Recursively enumerable sets and degrees: a study of computable functions and computably generated sets
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1987
|
| Series: | Perspectives in mathematical logic
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 437 S. graph. Darst. |
| ISBN: | 3540152997 0387152997 |
Staff View
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| 100 | 1 | |a Soare, Robert I. |e Verfasser |0 (DE-588)1067760717 |4 aut | |
| 245 | 1 | 0 | |a Recursively enumerable sets and degrees |b a study of computable functions and computably generated sets |c Robert I. Soare |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1987 | |
| 300 | |a XVIII, 437 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Perspectives in mathematical logic | |
| 650 | 4 | |a Fonctions calculables | |
| 650 | 4 | |a Fonctions récursives | |
| 650 | 4 | |a Computable functions | |
| 650 | 4 | |a Recursively enumerable sets | |
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Record in the Search Index
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| adam_text | Table of Contents
Introduction 1
Part A. The Fundamental Concepts of Recursion Theory .... 5
Chapter I. Recursive Functions 7
1. An Informal Description 7
2. Formal Definitions of Computable Functions 8
2.1. Primitive Recursive Functions 8
2.2. Diagonalization and Partial Recursive Functions . 10
2.3. Turing Computable Functions 11
3. The Basic Results 14
4. Recursively Enumerable Sets and Unsolvable Problems . . 18
5. Recursive Permutations and Myhill s Isomorphism
Theorem 24
Chapter II. Fundamentals of Recursively Enumerable Sets
and the Recursion Theorem 27
1. Equivalent Definitions of Recursively Enumerable Sets and
Their Basic Properties 27
2. Uniformity and Indices for Recursive and Finite Sets . . 32
3. The Recursion Theorem 36
4. Complete Sets, Productive Sets, and Creative Sets ... 40
Chapter III. Turing Reducibility and the Jump Operator .... 46
1. Definitions of Relative Computability 46
2. Turing Degrees and the Jump Operator 52
3. The Modulus Lemma and Limit Lemma 56
Chapter IV. The Arithmetical Hierarchy 60
1. Computing Levels in the Arithmetical Hierarchy .... 60
2. Post s Theorem and the Hierarchy Theorem 64
3. En Complete Sets 65
4. The Relativized Arithmetical Hierarchy and High and Low
Degrees 69
xiv Table of Contents
Part B. Post s Problem, Oracle Constructions and the Finite Injury
Priority Method 75
Chapter V. Simple Sets and Post s Problem 77
1. Immune Sets, Simple Sets and Post s Construction ... 77
2. Hypersimple Sets and Majorizing Functions 80
3. The Permitting Method 84
4. Effectively Simple Sets Are Complete 87
5. A Completeness Criterion for R.E. Sets 88
Chapter VI. Oracle Constructions of Non R.E. Degrees .... 92
1. A Pair of Incomparable Degrees Below 0 93
2. Avoiding Cones of Degrees 96
3. Inverting the Jump 97
4. Upper and Lower Bounds for Degrees 100
5.* Minimal Degrees 103
Chapter VII. The Finite Injury Priority Method 110
1. Low Simple Sets 110
2. The Original Friedberg Muchnik Theorem 118
3. Splitting Theorems 121
Part C. Infinitary Methods for Constructing R.E. Sets
and Degrees 127
Chapter VIII. The Infinite Injury Priority Method 129
1. The Obstacles in Infinite Injury and the Thickness
Lemma 130
2. The Injury and Window Lemmas and the Strong Thickness
Lemma 134
3. The Jump Theorem 137
4. The Density Theorem and the Sacks Coding Strategy . . 142
5.* The Pinball Machine Model for Infinite Injury 147
Chapter IX. The Minimal Pair Method and Embedding Lattices
into the R.E. Degrees 151
1. Minimal Pairs and Embedding the Diamond Lattice ... 152
2.* Embedding Distributive Lattices 157
3. The Non Diamond Theorem 161
4.* Nonbranching Degrees 168
5.* Noncappable Degrees 174
Table of Contents xv
Chapter X. The Lattice of R.E. Sets Under Inclusion 178
1. Ideals, Filters, and Quotient Lattices 178
2. Splitting Theorems and Boolean Algebras 181
3. Maximal Sets 187
4. Major Subsets and r Maximal Sets 190
5. Atomless r Maximal Sets 195
6. Atomless hh Simple Sets 199
7.* £3 Boolean Algebras Represented as Lattices of
Supersets 203
Chapter XI. The Relationship Between the Structure and
the Degree of an R.E. Set 207
1. Martin s Characterization of High Degrees in Terms of
Dominating Functions 207
2. Maximal Sets and High R.E. Degrees 215
3. Low R.E. Sets Resemble Recursive Sets 224
4. Non Low2 R.E. Degrees Contain Atomless R.E. Sets . . . 230
5.* Low2 R.E. Degrees Do Not Contain
Atomless R.E. Sets 233
Chapter XII. Classifying Index Sets of R.E. Sets 241
1. Classifying the Index Set G{A) = { x : Wx =T A } ... 242
2. Classifying the Index Sets G{ A), G{ A),
and G{ A) 246
3. Uniform Enumeration of R.E. Sets and £3 Index Sets . . 253
4. Classifying the Index Sets of the Highn, Lown, and
Intermediate R.E. Sets 259
5. Fixed Points up to Turing Equivalence 270
6. A Generalization of the Recursion Theorem and
the Completeness Criterion to All Levels of
the Arithmetical Hierarchy 272
Part D. Advanced Topics and Current Research Areas in the R.E.
Degrees and the Lattice of R.E. Sets 279
Chapter XIII. Promptly Simple Sets, Coincidence of Classes of
R.E. Degrees, and an Algebraic Decomposition of the
R.E. Degrees 281
I 1. Promptly Simple Sets and Degrees 282
xvi Table of Contents
2. Coincidence of the Classes of Promptly Simple Degrees,
Noncappable Degrees, and Effectively Noncappable
Degrees 288
3. A Decomposition of the R.E. Degrees Into the Disjoint
Union of a Definable Ideal and a Definable Filter .... 294
4. Cuppable Degrees and the Coincidence of Promptly Simple
and Low Cuppable Degrees 296
Chapter XIV. The Tree Method and (/ Priority Arguments ... 300
1. The Tree Method With O Priority Arguments 301
2. The Tree Method in Priority Arguments and the
Classification of 0 ,0 , and 0 Priority Arguments ... 304
3. The Tree Method With 0 Priority Arguments 308
3.1. Trees Applied to an Ordinary 0 Priority
Argument 308
3.2. A 0 Priority Argument Which Requires the Tree
Method 309
4. The Tree Method With a 0 Priority Argument:
The Lachlan Nonbounding Theorem 315
4.1. Preliminaries 315
4.2. The Basic Module for Meeting a Subrequirement . 316
4.3. The Priority Tree 320
4.4. Intuition for the Priority Tree
and the Proof 325
4.5. The Construction 327
4.6. The Verification 331
Chapter XV. Automorphisms of the Lattice of R.E. Sets .... 338
1. Invariant Properties 338
2. Some Basic Properties of Automorphisms of f 341
3. Noninvariant Properties 345
4. The Statement of the Extension Theorem
and Its Motivation 348
5. Satisfying the Hypotheses of the Extension Theorem
for Maximal Sets 354
6. The Proof of the Extension Theorem 359
6.1. The Machines 359
6.2. The Construction 362
6.3. The Requirements and the Motivation
for the Rules 363
6.4. The Rules 365
Table of Contents xvii
6.5. The Verification 368
Chapter XVI. Further Results and Open Questions
About R.E. Sets and Degrees 374
1. Automorphisms and Isomorphisms of the Lattice
of R.E. Sets 374
2. The Elementary Theory of f 379
3. The Elementary Theory of the R.E. Degrees 383
4. The Algebraic Structure of R 385
References 389
Notation Index 419
Subject Index 429
|
| any_adam_object | 1 |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3 |
| dewey-search | 511.3 |
| dewey-sort | 3511.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik Philosophie |
| format | Book |
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| indexdate | 2024-07-09T15:19:39Z |
| institution | BVB |
| isbn | 3540152997 0387152997 |
| language | English |
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| physical | XVIII, 437 S. graph. Darst. |
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| spelling | Soare, Robert I. Verfasser (DE-588)1067760717 aut Recursively enumerable sets and degrees a study of computable functions and computably generated sets Robert I. Soare Berlin [u.a.] Springer 1987 XVIII, 437 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Perspectives in mathematical logic Fonctions calculables Fonctions récursives Computable functions Recursively enumerable sets Berechenbarkeit (DE-588)4138368-0 gnd rswk-swf Rekursionstheorie (DE-588)4122329-9 gnd rswk-swf Berechenbare Funktion (DE-588)4135762-0 gnd rswk-swf Rekursive Funktion (DE-588)4138367-9 gnd rswk-swf Rekursiv aufzählbare Menge (DE-588)4135763-2 gnd rswk-swf Berechenbare Funktion (DE-588)4135762-0 s DE-604 Rekursiv aufzählbare Menge (DE-588)4135763-2 s Berechenbarkeit (DE-588)4138368-0 s Rekursive Funktion (DE-588)4138367-9 s Rekursionstheorie (DE-588)4122329-9 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000501457&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Soare, Robert I. Recursively enumerable sets and degrees a study of computable functions and computably generated sets Fonctions calculables Fonctions récursives Computable functions Recursively enumerable sets Berechenbarkeit (DE-588)4138368-0 gnd Rekursionstheorie (DE-588)4122329-9 gnd Berechenbare Funktion (DE-588)4135762-0 gnd Rekursive Funktion (DE-588)4138367-9 gnd Rekursiv aufzählbare Menge (DE-588)4135763-2 gnd |
| subject_GND | (DE-588)4138368-0 (DE-588)4122329-9 (DE-588)4135762-0 (DE-588)4138367-9 (DE-588)4135763-2 |
| title | Recursively enumerable sets and degrees a study of computable functions and computably generated sets |
| title_auth | Recursively enumerable sets and degrees a study of computable functions and computably generated sets |
| title_exact_search | Recursively enumerable sets and degrees a study of computable functions and computably generated sets |
| title_full | Recursively enumerable sets and degrees a study of computable functions and computably generated sets Robert I. Soare |
| title_fullStr | Recursively enumerable sets and degrees a study of computable functions and computably generated sets Robert I. Soare |
| title_full_unstemmed | Recursively enumerable sets and degrees a study of computable functions and computably generated sets Robert I. Soare |
| title_short | Recursively enumerable sets and degrees |
| title_sort | recursively enumerable sets and degrees a study of computable functions and computably generated sets |
| title_sub | a study of computable functions and computably generated sets |
| topic | Fonctions calculables Fonctions récursives Computable functions Recursively enumerable sets Berechenbarkeit (DE-588)4138368-0 gnd Rekursionstheorie (DE-588)4122329-9 gnd Berechenbare Funktion (DE-588)4135762-0 gnd Rekursive Funktion (DE-588)4138367-9 gnd Rekursiv aufzählbare Menge (DE-588)4135763-2 gnd |
| topic_facet | Fonctions calculables Fonctions récursives Computable functions Recursively enumerable sets Berechenbarkeit Rekursionstheorie Berechenbare Funktion Rekursive Funktion Rekursiv aufzählbare Menge |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000501457&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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