Computability: An Introduction to Recursive Function TheoryWhat 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 Gödel'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. |
Contents
I | 1 |
II | 2 |
III | 4 |
V | 5 |
VI | 7 |
VIII | 9 |
IX | 16 |
X | 22 |
XLI | 112 |
XLII | 121 |
XLIV | 123 |
XLV | 133 |
XLVI | 140 |
XLVII | 143 |
XLIX | 146 |
L | 149 |
XI | 23 |
XII | 25 |
XVI | 29 |
XVII | 32 |
XVIII | 42 |
XIX | 48 |
XXI | 49 |
XXII | 51 |
XXIII | 52 |
| 57 | |
XXV | 65 |
XXVI | 67 |
XXVII | 72 |
XXIX | 76 |
XXX | 79 |
XXXI | 81 |
XXXII | 85 |
XXXIII | 90 |
XXXIV | 93 |
XXXV | 100 |
XXXVI | 101 |
XXXVII | 106 |
XXXVIII | 107 |
XXXIX | 108 |
XL | 109 |
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Common terms and phrases
algorithm apply bijection calculate chapter characterisation characteristic function Church's thesis code number computation P(x configuration corollary creative set defined degree denote diophantine Dom(f Dom(g elementary functions equivalent example exercise flow diagram formal proof function f ƒ is computable Gödel Hence I₁ infinite informal jump instruction K₁ lemma m-complete mathematical max(x means minimalisation n-ary natural numbers notation notion obtained partial function partially decidable predicates Peano arithmetic polynomial Post-system predicate M(x primitive recursive functions productive proof of theorem provable prove r.e. m-degree r.e. sets Ran(f recursive operator recursive sets recursively inseparable result Rice's theorem rm(x s-m-n theorem second Recursion theorem sequence Show single number statement stops string subset Suppose that f(x symbol tape theorem 1.1 total computable function total function Turing degrees Turing machine unary computable function unary function undecidable undefined otherwise URM-computable write x-computable