Computability

Intends to lay a common basis for the different branches of
recursion theory. Leads from the very basic theory to modern
concepts of computability. Consists of three consecutive
parts: 1. Basic Concepts of Computability. 2. Traditional
Recursion Theory. 3. Unified Type 2 theory of constructivity
and computability on Baire's space including a general the-
ory of representations.

Prerequisites and Notation.- 1: Basic Concepts of Computability.- 1.1 Flowcharts and Machines.- 1.2 Register Machines and Register Computability.- 1.3 Primitive Recursive and ?-Recursive Functions.- 1.4 WHILE-Programs and WHILE-Computability.- 1.5 Tape Machines.- 1.6 Stack Machines.- 1.7 Comparison of Number and Word Functions, Church’s Thesis.- 1.8 Recursive and Recursively Enumerable Sets.- 1.9 The Standard Numbering ? of P(1).- 1.10 Some Unsolvable Problems.- 2: Type 1 Recursion Theory.- 2.1 The Basic Concepts of Computability Theory.- 2.2 Numberings.- 2.3 Recursive and Recursively Enumerable Sets (Continued).- 2.4 Many-one and One-one Reducibility.- 2.5 The Recursion Theorem.- 2.6 Creative, Productive, Complete Sets.- 2.7 Effective Numberings.- 2.8 Ordinal Trees and Computable Ordinals.- 2.9 Some Applications to Logic.- 2.10 Oracle Machines and Relativized Recursion Theory.- 2.11 Turing Reducibility and the Kleene Hierarchy.- 2.12 Computational Complexity.- 3: Type 2 Theory of Constructivity and Computability.- 3.1 Type 2 Computability Models.- 3.2 Recursion Theory on Baire’s Space.- 3.3 Representations.- 3.4 Effective Representations.- 3.5 Complete Partial Orders.- 3.6 Type 1 Computability and Type 2 Computability.- 3.7 Solving Domain Equations.- 3.8 Applications to Analysis.- Index of Notations.