MATHEMATICAL AND PHILOSOPHICAL PERSPECTIVES ON ALGORITHMIC RANDOMNESS A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Christopher P. Porter Peter Cholak, Co-Director Michael Detlefsen, Co-Director Curtis Franks, Co-Director Graduate Program in Mathematics and Philosophy Notre Dame, Indiana April 2012 c Copyright by Christopher P. Porter 2012 All Rights Reserved MATHEMATICAL AND PHILOSOPHICAL PERSPECTIVES ON ALGORITHMIC RANDOMNESS Abstract by Christopher P. Porter The mathematical portion of this dissertation is a study of the various interactions between definitions of algorithmic randomness, Turing functionals, and non-uniform probability measures on Cantor space. Chapters 1 and 2 introduce the main results and the relevant background for the remaining chapters. In Chapter 3, we study the connection between Turing functionals and a number of different definitions of randomness, culminating in a number of characterizations of these definitions of ran- domness in terms of a priori complexity, a notion of initial segment complexity given in terms of Turing functionals. In Chapter 4, we investigate possible generalizations of Demuth's Theorem, an important theorem in algorithmic randomness concerning the behavior of random sequences under truth-table reducibility. One technique de- veloped in this chapter, that of inducing a probability measure by means of a special type of truth-table functional that we call a tally functional, proves to be very use- ful. We use this technique to study randomness with respect to trivial computable measures in both Chapters 5 and 6. In the philosophical portion of this dissertation, we consider the problem of pro- Christopher P. Porter ducing a correct definition of randomness, as introduced in Chapter 7: Some have claim that one definition of randomness in particular, Martin-L¨ofrandomness, cap- tures the so-called intuitive conception of randomness, a claim known as the Martin- L¨of-ChaitinThesis, but some have offered alternative definitions as capturing our intuitions of randomness. Prior to evaluating the Martin-L¨of-ChaitinThesis and re- lated randomness-theoretic theses, Chapters 8 and 9 discuss two roles of definitions of randomness, both of which motivated much early work in the development of algo- rithmic randomness: the resolutory role of randomness, which is successfully filled by a definition of randomness that allows for the solution of problems in a specific theory of probability, and the exemplary role of randomness, which is successfully filled by a definition of randomness that counts as random those sequences that exemplify the properties typically held by sequences chosen at random. In Chapter 10, we lay out the status of the Martin-L¨of-Chaitin Thesis, discussing the evidence that has been offered in support of it, as well as the arguments that have been raised against it. In Chapter 11, we argue that the advocate of a claim like the Martin-L¨of-ChaitinThesis faces what we call the Justificatory Challenge: she must present a precise account of the so-called intuitive conception of randomness, so as to justify the claim that her preferred definition of randomness is the correct one and block the claim of cor- rectness made on behalf of alternative definitions of randomness. Lastly, in Chapter 12, we present two further roles for definitions of randomness to play, which we call the calibrative role of randomness and the limitative role of randomness, which can be successfully filled by multiple definitions of randomness. Definitions filling the calibrative role allow us to calibrate the level of randomness necessary and sufficient Christopher P. Porter for certain \almost-everywhere" results in classical mathematics to hold, while def- initions filling the limitative role illuminate a phenomenon known as the indefinite contractibility of the notion of randomness. Moreover, we argue that in light of the fact that many definitions can successfully fill these two roles, we should accept what we call the No-Thesis Thesis: there is no definition of randomness that (i) yields a well-defined, definite collection of random sequences and (ii) captures everything that mathematicians have taken to be significant about the concept of randomness. To Laura ii CONTENTS FIGURES . viii PREFACE . ix ACKNOWLEDGMENTS . .x CHAPTER 1: MATHEMATICAL PERSPECTIVES ON ALGORITHMIC RANDOMNESS . .1 CHAPTER 2: MATHEMATICAL BACKGROUND . .5 2.1 Introduction . .5 2.2 Basic Notation . .6 2.3 Computability Essentials . .9 2.3.1 Computability on ! ........................9 2.3.2 Computability on 2<! ...................... 10 2.3.3 Computability on 2! ....................... 11 2.3.4 The Turing Degrees . 13 2.3.5 Strong Reducibilities . 15 2.4 Topology and Measure on Cantor Space . 16 2.4.1 Topological Considerations . 16 2.4.2 Computable Measures . 18 2.4.3 Turing Functionals and Induced Measures . 21 2.5 Notions of Algorithmic Randomness . 24 2.5.1 Martin-L¨ofRandomness . 25 2.5.2 Weaker Definitions of Algorithmic Randomness . 47 2.5.3 Stronger Definitions of Randomness . 53 2.5.4 An Open Case: Kolmogorov-Loveland Randomness . 57 2.6 Several Useful Theorems . 60 iii CHAPTER 3: THE FUNCTIONAL EXISTENCE THEOREM . 62 3.1 Introduction . 62 3.1.1 Motivation . 62 3.1.2 The Machine Existence Theorem and Discrete Semimeasures . 65 3.1.3 An Analogue for Continuous Semimeasures? . 66 3.2 The Construction . 67 3.2.1 A General Overview . 67 3.2.2 The Formal Details . 68 3.3 Applications of the Functional Existence Theorem . 75 3.3.1 Measure-Boundedness . 75 3.3.2 Characterizing Notions of Randomness via Truth-Table Func- tionals . 84 3.3.3 A Priori Complexity Characterizations of Notions of Randomness 87 CHAPTER 4: DEMUTH'S THEOREM: VARIANTS AND APPLICATIONS 94 4.1 Introduction . 94 4.2 Proving Demuth's Theorem . 95 4.3 Demuth's Theorem for Other Notions of Randomness . 103 4.4 The failure of Demuth's theorem for wtt-reducibility . 107 4.5 Some Applications . 117 4.5.1 Random Turing Degrees . 118 4.5.2 Random Computably Enumerable Sets . 123 CHAPTER 5: TALLY FUNCTIONALS, TRIVAL MEASURES, AND DIMINU- TIVE MEASURES . 127 5.1 Introduction . 127 5.2 Tally Functionals . 127 5.3 Trivial and Diminutive Measures . 130 5.3.1 On Trivial Measures . 130 5.3.2 On Diminutive Measures . 135 5.3.3 Some Questions . 138 5.4 Separating Classes of Non-Uniform Randomness . 139 CHAPTER 6: TRIVIAL MEASURES AND FINITE DISTRIBUTIVE LAT- TICES . 145 6.1 Trivial Measures and Finite Distributive Lattices . 145 6.2 Open Questions . 162 iv CHAPTER 7: PHILOSOPHICAL PERSPECTIVES ON ALGORITHMIC RANDOMNESS . 164 7.1 Motivating the Problem . 164 7.2 A Conceptual Analysis of Randomness? . 169 7.3 Outline of the Chapters . 171 CHAPTER 8: THE RESOLUTORY ROLE OF RANDOMNESS . 175 8.1 Introduction . 175 8.2 Von Mises' Account of Probability . 177 8.2.1 Motivating von Mises' Definition . 177 8.2.2 Von Mises' Axioms of Collectives . 180 8.3 Objections and Replies . 187 8.3.1 The Admissibility Objection . 188 8.3.2 Von Mises' Response to the Admissibility Objection . 190 8.3.3 The Undefinability Objection . 191 8.3.4 Von Mises' Response to the Undefinability Objection . 194 8.3.5 Copeland's Response to the Two Objections . 195 8.4 Wald's Theorems and von Mises' Modified Account . 197 8.4.1 Wald's Two Problems and their Solutions . 198 8.4.2 Von Mises' Modified Account . 202 8.5 Von Mises' Approach to Solving Problems in the Probability Calculus 204 8.5.1 Four Fundamental Operations on Collectives . 204 8.5.2 Problems and Solutions . 207 8.6 An Ideal of Completeness . 209 8.6.1 Formulating the Resolutory Ideal . 210 8.6.2 Attaining the Resolutory Ideal . 213 8.7 Church on the Resolutory Ideal . 216 8.7.1 Church's Restricted Definition of Randomness . 218 8.7.2 Church's Restricted Version of the Resolutory Ideal . 220 CHAPTER 9: THE EXEMPLARY ROLE OF RANDOMNESS . 226 9.1 Introduction . 226 9.2 Ville's Putative Counterexample to von Mises' Definition . 230 9.2.1 Ville's Theorem . 230 9.2.2 Consequences of Ville's Theorem? . 232 9.3 Ville's Alternative Formalization of a Betting Strategy . 237 9.3.1 The Definition of Martingale . 239 v 9.3.2 Ville's Correspondence between Martingales and Null Sets . 243 9.4 The Exemplary ideal of Completeness . 245 9.4.1 In Search of an Improved Axiom of Irregularity . 245 9.4.2 The Exemplary Ideal as Unattainable . 249 9.5 Martin-L¨of's Definition of Randomness . 255 9.5.1 Kolmogorov's Definition of Random Finite Strings . 256 9.5.2 Martin-L¨ofon Kolmogorov's Definition . 259 9.5.3 The Definition of Martin-L¨ofRandomness . 264 9.6 Schnorr's Alternative Definition . 268 9.6.1 The Rationale for Schnorr Randomness . 268 9.6.2 An Alternative Formulation of the Exemplary Ideal? . 270 9.6.3 Martin-L¨of's Response to Schnorr's Objection? . 271 9.7 Summing Up . 272 CHAPTER 10: THE STATUS OF THE MARTIN-LOF-CHAITIN¨ THESIS . 274 10.1 Introduction . 274 10.2 Some Clarificatory Remarks . 276 10.2.1 The So-Called Intuitive Conception of Randomness? . 276 10.2.2 \Capturing" the Prevailing Intuitive Conception of Randomness?277 10.2.3 Establishing the Defectiveness of a Definition of Randomness . 280 10.3 On the Evidence for the MLCT ..................... 286 10.3.1 Three Convergent Definitions . 287 10.3.2 An Initial Worry . 289 10.3.3 Kreisel's Concern . 291 10.3.4 Supplementing the Convergence Results . 295 10.4 The Challenge of Schnorr Randomness . 299 10.4.1 Is Martin-L¨ofRandomness Type I Defective? . 299 10.4.2 Putative Counterexamples to ST ...............