computability and randomness jan 2009

[...]... Relativizing C and K Basics of relative ML -randomness Symmetry of relative Martin-L¨f randomness o Computational complexity, and relative randomness The halting probability Ω relative to an oracle Notions weaker than ML -randomness Weak randomness Schnorr randomness Computable measure machines Notions stronger than ML -randomness Weak 2 -randomness 2 -randomness and initial segment complexity 2 -randomness and being... Examples of Martin-L¨f random sets o Facts about ML-random sets Left-c.e ML-random reals and Solovay reducibility Randomness on reals, and randomness for bases other than 2 A nonempty Π0 subclass of MLR has ML-random measure 1 Martin-L¨f randomness and reduction procedures o Each set is weak truth-table reducible to a ML-random set Autoreducibility and indifferent sets Martin-L¨f randomness relative to... randomness and K-triviality o Π1 Machines and prefix-free complexity 1 A version of Martin-L¨f randomness based on Π1 sets o 1 An analog of K-triviality Lowness for Π1 -ML -randomness 1 ∆1 -randomness and Π1 -randomness 1 1 Notions that coincide with ∆1 -randomness 1 More on Π1 -randomness 1 Lowness properties in higher computability theory Hyp-dominated sets Traceability Solutions to the exercises Solutions... 361 361 363 364 9 Higher computability and randomness 9.1 Preliminaries on higher computability theory Π1 and other relations 1 365 366 366 Contents 9.2 9.3 9.4 Well-orders and computable ordinals Representing Π1 relations by well-orders 1 Π1 classes and the uniform measure 1 Reducibilities A set theoretical view Analogs of Martin-L¨f randomness and K-triviality o Π1 Machines and prefix-free complexity... Existence Theorem and a characterization of K The Coding Theorem 2.3 Conditional descriptive complexity Basics An expression for K(x, y) 2.4 Relating C and K Basic interactions Solovay’s equations 2.5 Incompressibility and randomness for strings Comparing incompressibility notions Randomness properties of strings 3 Martin-L¨f randomness and its variants o 3.1 A mathematical definition of randomness for... degree of nonrandomness in ML-random sets 7.3 Computable supermartingales Schnorr randomness and martingales Preliminaries on computable martingales 7.4 How to build a computably random set Three preliminary Theorems: outline Partial computable martingales A template for building a computably random set Computably random sets and initial segment complexity The case of a partial computably random set 7.5... 243 244 247 247 249 251 253 256 258 7 Randomness and betting strategies 7.1 Martingales Formalizing the concept of a betting strategy Supermartingales Some basics on supermartingales Sets on which a supermartingale fails Characterizing null classes by martingales 7.2 C.e supermartingales and ML -randomness Computably enumerable supermartingales Characterizing ML -randomness via c.e supermartingales Universal... betting strategy Basics of selection rules Stochasticity Stochasticity and initial segment complexity Nonmonotonic betting strategies Muchnik’s splitting technique Kolmogorov–Loveland randomness 288 288 288 289 294 295 297 8 Classes of computational complexity 8.1 The class Low(Ω) The Low(Ω) basis theorem Being weakly low for K 2 -randomness and strong incompressibilityK Each computably dominated set in Low(Ω)... function Sets of PA degree Martin-L¨f random sets of PA degree o Turing degrees of Martin-L¨f random sets o Relating n -randomness and higher fixed point freeness xi 105 106 106 107 107 108 109 113 115 116 117 117 119 120 121 122 122 124 125 127 128 129 131 133 134 136 140 141 144 145 145 147 148 150 151 152 154 155 156 157 159 160 xii 5 Contents Lowness properties and K-triviality 5.1 Equivalent lowness... 147 148 150 151 152 154 155 156 157 159 160 xii 5 Contents Lowness properties and K-triviality 5.1 Equivalent lowness properties Being low for K Lowness for ML -randomness When many oracles compute a set Bases for ML -randomness Lowness for weak 2 -randomness 5.2 K-trivial sets Basics on K-trivial sets K-trivial sets are ∆0 2 The number of sets that are K-trivial for a constant b Closure properties of K . 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 -randomness and initial. Roman Kossak and James Schmerl: The Structure of Models of Peano Arithmetic 51. Andr´e Nies: Computability and Randomness Computability and Randomness Andr´e Nies The University of Auckland 1 3 Great. Martin-L¨of randomness based on Π 1 1 sets 376 An analog of K-triviality 376 Lowness for Π 1 1 -ML -randomness 377 9.3 ∆ 1 1 -randomness and Π 1 1 -randomness 378 Notions that coincide with ∆ 1 1 -randomness