Handbook of Computability Theory, Volume 140

Part 1: Fundamentals of Computability Theory.
1. The history and concept of computability (R.I. Soare).
2. &pgr;01 classes in recursion theory (D. Cenzer).
Part 2: Reducibilities and Degrees.
3. Reducibilities (P. Odifreddi).
4. Local degree theory (S.B. Cooper).
5. The global structure of the turing degrees (T.A. Slaman).
6. The recursively enumerable degrees (R.A. Shore).
7. An overview of the computably enumerable sets (R.I Soare).
Part 3: Generalized Computability Theory.
8. The continuous functionals (D. Normann).
9. Ordinal recursion theory (C.T. Chong, S.D. Friedman).
10. E-recursion (G.E. Sacks).
11. Recursion on abstract structures (P.G. Hinman).
Part 4: Mathematics and Computability Theory.
12. Computable rings and fields (V. Stoltenberg-Hansen, J.V. Tucker).
13. The structure of computability (M.B. Pour-El).
14. Theory of numberings (Y.L. Ershov).
Part 5: Logic and Computability Theory.
15. Pure recursive model theory (T.S. Millar).
16. Classifying recursive functions (H. Schwichtenberg).
Part 6: Computer Science and Computability Theory.
17. Computation models and function algebras (P. Clote).
18. Polynomial time reducibilities and degrees (K. Ambos-Spies).

The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.