Computability : An Introduction to Recursive Function Theory

What Can Computers Do In Principle? What Are Their Inherent Theoretical Limitations? These Are Questions To Which Computer Scientists Must Address Themselves. The Theoretical Framework Which Enables Such Questions To Be Answered Has Been Developed Over The Last Fifty Years From The Idea Of A Computable Function: Intuitively A Function Whose Values Can Be Calculated In An Effective Or Automatic Way. This Book Is An Introduction To Computability Theory (or Recursion Theory As It Is Traditionally Known To Mathematicians). Dr Cutland Begins With A Mathematical Characterisation Of Computable Functions Using A Simple Idealised Computer (a Register Machine); After Some Comparison With Other Characterisations, He Develops The Mathematical Theory, Including A Full Discussion Of Non-computability And Undecidability, And The Theory Of Recursive And Recursively Enumerable Sets. The Later Chapters Provide An Introduction To More Advanced Topics Such As Gildel's Incompleteness Theorem, Degrees Of Unsolvability, The Recursion Theorems And The Theory Of Complexity Of Computation. Computability Is Thus A Branch Of Mathematics Which Is Of Relevance Also To Computer Scientists And Philosophers. Mathematics Students With No Prior Knowledge Of The Subject And Computer Science Students Who Wish To Supplement Their Practical Expertise With Some Theoretical Background Will Find This Book Of Use And Interest. --publisher's Description. Prologue. Prerequisites And Notation -- Sets -- Functions -- Relations And Predicates -- Logical Notation -- Computable Functions -- Algorithms, Or Effective Procedures -- The Unlimited Register Machine -- Urm-computable Functions -- Decidable Predicates And Problems -- Computability On Other Domains -- Generating Computable Functions -- The Basic Functions -- Joining Programs Together -- Substitution -- Recursion -- Minimalisation -- Other Approaches To Computability: Church's Thesis -- Other Approaches To Computability -- Partial Recursive Functions (godel-kleene) -- A Digression: The Primitive Recursive Functions -- Turing-computability -- Symbol Manipulation Systems Of Post And Markov -- Computability On Domains Other Than N -- Church's Thesis -- Numbering Computable Functions -- Numbering Programs -- Numbering Computable Functions -- Discussion: The Diagonal Method -- The S-m-n Theorem -- Universal Programs -- Universal Functions And Universal Programs --^ Two Applications Of The Universal Program -- Effective Operations On Computable Functions -- Computability Of The Function [sigma Subscript N] -- Decidability, Undecidability And Partial Decidability -- Undecidable Problems In Computability -- The Word Problem For Groups -- Diophantine Equations -- Sturm's Algorithm -- Mathematical Logic -- Partially Decidable Predicates -- Recursive And Recursively Enumerable Sets -- Recursive Sets -- Recursively Enumerable Sets -- Productive And Creative Sets -- Simple Sets -- Arithmetic And Godel's Incompleteness Theorem -- Formal Arithmetic -- Incompleteness -- Godel's Incompleteness Theorem -- Undecidability -- Reducibility And Degrees -- Many-one Reducibility -- Degrees -- M-complete R.e. Sets -- Relative Computability -- Turing Reducibility And Turing Degrees -- Effective Operations On Partial Functions -- Recursive Operators -- Effective Operations On Computable Functions -- The First Recursion Theorem --^ An Application To The Semantics Of Programming Languages -- The Second Recursion Theorem -- The Second Recursion Theorem -- Myhill's Theorem -- Complexity Of Computation -- Complexity And Complexity Measures -- The Speed-up Theorem -- Complexity Classes -- The Elementary Functions -- Further Study. Nigel Cutland. Includes Index. Bibliography: P. [239]-240. Preface Prologue, prerequisites and notation 1. Computable functions 2. Generating computable functions 3. Other approaches to computability: Church's thesis 4. Numbering computable functions 5. Universal programs 6. Decidability, undecidability and partical decidability 7. Recursive and recursively enumerable sets 8. Arithmetic and Godel's incompleteness theorem 9. Reducibility and degrees 10. Effective operations on partial functions 11. The second recursion theorem 12. Complexity of computation 13. Further study.