Computability Theory and Its Applications: Current Trends and Open Problems
About this Volume
Edited by: Peter A. Cholak, Steffen Lempp, Manuel Lerman and Richard A. Shore
2000: Volume: 257
ISBNs: 978-0-8218-1922-7 (print); 978-0-8218-7847-7 (online)
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 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 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. Kučera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A. Shlapentokh).
Graduate students and mathematicians working in or interested in computability theory and its applications.
Table of Contents
- Klaus Ambos-Spies and Antonín Kučera – Randomness in computability theory
- Marat Arslanov – Open questions about the $n$-c.e. degrees
- Serikzhan Badaev and Sergey Goncharov – The theory of numberings: open problems
- Douglas Cenzer and Carl G. Jockusch, Jr. – $\Pi _1^0$ classes—structure and applications
- Peter A. Cholak – The global structure of computably enumerable sets
- C. T. Chong and Yue Yang – Computability theory in arithmetic: provability, structure and techniques
- Randall Dougherty and Alexander S. Kechris – How many Turing degrees are there?
- Rod Downey and J. B. Remmel – Questions in computable algebra and combinatorics
- Harvey Friedman and Stephen G. Simpson – Issues and problems in reverse mathematics
- Sergey Goncharov and Bakhadyr Khoussainov – Open problems in the theory of constructive algebraic systems
- Marcia Groszek – Independence results from ZFC in computability theory: some open problems
- Julia F. Knight – Problems related to arithmetic
- Manuel Lerman – Embeddings into the computably enumerable degrees
- André Nies – Definability in the c.e. degrees: questions and results
- Piergiorgio Odifreddi – Strong reducibilities, again
- Mikhail Peretyat′kin – Finitely axiomatizable theories and Lindenbaum algebras of semantic classes
- Alexandra Shlapentokh – Towards an analog of Hilbert’s tenth problem for a number field
- Richard A. Shore – Natural definability in degree structures
- Theodore A. Slaman – Recursion theory in set theory
- Robert I. Soare – Extensions, automorphisms, and definability
- Andrea Sorbi – Open problems in the enumeration degrees