## 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 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. |

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### Contents

I | 1 |

II | 2 |

III | 4 |

V | 5 |

VI | 7 |

VIII | 9 |

IX | 16 |

X | 22 |

XXXIX | 112 |

XL | 121 |

XLII | 123 |

XLIII | 133 |

XLIV | 140 |

XLV | 143 |

XLVII | 146 |

XLVIII | 149 |

XI | 23 |

XII | 25 |

XIII | 29 |

XIV | 32 |

XV | 42 |

XVI | 48 |

XVIII | 49 |

XIX | 51 |

XX | 52 |

57 | |

XXII | 65 |

XXIII | 67 |

XXIV | 72 |

XXVI | 76 |

XXVII | 79 |

XXVIII | 81 |

XXIX | 85 |

XXXI | 90 |

XXXII | 93 |

XXXIII | 100 |

XXXIV | 101 |

XXXV | 106 |

XXXVI | 107 |

XXXVII | 108 |

XXXVIII | 109 |

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### Common terms and phrases

algorithm apply basic calculate chapter characterisation characteristic function Church's thesis code number computation P,(x configuration construction corollary creative set defined degree denote diagonal diophantine domain elementary functions equivalent example exercise flow diagram formal proof function f(x function g g is computable Godel Hence infinite informal jump instruction lemma m-complete mathematical means minimalisation n-ary natural numbers notation obtained partial function partial recursive 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 recursive operator recursive sets recursively enumerable recursively inseparable registers result Rice's theorem s-m-n theorem second Recursion theorem sequence Show single number Speed-up theorem statement stops string subset symbol tape theorem 1.1 total computable function total function Turing degrees Turing machine unary computable function unary function undecidable undefined otherwise URM-computable URMO program write