Holdings: Computability theory

Computability theory:

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Bibliographic Details
Main Author:	Cooper, Stuart B. 1943-2015 (Author)
Format:	Book
Language:	English
Published:	Boca Raton [u.a.] Chapman & Hall/CRC 2004
Series:	Chapman & Hall/CRC mathematics
Subjects:	Computable functions Berechenbare Funktion Berechnungstheorie
Online Access:	Inhaltsverzeichnis
Physical Description:	IX, 409 S. graph. Darst.
ISBN:	1584882379

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MARC


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adam_text	Contents Part I Computability and Unsolvable Problems 1 1 Hilbert and the Origins of Computability Theory 3 1.1 Algorithms and Algorithmic Content ...................... 3 1.2 Hilbert’s Programme ..................................... 5 1.3 Gödel and the Discovery of Incomputability .............. 7 1.4 Computability and Unsolvability in the Real World ....... 8 2 Models of Computability and the Church—Turing Thesis 11 2.1 The Recursive Functions ................................ 12 2.2 Church’s Thesis and the Computability of Sets and Relations 19 2.3 Unlimited Register Machines............................. 25 2.4 Turing’s Machines ...................................... 34 2.5 Church, Turing, and the Equivalence of Models........... 42 3 Language, Proof and Computable Functions 45 3.1 Peano Arithmetic and Its Models ........................ 45 3.2 What Functions Can We Describe in a Theory? ............ 56 4 Coding, Self-Reference and the Universal Turing Machine 61 4.1 Russell’s Paradox ...................................... 61 4.2 Gödel Numberings........................................ 62 4.3 A Universal Turing Machine ............................. 65 4.4 The Fixed Point Theorem................................. 66 4.5 Computable Approximations............................... 67 5 Enumerability and Computability 69 5.1 Basic Notions .......................................... 69 5.2 The Normal Form Theorem................................. 73 5.3 Incomputable Sets and the Unsolvability of the Halting Prob- lem for Turing Machines ..................................... 78 5.4 The Busy Beaver Function ............................... 81 6 The Search for Natural Examples of Incomputable Sets 87 6.1 The Ubiquitous Creative Sets............................ 88 6.2 Some Less Natural Examples of Incomputable Sets ........ 90 6.3 Hilbert’s Tenth Problem and the Search for Really Natural Examples................................................ 94 7 Comparing Computability and the Ubiquity of Creative Sets 101 7.1 Many-One Reducibility.................................. 101 7.2 The Non-Computable Universe and Many-One Degrees . . . 106 7.3 Creative Sets Revisited ............................... 112 8 Godel’s Incompleteness Theorem 117 8.1 Semi-Representability and C.E. Sets.................... 117 8.2 Incomputability and Godel’s Theorem.................... 122 9 Decidable and Undecidable Theories 127 9.1 PA is Undecidable...................................... 127 9.2 Other Undecidable Theories and Their Many-One Equivalence 128 9.3 Some Decidable Theories ............................... 132 Part II Incomputability and Information Content 137 10 Computing with Oracles 139 10.1 Oracle Turing Machines................................. 139 10.2 Relativising, and Listing the Partial Computable Functionals 142 10.3 Introducing the Turing Universe ....................... 144 10.4 Enumerating with Oracles, and the Jump Operator ....... 147 ՜ 10.5 The Arithmetical Hierarchy and Post’s Theorem.......... 154 10.6 The Structure of the Turing Universe .................. 161 11 Nondeterminism, Enumerations and Polynomial Bounds 173 11.1 Oracles versus Enumerations of Data .................. 173 11.2 Enumeration Reducibility and the Scott Model for Lambda Calculus .............................................. 180 11.3 The Enumeration Degrees and the Natural Embedding of the Turing Degrees......................................... 191 11.4 The Structure of T e and the Arithmetical Hierarchy... 199 11.5 The Medvedev Lattice................................... 202 11.6 Polynomial Bounds and P =? NP ........................ 205 Part III More Advanced Topics 217 12 Post’s Problem; Immunity and Priority 219 12.1 Information Content and Structure ..................... 219 12.2 Immunity Properties ................................... 226 12.3 Approximation and Priority ............................ 237 12.4 Sacks’ Splitting Theorem and Cone Avoidance ............ 246 12.5 Minimal Pairs and Extensions of Embeddings ............. 252 12.6 The II3 Theory — Information Content Regained .......... 260 12.7 Higher Priority and Maximal Sets ....................... 266 13 Forcing and Category 273 13.1 Forcing in Computability Theory ........................ 273 13.2 Baire Space, Category and Measure ...................... 278 13.3 n-Genericity and Applications ........................... 287 13.4 Forcing with Trees and Minimal Degrees ................. 299 14 Applications of Determinacy 311 14.1 Gale֊Stewart Games ...................................... 311 14.2 An Upper Cone of Minimal Covers......................... 315 14.3 Borel and Projective Determinacy, and the Global Theory of T 318 15 The Computability of Theories 321 15.1 Feferman’s Theorem....................................... 321 15.2 Truth versus Provability ................................ 323 15.3 Complete Extensions of Peano Arithmetic and n^-Classes . . 324 15.4 The Low Basis Theorem .................................. 329 15.5 Arslanov’s Completeness Criterion........................ 331 15.6 A Priority-Free Solution to Post’s Problem ............. 333 15.7 Randomness .............................................. 335 16 Computability and Structure 343 16.1 Computable Models........................................ 343 16.2 Computability and Mathematical Structures ............... 348 16.3 Effective Ramsey Theory ................................. 362 16.4 Computability in Analysis ............................... 371 16.5 Computability and Incomputability in Science ............ 379 Further Reading 383 Index 389
any_adam_object	1
author	Cooper, Stuart B. 1943-2015
author_GND	(DE-588)133309096
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ctrlnum	(OCoLC)248800776 (DE-599)BVBBV019668177
dewey-full	511.3 511.322
dewey-hundreds	500 - Natural sciences and mathematics
dewey-ones	511 - General principles of mathematics
dewey-raw	511.3 511.3 22
dewey-search	511.3 511.3 22
dewey-sort	3511.3
dewey-tens	510 - Mathematics
discipline	Informatik Mathematik
format	Book
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spelling	Cooper, Stuart B. 1943-2015 Verfasser (DE-588)133309096 aut Computability theory S. Barry Cooper Boca Raton [u.a.] Chapman & Hall/CRC 2004 IX, 409 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC mathematics Computable functions Berechenbare Funktion (DE-588)4135762-0 gnd rswk-swf Berechnungstheorie (DE-588)4005581-4 gnd rswk-swf Berechenbare Funktion (DE-588)4135762-0 s Berechnungstheorie (DE-588)4005581-4 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012996455&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Cooper, Stuart B. 1943-2015 Computability theory Computable functions Berechenbare Funktion (DE-588)4135762-0 gnd Berechnungstheorie (DE-588)4005581-4 gnd
subject_GND	(DE-588)4135762-0 (DE-588)4005581-4
title	Computability theory
title_auth	Computability theory
title_exact_search	Computability theory
title_full	Computability theory S. Barry Cooper
title_fullStr	Computability theory S. Barry Cooper
title_full_unstemmed	Computability theory S. Barry Cooper
title_short	Computability theory
title_sort	computability theory
topic	Computable functions Berechenbare Funktion (DE-588)4135762-0 gnd Berechnungstheorie (DE-588)4005581-4 gnd
topic_facet	Computable functions Berechenbare Funktion Berechnungstheorie
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