Turing Computability Theory and Applications
This book emphasizes three very important concepts: computability, as opposed to recursion or induction; classical computability; and the art of computability, a skill to be practiced but also important in an esthetic sense of beauty and taste in mathematics.
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Berlin, Heidelberg
Springer Nature
2016
Springer Springer Berlin / Heidelberg Springer Berlin Heidelberg |
| Edition | 1 |
| Series | Theory and Applications of Computability |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9783642319334 3642319335 9783642319327 9783662568583 3642319327 3662568586 |
| ISSN | 2190-619X 2190-6203 |
| DOI | 10.1007/978-3-642-31933-4 |
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| Abstract | This book emphasizes three very important concepts: computability, as opposed to recursion or induction; classical computability; and the art of computability, a skill to be practiced but also important in an esthetic sense of beauty and taste in mathematics. |
|---|---|
| AbstractList | Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers and to computability theory, the study of what machines can and cannot compute. This book presents classical computability theory from Turing and Post to current results and methods, and their use in studying the information content of algebraic structures, models, and their relation to Peano arithmetic. The author presents the subject as an art to be practiced, and an art in the aesthetic sense of inherent beauty which all mathematicians recognize in their subject. Part I gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic. Part II includes the study of computably open and closed sets of reals and basis and nonbasis theorems for effectively closed sets. Part III covers minimal Turing degrees. Part IV is an introduction to games and their use in proving theorems. Finally, Part V offers a short history of computability theory.The author has honed the content over decades according to feedback from students, lecturers, and researchers around the world. Most chapters include exercises, and the material is carefully structured according to importance and difficulty. The book is suitable for advanced undergraduate and graduate students in computer science and mathematics and researchers engaged with computability and mathematical logic. This book emphasizes three very important concepts: computability, as opposed to recursion or induction; classical computability; and the art of computability, a skill to be practiced but also important in an esthetic sense of beauty and taste in mathematics. |
| Author | Soare, Robert I |
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| ISSN | 2190-619X |
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| Notes | Includes bibliographical references |
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| PublicationPlace | Berlin, Heidelberg |
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| PublicationSeriesTitle | Theory and Applications of Computability |
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| Publisher | Springer Nature Springer Springer Berlin / Heidelberg Springer Berlin Heidelberg |
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| RelatedPersons | Bonizzoni, Paola Brattka, Vasco Mayordomo, Elvira Panangaden, Prakash |
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| Snippet | This book emphasizes three very important concepts: computability, as opposed to recursion or induction; classical computability; and the art of computability,... Turing's famous 1936 paper introduced a formal definition of a computing machine, a Turing machine. This model led to both the development of actual computers... |
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| SubjectTerms | Computable functions Computer Science Data processing Computer science Logic, Symbolic and mathematical Mathematical Logic and Foundations Mathematics of Computing Theory of Computation |
| Subtitle | Theory and Applications |
| TableOfContents | Myhill Isomorphism Theorem -- Acceptable Numberings -- Exercises -- 2 Computably Enumerable Sets -- 2.1 ** Characterizations of C.E. Sets -- 2.1.1 The Σ01 Normal Form for C.E. Sets -- 2.1.2 The Uniformization Theorem -- 2.1.3 The Listing Theorem for C.E. Sets -- 2.1.4 The C.E. and Computable Sets as Lattices -- 2.1.5 Exercises -- 2.2 * Recursion Theorem (Fixed Point Theorem) -- 2.2.1 Fixed Points in Mathematics -- 2.2.2 Operating on Indices -- 2.2.3 ** A Direct Proof of the Recursion Theorem -- 2.2.4 A Diagonal Argument Which Fails -- 2.2.5 Informal Applications of the Recursion Theorem -- 2.2.6 Other Properties of the Recursion Theorem -- 2.2.7 Exercises -- 2.3 Indexing Finite and Computable Sets -- 2.3.1 Computable Sets and Δ0 and Δ1 Indices -- 2.3.2 Canonical Index y for Finite Set Dy and String σy -- 2.3.3 Acceptable Numberings of Partial Computable Functions -- 2.3.4 Exercises -- 2.4 * Complete Sets and Creative Sets -- 2.4.1 Productive Sets -- 2.4.2 ** Creative Sets Are Complete -- 2.4.3 Exercises -- 2.5 ** Elementary Lachlan Games -- 2.5.1 The Definition of a Lachlan Game -- 2.5.2 Playing Partial Computable (P.C.) Functions -- 2.5.3 Some Easy Examples of Lachlan Games -- 2.5.4 Practicing Lachlan Games -- 2.5.5 The Significance of Lachlan Games -- 2.5.6 Exercises on Lachlan Games -- 2.6 The Order of Enumeration of C.E. Sets -- 2.6.1 Uniform Sequences and Simultaneous Enumerations -- 2.6.2 Static and Dynamic Properties of C.E. Sets -- 2.6.3 Exercises -- 2.7 The Friedberg Splitting Theorem -- 2.7.1 The Priority Ordering of Requirements -- 2.7.2 Exercises -- 3 Turing Reducibility -- 3.1 The Concept of Relative Computability -- 3.1.1 Turing Suggests Oracle Machines (o-Machines) -- 3.1.2 Post Develops Relative Computability -- 3.2 ** Turing Computability -- 3.2.1 An o-Machine Model for Relative Computability 3.2.2 Turing Computable Functionals Φe -- 3.3 * Oracle Graphs of Turing Functional Φe -- 3.3.1 The Prefix-Free Graph Fe of Functional Φe -- 3.3.2 The Oracle Graph Ge of Functional Φe -- 3.3.3 The Use Principle for Turing Functionals -- 3.3.4 Permitting Constructions -- 3.3.5 Lachlan Notation for Approximation by Stages -- 3.3.6 Standard Theorems Relativized to A -- 3.3.7 Exercises -- 3.4 * Turing Degrees and the Jump Operator -- 3.4.1 The Structure of the Turing Degrees -- 3.4.2 The Jump Theorem -- 3.4.3 Exercises -- 3.5 * Limit Computable Sets and Domination -- 3.5.1 Domination and Quantifiers ( x) and (x) -- 3.5.2 Uniformly Computable Sequences -- 3.5.3 Limit Computable Sets -- 3.5.4 Exercises -- 3.6 * The Limit Lemma -- 3.6.1 The Modulus Lemma for C.E. Sets -- 3.6.2 The Ovals of 1 and 2 Degrees -- 3.6.3 Reaching With the Jump: Low and High Sets -- 3.6.4 Exercises -- 3.7 * Trees and the Low Basis Theorem -- 3.7.1 Notation for Trees -- 3.7.2 * The Low Basis Theorem for 01 Classes -- 3.7.3 Exercises -- 3.8 Bounded Reducibilities and n-C.E. Sets -- 3.8.1 A Matrix Mx for Bounded Reducibilities -- 3.8.2 Bounded Turing Reducibility -- 3.8.3 Truth-Table Reductions -- 3.8.4 Difference of C.E., n-c.e., and -c.e. Sets -- 3.8.5 Exercises -- 4 The Arithmetical Hierarchy -- 4.1 Levels in the Arithmetical Hierarchy -- 4.1.1 Quantifier Manipulation -- 4.1.2 Placing a Set in Σn or Πn -- 4.1.3 Exercises -- 4.2 ** Post's Theorem and the Hierarchy Theorem -- 4.2.1 Post's Theorem Relating n to (n) -- 4.2.2 Exercises -- 4.3 * Σn-Complete Sets and Πn-Complete Sets -- 4.3.1 Classifying 2 and 2 Sets: Fin, Inf, and Tot -- 4.3.2 Constructions with Movable Markers -- 4.3.3 Classifying Cof as Σ3-Complete -- 4.3.4 Classifying Rec as Σ3-Complete -- 4.3.5 Σ3-Representation Theorems -- 4.3.6 Exercises -- 4.4 Relativized Hierarchy: Lown and Highn Sets Intro -- Contents -- Preface -- The Art of Classical Computability -- The Great Papers of Computability -- Turing's 1936 Paper, Especially 9 -- Post's 1944 Paper, Especially 11 -- Introduction -- Turing Machines -- Oracle Machines and Turing Reducibility -- Computable Enumerable Sets -- Bounded Turing Reductions -- Finally Understanding Turing Reducibility -- Priority Arguments -- Other Parts of This Book -- How to Read This Book -- Part I: Foundations of Computability -- Part II: Trees and Π01Classes -- Part III: Minimal Degrees -- Part IV: Games in Computability Theory -- Symbols Marking Importance and Di culty -- Notation -- Notation for Sets -- Logical Notation -- Lattices and Boolean Algebras -- Notation for Strings and Functionals -- Gödel Numbering of Finite Objects -- Standard Pairing Function -- Effective Numbering of Finite Sets and Strings -- Partial Computable (P.C.) Functions -- Turing Functionals ΦAe -- Lachlan Notation -- Acknowledgements -- Part I Foundations of Computability -- 1 Defining Computability -- 1.1 Algorithmically Computable Functions -- Algorithms in Mathematics -- The Obstacle of Diagonalization and Partial Functions -- The Quest for a Characterization -- Turing's Breakthrough -- 1.2 ** Turing Defines Effectively Calculable -- 1.3 ** Turing's Thesis, Turing's Theorem, TT -- 1.4 ** Turing Machines -- 1.4.1 Exercises on Turing Machines -- 1.5 ** The Basic Results -- 1.5.1 Numbering Turing Programs Pe -- 1.5.2 Numbering Turing Computations -- 1.5.3 The Enumeration Theorem and Universal Machine -- 1.5.4 The Parameter Theorem or s-m-n Theorem -- 1.6 ** Unsolvable Problems -- 1.6.1 Computably Enumerable Sets -- 1.6.2 Noncomputable C.E. Sets -- 1.6.3 The Index Set Theorem and Rice's Theorem -- 1.6.4 Computable Approximations to Computations -- 1.6.5 Exercises -- 1.7 * Computable Permutations and Isomorphisms 4.4.1 Relativized Post's Theorem -- 4.4.2 Lown and Highn Sets -- 4.4.3 Common Jump Classes of Degrees -- 4.4.4 Syntactic Properties of Highn and Lown Sets -- 4.4.5 Exercises -- 4.5 * Domination and Escaping Domination -- 4.5.1 Domination Properties -- 4.5.2 Martin's High Domination Theorem -- 4.5.3 Exercises -- 4.6 Characterizing Nonlow2 Sets A ≤Tθ' -- 4.6.1 Exercises -- 4.7 Domination, Escape, and Classes of Degrees -- 4.8 Uniform Enumerations of Functions and Sets -- 4.8.1 Limits of Functions -- 4.8.2 A-uniform Enumeration of the Computable Functions -- 4.9 Characterizing Low2 Sets A T' -- 4.9.1 Exercises -- 5 Classifying C.E. Sets -- 5.1 * Degrees of Computably Enumerable Sets -- 5.1.1 Post's Problem and Post's Program -- 5.1.2 Dynamic Turing Reductions on C.E. Sets -- 5.2 * Simple Sets and the Permitting Method -- 5.2.1 Post's Simple Set Construction -- 5.2.2 The Canonical Simple Set Construction -- 5.2.3 Domination and a Complete Simple Set -- 5.2.4 Simple Permitting and Simple Sets -- 5.2.5 Permitting as a Game -- 5.2.6 Exercises -- 5.3 * Hypersimple Sets and Dominating Functions -- 5.3.1 Weak and Strong Arrays of Finite Sets -- 5.3.2 Dominating Functions and Hyperimmune Sets -- 5.3.3 Degrees of Hypersimple Sets and Dekker's Theorem -- 5.3.4 Exercises -- 5.4 * The Arslanov Completeness Criterion -- 5.4.1 Effectively Simple Sets Are Complete -- 5.4.2 Arslanov's Completeness Criterion for C.E. Sets -- 5.4.3 Exercises -- 5.5 More General Permitting -- 5.5.1 Standard and General Permitting -- 5.5.2 Reverse Permitting -- 5.5.3 Building a Turing Functional ΘC = A -- 5.6 Hyperimmune-Free Degrees -- 5.6.1 Two Downward Closure Properties of Domination -- 5.6.2 Δ2 Degrees Are Hyperimmune -- 3.6.3 Σ2 Approximations and Domination -- 5.7 Historical Remarks and Research References -- 5.7.1 Δ2-Permitting -- 6 Oracle Constructions and Forcing 6.1 * Kleene-Post Finite Extensions -- 6.1.1 Exercises -- 6.2 Minimal Pairs and Avoiding Cones -- 6.2.1 Exercises -- 6.3 * Generic Sets -- 6.3.1 1-Generic Sets -- 6.3.2 Forcing the Jump -- 6.3.3 Doing Many Constructions at Once -- 6.3.4 Exercises -- 6.4 * Inverting the Jump -- 6.4.1 Exercises -- 6.5 Upper and Lower Bounds for Degrees -- 6.5.1 Exercises -- 7 The Finite Injury Method -- 7.1 A Solution to Post's Problem -- 7.1.1 The Intuition Behind Finite Injury -- 7.1.2 The Injury Set for Requirement Ne -- 7.2 * Low Simple Sets -- 7.2.1 The Requirements for a Low Simple Set A -- 7.2.2 A Computable ĝ(e,s) with g(e) = lims ĝ(e,s) = A'(e) -- 7.2.3 The Construction of a Low Simple Set A -- 7.2.4 The Verification of a Low Simple Set A -- 7.2.5 The Restraint Functions r(e,s) as Walls -- 7.2.6 Exercises -- 7.3 * The Friedberg-Muchnik Theorem -- 7.3.1 Renumbering the Requirements -- 7.3.2 The Basic Module to Meet Re for e Even -- 7.3.3 The Full Construction -- 7.3.4 The Verification -- 7.3.5 Exercises -- 7.4 * Preservation Strategy to Avoid Upper Cones -- 7.4.1 The Notation -- 7.4.2 The Basic Module for Requirement Ne -- 7.4.3 The Construction of A -- 7.4.4 The Verification -- 7.5 Sacks Splitting Theorem -- 7.6 Avoiding the Cone Above a Δ2 Set C > -- T 0 -- 7.6.1 Exercises -- Part II Trees and Π01 Classes -- 8 Open and Closed Classes -- 8.1 Open Classes in Cantor Space -- 8.2 Closed Classes in Cantor Space -- 8.3 The Compactness Theorem -- 8.4 Notation for Trees -- 8.5 Effective Compactness Theorem -- 8.6 Dense Open Subsets of Cantor Space -- 8.7 Exercises -- 9 Basis Theorems -- 9.1 Bases and Nonbases for Π01-Classes -- 9.2 Previous Basis Theorems for Π01-Classes -- 9.3 Nonbasis Theorems for 01-Classes -- 9.4 The Super Low Basis Theorem (SLBT) -- 9.5 The Computably Dominated Basis Theorem -- 9.6 Low Antibasis Theorem -- 9.7 Proper Lown Basis Theorem 10 Peano Arithmetic and 01-Classes |
| Title | Turing Computability |
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