Turing Computability Theory and Applications

• Main Author: Soare, Robert I
• 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: Computable functions; Computer Science; Data processing Computer science; Logic, Symbolic and mathematical; Mathematical Logic and Foundations; Mathematics of Computing; Theory of Computation

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.

Table of Contents:

• 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 &gt -- 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