Verfügbarkeit: Computability and randomness

Computability and randomness:

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1. Verfasser:	Nies, André (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Oxford Oxford Univ. Press 2009
Ausgabe:	1. publ.
Schriftenreihe:	Oxford logic guides 51
Schlagworte:	Numbers, Random Computational complexity Zufallszahlen Berechnungskomplexität
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XV, 433 S.

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Datensatz im Suchindex

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adam_text	Contents The complexity of sets 1 1.1 The basic concepts 3 Partial computable functions 3 Computably enumerable sets 6 Indices and approximations 7 1.2 Relative computational complexity of sets 8 Many-one reducibility 8 Turing reducibility 9 Relativization and the jump operator 10 Strings over {0.1} 12 Approximating the functionals Ф,., and the use principle 13 Weak truth-table reducibility and truth-table reducibility 14 Degree structures 16 1.3 Sets of natural numbers 16 1.4 Descriptive complexity of sets 18 Δ2 sets and the Shoenfield Limit Lemma 18 Sets and functions that are »-ce. or cj-c.e. 19 Degree structures on particular classes 20 The arithmetical hierarchy 21 1.5 Absolute computational complexity of sets 24 Sets that are low,, 26 Computably dominated sets 27 Sets that are high,, 28 1.6 Post s problem 29 Turing incomparable Δί, -sets 30 Simple sets 31 A e.e. set that is neither computable nor Turing complete 32 Is there a natural solution to Post s problem? 34 Turing incomparable ce. sets 35 1.7 Properties of ce. sets 37 Each incomputable ce. wtt-degrec contains a simple set 38 Hypersimple sets 39 Promptly simple sets 40 Minimal pairs and promptly simple sets 41 Creative sets * 43 1.8 Cantor space 45 Open sets 47 Binary trees and closed sets 47 Representing open sets 48 Contents Compactness and clopen sets 48 The correspondence between subsets of N and real numbers 49 Effectivity notions for real numbers 50 Effectivity notions for classes of sets 52 Examples of Π? classes 55 Isolated points and perfect sets 56 The Low Basis Theorem 56 The basis theorem for computably dominated sets 59 Weakly l-generic sets 61 l-generic sets 63 The arithmetical hierarchy of classes 64 Comparing Cantor space with Baire space 67 1.9 Measure and probability 68 Outer measures 68 Measures 69 Uniform measure and null classes 70 Uniform measure of arithmetical classes 71 Probability theory 73 The descriptive complexity of strings 74 Comparing the growth rate of functions 75 2.1 The plain descriptive complexity С 75 Machines and descriptions 75 The counting condition, and incompressible strings 77 Invariance, continuity, and growth of С 79 Algorithmic properties of С 81 2.2 The prefix-free complexity Л 82 Drawbacks of С 83 Prefix-free machines 83 The Machine Existence Theorem and a characterization of A 86 The Coding Theorem !)1 2.3 Conditional descriptive complexity 92 Basics 92 An expression for K(.r.y) * 93 2.4 Relating С and A 94 Basic interactions 94 Solovay s equations * 95 2.5 Incompressibility and randomness for strings 97 Comparing incompressibility notions 98 Randomness properties of strings 99 Martin-Löf randomness and its variants 102 3.1 A mathematical definition of randomness for sets 102 Martin-Löf tests and their variants 104 Schnorr s Іііеопчп and universal Martin-Löf tests 105 Contents xi The initial segment approach 105 3.2 Martin-Löf randomness 106 The test concept 106 A universal Martin-Löf test 107 Characterization of MLR via the initial segment complexity 107 Examples of Martin-Löf random sets 108 Facts about ML-random sets 109 Left-c.e. ML-random reals and Solovay reducibility 113 Randomness on reals, and randomness for bases other than 2 115 A nonempty Пј subclass of MLR has ML-random measure * 116 3.3 Martin-Löf randomness and reduction procedures 117 Each set is weak truth-table reducible to a ML-random set 117 Autoreducibility and indifferent sets * 119 3.4 Martin-Löf randomness relative to an oracle 120 Relativizing С and A 121 Basics of relative ML-randonmess 122 Symmetry of relative Martin-Löf randomness 122 Computational complexity, and relative randomness 124 The halting probability Ω relative to an oracle * 125 3.5 Notions weaker than ML-randomness 127 Weak randomness 128 Schnorr randomness 129 Computable measure machines 131 3.6 Notions stronger than ML-randomness 133 Weak 2-randomness 134 2-randonmess and initial segment complexity 136 2-randomness and being low for Ω 140 Demut h randomness * 141 4 Diagonally noncomputable functions 144 4.1 D.n.c. functions and sets of d. u.c. degree 145 Basics on d.n.c. functions and fixed point freeness 145 The initial segment complexity of sets of d.n.c. degree 147 A completeness criterion for ce. sets 148 4.2 Injury-free constructions of ce. sets 150 Each Λ set of d.n.c. degree bounds a promptly simple set 151 Variants of Kučera s Theorem 152 An injury-free proof of the Friedberg Muchnik Theorem * 154 4.3 Strengthening the notion of a d.n.c. function 155 Sets of PA degree 1Г)(І Martin-Löf random sets of PA degree 157 Turing degrees of Martin-Löf random sets * 15!) Relating //-randomness and higher fixed point frceness ІШ xii Contents 5 Lowness properties and A -triviality 163 5.1 Equivalent lowness properties 165 Being low for A 105 Lowness for ML-randomness 167 When many oracles compute a set 168 Bases for ML-randomness 170 Lowness for weak 2-randomness 174 5.2 A -trivial sets 176 Basics on A -trivial sets 176 A -trivial sets are Aj 177 The number of sets that are A -trivial for a constant b * 179 Closure properties of K, 181 C-trivial sets 182 Replacing the constant by a slowly growing function * 183 5.3 The cost function method 184 The basics of cost functions 186 A cost function criterion for A -triviality 188 Cost functions and injury-free solutions to Post s problem 189 Construction of a promptly simple Turing lower bound 190 A -trivial sets and Σ ι -induction * 192 Avoiding to be Turing reducible to a given low c.e. set 193 Necessity of the cost function method for c.e. A -trivial sets 195 Listing the (j- -cc.) A -trivial sets with constants 196 Adaptive cost functions 198 5.4 Each A-trivial set is low for A 200 Introduction to the proof 201 The formal proof of Theorem 5.4.1 210 5.5 Properties of the class of A -trivial sets 215 A Main Lemma derived from the golden run method 215 The standard cost function characterizes the A -trivial sets 217 The number of changes * 219 Ω- 1 for A -trivial .4 221 Each A -trivial set is low for weak 2-randomnes.s 223 5.6 The weak reduci bili ty associated with Low(MLR) 224 Preordcrings coinciding with ¿.R-redueibility 226 A stronger result under the extra hypothesis that A <j- B 228 The size of lower and upper cones for <ля * 230 Operators obtained by relativizing classes 231 Studying <lr by applying the operator К 232 Comparing the operators S ¡.r and К * 233 Uniformly almost everywhere dominating sets 234 0 <lh C if an«! only if С is uniformly a.e. dominating 235 Contents xiii Some advanced computability theory 238 6.1 Enumerating Turing functionals 239 Basics and a first example 239 C.e. oracles, markers, and a further example 240 6.2 Promptly simple degrees and low cuppability 242 C.e. sets of promptly simple degree 243 A c.e. degree is promptly simple iff it is low cuppable 244 6.3 C.e. operators and highness properties 247 The basics of c.e. operators 247 Pseudojump inversion 249 Applications of pseudojump inversion 251 Inversion of a c.e. operator via a ML-random set 253 Separation of highness properties 256 Minimal pairs and highness properties * 258 Randomness and betting strategies 259 7.1 Martingales 260 Formalizing the concept of a betting strategy 260 Supermartingales 261 Some basics on supennartingales 262 Sets on which a supermartingale fails 263 Characterizing null classes by martingales 264 7.2 C.e. supennartingales and ML-randomness 264 Computably enumerable supermartingales 265 Characterizing ML-randonmess via c.e. supermartingales 265 Universal c.e. supermartingales 266 The degree of nonrandoniness in .ML-random sets + 266 7.3 Computable supennartingales 26S Schnorr randomness and martingales 26S Preliminaries on computable martingales 270 7.4 How to build a coniputably random set 271 Three preliminary Theorems: outline 2 1 2 Partial computable martingales 273 A template for building a comprit ably random set 271 Computably random sets and initial segment complexity 275 The case of a partial computably random set 277 7.5 Each high degree contains a computably random set 279 Martingales that dominate -~9 Each high c.e. degree contains a coniputablv random left-c.e. set ■><> A computably random set that is not partial computably ι >Sl random - A >rrictly computably random set in each high degree 2M5 A strict lv Schnorr random set in each high degree 2s5 Contents 7.6 Varying the concept of a betting strategy 288 Basics of selection rules 288 Stochasticity 288 Stochasticity and initial segment complexity 289 Nonmonotonic betting strategies 294 Muchnik s splitting technique 295 Kolmogorov-Loveland randomness 297 Classes of computational complexity 301 8.1 The class Low(H) 303 The Low(O) basis theorem 303 Being weakly low for К 305 2-randomness and strong incompressibilitVK- 308 Each computably dominated set in Low(ň) is computable 309 A related result on computably dominated sets in GLi 311 8.2 Traceability 312 C.e. traceable sets and array computable sets 313 Computably traceable sets 316 Lowness for computable measure machines 318 Facile sets as an analog of the ^ -trivial sets * 319 8.3 Lowness for randomness notions 321 Lowness for С -null classes 322 The class Low(MLR. SR) 323 Classes that coincide with Low(SR) 326 Low(MLR. CR) coincides with being low for К 328 8.4 Jump traceability 336 Basics of jump traceability. and existence theorems 336 Jump traceability and descriptive string complexity 338 The weak reducibility associated with jump traceability 339 Jump traceability and superlowness are equivalent for c.e. sets 341 More on weak reducibilities 343 Strong jump traceability 343 Strong superlowness * 346 8.5 Subclasses of the iť-trivial sets 348 Some A -trivial c.e. set is not strongly jump traceable 348 Strongly jump traceable c.e. sets and benign cost functions 351 The diamond operator 356 8.6 Summary and discussion 361 A diagram of downward closed properties 361 Computational complexity versus randomness 363 Some updates 364 Higher computability and randomness 365 9.1 Preliminaries on higher computability theory 366 II] and other relations 366 Contents xv Well-orders and computable ordinals 367 Representing П} relations by well-orders 367 П} classes and the uniform measure 369 Reducibilities 370 A set theoretical view 371 9.2 Analogs of Martin-Löf randomness and K-triviality 372 П} Machines and prefix-free complexity 373 A version of Martin-Löf randomness based on П} sets 376 An analog of A -triviality 376 Lowness for nJ-ML-randomness 377 9.3 Δ {-randomness and Π } -randomness 378 Notions that coincide with Δ\| -randomness 37Í) More on n}-randonmess 380 9.4 Lowness properties in higher computability theory 381 Hyp-dominated sets 381 Traceability 382 Solutions to the exercises 385 Solutions to Chapter 1 385 Solutions to Chapter 2 389 Solutions to Chapter 3 391 Solutions to Chapter 4 393 Solutions to Chapter 5 395 Solutions to Chapter 6 399 Solutions to Chapter 7 400 Solutions to Chapter 8 402 Solutions to Chapter 9 408 References H Notation Index -Ч^ Index 12;5
adam_txt	Contents The complexity of sets 1 1.1 The basic concepts 3 Partial computable functions 3 Computably enumerable sets 6 Indices and approximations 7 1.2 Relative computational complexity of sets 8 Many-one reducibility 8 Turing reducibility 9 Relativization and the jump operator 10 Strings over {0.1} 12 Approximating the functionals Ф,., and the use principle 13 Weak truth-table reducibility and truth-table reducibility 14 Degree structures 16 1.3 Sets of natural numbers 16 1.4 Descriptive complexity of sets 18 Δ2 sets and the Shoenfield Limit Lemma 18 Sets and functions that are »-ce. or cj-c.e. 19 Degree structures on particular classes 20 The arithmetical hierarchy 21 1.5 Absolute computational complexity of sets 24 Sets that are low,, 26 Computably dominated sets 27 Sets that are high,, 28 1.6 Post's problem 29 Turing incomparable Δί,' -sets 30 Simple sets 31 A e.e. set that is neither computable nor Turing complete 32 Is there a natural solution to Post's problem? 34 Turing incomparable ce. sets 35 1.7 Properties of ce. sets 37 Each incomputable ce. wtt-degrec contains a simple set 38 Hypersimple sets 39 Promptly simple sets 40 Minimal pairs and promptly simple sets 41 Creative sets * 43 1.8 Cantor space 45 Open sets 47 Binary trees and closed sets 47 Representing open sets 48 Contents Compactness and clopen sets 48 The correspondence between subsets of N and real numbers 49 Effectivity notions for real numbers 50 Effectivity notions for classes of sets 52 Examples of Π? classes 55 Isolated points and perfect sets 56 The Low Basis Theorem 56 The basis theorem for computably dominated sets 59 Weakly l-generic sets 61 l-generic sets 63 The arithmetical hierarchy of classes 64 Comparing Cantor space with Baire space 67 1.9 Measure and probability 68 Outer measures 68 Measures 69 Uniform measure and null classes 70 Uniform measure of arithmetical classes 71 Probability theory 73 The descriptive complexity of strings 74 Comparing the growth rate of functions 75 2.1 The plain descriptive complexity С 75 Machines and descriptions 75 The counting condition, and incompressible strings 77 Invariance, continuity, and growth of С 79 Algorithmic properties of С 81 2.2 The prefix-free complexity Л 82 Drawbacks of С 83 Prefix-free machines 83 The Machine Existence Theorem and a characterization of A" 86 The Coding Theorem !)1 2.3 Conditional descriptive complexity 92 Basics 92 An expression for K(.r.y) * 93 2.4 Relating С and A' 94 Basic interactions 94 Solovay's equations * 95 2.5 Incompressibility and randomness for strings 97 Comparing incompressibility notions 98 Randomness properties of strings 99 Martin-Löf randomness and its variants 102 3.1 A mathematical definition of randomness for sets 102 Martin-Löf tests and their variants 104 Schnorr's Іііеопчп and universal Martin-Löf tests 105 Contents xi The initial segment approach 105 3.2 Martin-Löf randomness 106 The test concept 106 A universal Martin-Löf test 107 Characterization of MLR via the initial segment complexity 107 Examples of Martin-Löf random sets 108 Facts about ML-random sets 109 Left-c.e. ML-random reals and Solovay reducibility 113 Randomness on reals, and randomness for bases other than 2 115 A nonempty Пј subclass of MLR has ML-random measure * 116 3.3 Martin-Löf randomness and reduction procedures 117 Each set is weak truth-table reducible to a ML-random set 117 Autoreducibility and indifferent sets * 119 3.4 Martin-Löf randomness relative to an oracle 120 Relativizing С and A" 121 Basics of relative ML-randonmess 122 Symmetry of relative Martin-Löf randomness 122 Computational complexity, and relative randomness 124 The halting probability Ω relative to an oracle * 125 3.5 Notions weaker than ML-randomness 127 Weak randomness 128 Schnorr randomness 129 Computable measure machines 131 3.6 Notions stronger than ML-randomness 133 Weak 2-randomness 134 2-randonmess and initial segment complexity 136 2-randomness and being low for Ω 140 Demut h randomness * 141 4 Diagonally noncomputable functions 144 4.1 D.n.c. functions and sets of d. u.c. degree 145 Basics on d.n.c. functions and fixed point freeness 145 The initial segment complexity of sets of d.n.c. degree 147 A completeness criterion for ce. sets 148 4.2 Injury-free constructions of ce. sets 150 Each Λ" set of d.n.c. degree bounds a promptly simple set 151 Variants of Kučera's Theorem 152 An injury-free proof of the Friedberg Muchnik Theorem * 154 4.3 Strengthening the notion of a d.n.c. function 155 Sets of PA degree 1Г)(І Martin-Löf random sets of PA degree 157 Turing degrees of Martin-Löf random sets * 15!) Relating //-randomness and higher fixed point frceness ІШ xii Contents 5 Lowness properties and A'-triviality 163 5.1 Equivalent lowness properties 165 Being low for A' 105 Lowness for ML-randomness 167 When many oracles compute a set 168 Bases for ML-randomness 170 Lowness for weak 2-randomness 174 5.2 A'-trivial sets 176 Basics on A'-trivial sets 176 A'-trivial sets are Aj 177 The number of sets that are A'-trivial for a constant b * 179 Closure properties of K, 181 C-trivial sets 182 Replacing the constant by a slowly growing function * 183 5.3 The cost function method 184 The basics of cost functions 186 A cost function criterion for A'-triviality 188 Cost functions and injury-free solutions to Post's problem 189 Construction of a promptly simple Turing lower bound 190 A'-trivial sets and Σ ι -induction * 192 Avoiding to be Turing reducible to a given low c.e. set 193 Necessity of the cost function method for c.e. A'-trivial sets 195 Listing the (j-'-cc.) A'-trivial sets with constants 196 Adaptive cost functions 198 5.4 Each A-trivial set is low for A' 200 Introduction to the proof 201 The formal proof of Theorem 5.4.1 210 5.5 Properties of the class of A'-trivial sets 215 A Main Lemma derived from the golden run method 215 The standard cost function characterizes the A'-trivial sets 217 The number of changes * 219 Ω-"1 for A'-trivial .4 221 Each A'-trivial set is low for weak 2-randomnes.s 223 5.6 The weak reduci bili ty associated with Low(MLR) 224 Preordcrings coinciding with ¿.R-redueibility 226 A stronger result under the extra hypothesis that A <j- B' 228 The size of lower and upper cones for <ля * 230 Operators obtained by relativizing classes 231 Studying <lr by applying the operator К 232 Comparing the operators S ¡.r and К * 233 Uniformly almost everywhere dominating sets 234 0' <lh C' if an«! only if С is uniformly a.e. dominating 235 Contents xiii Some advanced computability theory 238 6.1 Enumerating Turing functionals 239 Basics and a first example 239 C.e. oracles, markers, and a further example 240 6.2 Promptly simple degrees and low cuppability 242 C.e. sets of promptly simple degree 243 A c.e. degree is promptly simple iff it is low cuppable 244 6.3 C.e. operators and highness properties 247 The basics of c.e. operators 247 Pseudojump inversion 249 Applications of pseudojump inversion 251 Inversion of a c.e. operator via a ML-random set 253 Separation of highness properties 256 Minimal pairs and highness properties * 258 Randomness and betting strategies 259 7.1 Martingales 260 Formalizing the concept of a betting strategy 260 Supermartingales 261 Some basics on supennartingales 262 Sets on which a supermartingale fails 263 Characterizing null classes by martingales 264 7.2 C.e. supennartingales and ML-randomness 264 Computably enumerable supermartingales 265 Characterizing ML-randonmess via c.e. supermartingales 265 Universal c.e. supermartingales 266 The degree of nonrandoniness in .ML-random sets + 266 7.3 Computable supennartingales 26S Schnorr randomness and martingales 26S Preliminaries on computable martingales 270 7.4 How to build a coniputably random set 271 Three preliminary Theorems: outline '2 1 '2 Partial computable martingales 273 A template for building a comprit ably random set 271 Computably random sets and initial segment complexity 275 The case of a partial computably random set 277 7.5 Each high degree contains a computably random set 279 Martingales that dominate -~9 Each high c.e. degree contains a coniputablv random left-c.e. set " ■><> A computably random set that is not partial computably ι ' '>Sl random - ' A >rrictly computably random set in each high degree 2M5 A strict lv Schnorr random set in each high degree 2s5 Contents 7.6 Varying the concept of a betting strategy 288 Basics of selection rules 288 Stochasticity 288 Stochasticity and initial segment complexity 289 Nonmonotonic betting strategies 294 Muchnik's splitting technique 295 Kolmogorov-Loveland randomness 297 Classes of computational complexity 301 8.1 The class Low(H) 303 The Low(O) basis theorem 303 Being weakly low for К 305 2-randomness and strong incompressibilitVK- 308 Each computably dominated set in Low(ň) is computable 309 A related result on computably dominated sets in GLi 311 8.2 Traceability 312 C.e. traceable sets and array computable sets 313 Computably traceable sets 316 Lowness for computable measure machines 318 Facile sets as an analog of the ^"-trivial sets * 319 8.3 Lowness for randomness notions 321 Lowness for С -null classes 322 The class Low(MLR. SR) 323 Classes that coincide with Low(SR) 326 Low(MLR. CR) coincides with being low for К 328 8.4 Jump traceability 336 Basics of jump traceability. and existence theorems 336 Jump traceability and descriptive string complexity 338 The weak reducibility associated with jump traceability 339 Jump traceability and superlowness are equivalent for c.e. sets 341 More on weak reducibilities 343 Strong jump traceability 343 Strong superlowness * 346 8.5 Subclasses of the iť-trivial sets 348 Some A'-trivial c.e. set is not strongly jump traceable 348 Strongly jump traceable c.e. sets and benign cost functions 351 The diamond operator 356 8.6 Summary and discussion 361 A diagram of downward closed properties 361 Computational complexity versus randomness 363 Some updates 364 Higher computability and randomness 365 9.1 Preliminaries on higher computability theory 366 II] and other relations 366 Contents xv Well-orders and computable ordinals 367 Representing П} relations by well-orders 367 П} classes and the uniform measure 369 Reducibilities 370 A set theoretical view 371 9.2 Analogs of Martin-Löf randomness and K-triviality 372 П} Machines and prefix-free complexity 373 A version of Martin-Löf randomness based on П} sets 376 An analog of A'-triviality 376 Lowness for nJ-ML-randomness 377 9.3 Δ {-randomness and Π } -randomness 378 Notions that coincide with Δ\| -randomness 37Í) More on n}-randonmess 380 9.4 Lowness properties in higher computability theory 381 Hyp-dominated sets 381 Traceability 382 Solutions to the exercises 385 Solutions to Chapter 1 385 Solutions to Chapter 2 389 Solutions to Chapter 3 391 Solutions to Chapter 4 393 Solutions to Chapter 5 395 Solutions to Chapter 6 399 Solutions to Chapter 7 400 Solutions to Chapter 8 402 Solutions to Chapter 9 408 References H" Notation Index -Ч^ Index 12;5
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indexdate	2024-07-09T21:21:36Z
institution	BVB
language	English
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physical	XV, 433 S.
publishDate	2009
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publisher	Oxford Univ. Press
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series	Oxford logic guides
series2	Oxford logic guides
spelling	Nies, André Verfasser (DE-588)113308426 aut Computability and randomness André Nies 1. publ. Oxford Oxford Univ. Press 2009 XV, 433 S. txt rdacontent n rdamedia nc rdacarrier Oxford logic guides 51 Numbers, Random Computational complexity Zufallszahlen (DE-588)4124968-9 gnd rswk-swf Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Zufallszahlen (DE-588)4124968-9 s Berechnungskomplexität (DE-588)4134751-1 s DE-604 Oxford logic guides 51 (DE-604)BV000013997 51 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016742636&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Nies, André Computability and randomness Oxford logic guides Numbers, Random Computational complexity Zufallszahlen (DE-588)4124968-9 gnd Berechnungskomplexität (DE-588)4134751-1 gnd
subject_GND	(DE-588)4124968-9 (DE-588)4134751-1
title	Computability and randomness
title_auth	Computability and randomness
title_exact_search	Computability and randomness
title_exact_search_txtP	Computability and randomness
title_full	Computability and randomness André Nies
title_fullStr	Computability and randomness André Nies
title_full_unstemmed	Computability and randomness André Nies
title_short	Computability and randomness
title_sort	computability and randomness
topic	Numbers, Random Computational complexity Zufallszahlen (DE-588)4124968-9 gnd Berechnungskomplexität (DE-588)4134751-1 gnd
topic_facet	Numbers, Random Computational complexity Zufallszahlen Berechnungskomplexität
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016742636&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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work_keys_str_mv	AT niesandre computabilityandrandomness

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