Computability Theory

Computability Theory

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Computability theory originated with the seminal work of Godel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.show more

Product details

  • Hardback | 420 pages
  • 160.02 x 231.14 x 27.94mm | 725.74g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 35 black & white illustrations
  • 1584882379
  • 9781584882374
  • 1,567,027

Review quote

"A very nice volume indeed. Although primarily a textbook, it lives up to the author's aim to have 'plenty here to interest and inform everyone, from the beginner to the expert.' Cooper writes in an informal style, emphasizing the ideas underlying the techniques. All the standard topics and classic results are here. Students will find useful pointers to the literature and an abundance of exercises woven into the text." - Zentralblatt MATH, 1041 "[It] provides not only a reference repository of well-crafted proofs or proof-outlines for a large number of basic and beyond-basic facts in several areas of computability theory, but can also serve well as the textual basis for a course on the subject" - Mathematical Reviews, 2005hshow more

Table of contents

SECTION I: COMPUTABILITY, AND UNSOLVABLE PROBLEMS HILBERT AND THE ORIGINS OF COMPUTABILITY THEORY Algorithms and Algorithmic Content Hilbert's Programme Godel, and the Discovery of Incomputability Computability and Unsolvability in the Real World MODELS OF COMPUTABILITY AND THE CHURCH-TURING THESIS The Recursive Functions Church's Thesis, and the Computability of Sets and Relations Unlimited Register Machines Turing's Machines Church, Turing, and the Equivalence of Models LANGUAGE, PROOF AND COMPUTABLE FUNCTIONS Peano Arithmetic and its Models What Functions Can We Describe in a Theory? CODING, SELF-REFERENCE AND THE UNIVERSAL TURING MACHINE Russell's Paradox Godel Numberings A Universal Turing Machine The Fixed Point Theorem Computable Approximations ENUMERABILITY AND COMPUTABILITY Basic Notions The Normal Form Theorem Incomputable Sets and the Unsolvability of the Halting Problem for Turing Machines The Busy Beaver function THE SEARCH FOR NATURAL EXAMPLES OF INCOMPUTABLE SETS The Ubiquitous Creative Sets Some Less Natural Examples of Incomputable Sets Hilbert's Tenth Problem and the Search for Really Natural Examples COMPARING COMPUTABILITY Many-One Reducibility The Non-Computable Universe and Many-One Degrees Creative Sets Revisited GODEL'S INCOMPLETENESS THEOREM Semi-Representability and C.E. Sets Incomputability and Godel's Theorem DECIDABLE AND UNDECIDABLE THEORIES PA is Undecidable Other Undecidable Theories, and their Many-One Equivalence Some Decidable Theories SECTION II: INCOMPUTABILITY AND INFORMATION CONTENT COMPUTING WITH ORACLES Oracle Turing Machines Relativising, and Listing the Partial Computable Functionals Introducing the Turing Universe Enumerating with Oracles, and the Jump Operator The Arithmetical Hierarchy and Post's Theorem The Structure of the Turing Universe NONDETERMINISM, ENUMERATIONS AND POLYNOMIAL BOUNDS Oracles versus Enumerations of Data Enumeration Reducibility and the Scott Model for Lambda Calculus The Enumeration Degrees,and the Natural Embedding of the Turing Degrees The Structure of De and the Arithmetical Hierarchy The Medvedev Lattice Polynomial Bounds and P =?NP SECTION III: MORE ADVANCED TOPICS POST'S PROBLEM: IMMUNITY AND PRIORITY Information Content and Structure Immunity Properties Approximation and Priority Sacks Splitting Theorem and Cone Avoidance Minimal Pairs and Extensions of Embeddings The |3 Theory - Information Content Regained Higher Priority and Maximal Sets The Computability of Theories FORCING AND CATEGORY Forcing in Computability Theory Baire Space, Category and Measure n-Genericity and Applications Forcing with Trees, and Minimal Degrees APPLICATIONS OF DETERMINACY Gale-Stewart Games An Upper Cone of Minimal Covers Borel and Projective Determinacy, and the Global Theory of D THE COMPUTABILITY OF THEORIES Feferman's Theorem Truth versus Provability Complete extensions of Peano Arithmetic and Classes The Low Basis Theorem Arslanov's Completeness Criterion A Priority-Free Solution to Post's Problem Randomness COMPUTABILITY AND STRUCTURE Computable Models Computability and Mathematical Structures Effective Ramsey Theory Computability in Analysis Computability and Incomputability in Science FURTHER READING INDEXshow more

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11 ratings
4.63 out of 5 stars
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4 18% (2)
3 9% (1)
2 0% (0)
1 0% (0)
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