Definability in the c.e. degrees: Questions and results
Andre Nies
Strong reducibilities, again
Piergiorgio Odifreddi
Finitely axiomatizable theories and Lindenbaum algebras of semantic classes
Mikhail Peretyat'kin
Towards an analog of Hilbert's tenth problem for a number field
Alexandra Shlapentokh
Natural definability in degree structures
Richard A. Shore
Recursion theory in set theory
Theodore A. Slaman
Extensions, automorphisms, and definability
Robert I. Soare
Open problems in the enumeration degrees
Andrea Sorbi

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