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The Realizability Approach to Computable Analysis and Topology

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The Realizability Approach to Computable Analysis and Topology Andrej Bauer September 2000 CMU-CS-00-164 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee Dana S. Scott, chair Steven M. Awodey Lenore Blum Abbas Edalat Frank Pfenning Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy This research was sponsored by the National Science Foundation (NSF) under Grant No. CCR-9217549 and the NAFSA Association of International Educators through a generous fellowship. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. Keywords: computability, realizability, modest sets, constructive and computable analysis and topology To disertacijo posveˇcammojemu oˇcetu, ki je moj prvi in najvplivnejˇsiuˇciteljmatematike. I dedicate this dissertation to my father, who is my first and most influential teacher of mathematics. Abstract In this dissertation, I explore aspects of computable analysis and topology in the frame- work of relative realizability. The computational models are partial combinatory alge- bras with subalgebras of computable elements, out of which categories of modest sets are constructed. The internal logic of these categories is suitable for developing a theory of computable analysis and topology, because it is equipped with a computability pred- icate and it supports many constructions needed in topology and analysis. In addition, a number of previously studied approaches to computable topology and analysis are special cases of the general theory of modest sets. In the first part of the dissertation, I present categories of modest sets and axiomatize their internal logic, including the computability predicate. The logic is a predicative intuitionistic first-order logic with dependent types, subsets, quotients, inductive and coinductive types. The second part of the dissertation investigates examples of categories of modest sets. I focus on equilogical spaces, and their relationship with domain theory and Type Two Effectivity (TTE). I show that domains with totality embed in equilogical spaces, and that the embedding preserves both simple and dependent types. I relate equilogical spaces and TTE in three ways: there is an applicative retraction between them, they share a common cartesian closed subcategory that contains all countably based T0- spaces, and they are related by a logical transfer principle. These connections explain why domain theory and TTE agree so well. In the last part of the dissertation, I demonstrate how to develop computable analysis and topology in the logic of modest sets. The theorems and constructions performed in this logic apply to all categories of modest sets. Furthermore, by working in the internal logic, rather than directly with specific examples of modest sets, we argue abstractly and conceptually about computability in analysis and topology, avoiding the unpleasant details of the underlying computational models, such as G¨odelencodings and representations by sequences. Acknowledgements I thank Professor Dana Scott for being my doctoral advisor. He has treated me with a great deal of kindness, wisdom, and patience. I can only hope to be able to pass the favor on to a student of mine one day. I thank my doctoral committee members for their work and support. I thank Profes- sor Lenore Blum for stimulating discussions and excellent suggestions for future topics of research|I wish I had the time to include them in this dissertation. I thank Professor Abbas Edalat for supporting my work, and for his hospitality when I visited him at the Imperial College in London. I thank Professor Steve Awodey for his friendship and self- less and excellent teaching of category theory and categorical logic. I thank Professor Frank Pfenning for introducing me to constructive and intuitionistic logic, and con- vincing me with exquisite elegance and clarity of thought to embrace the intuitionistic logic. There are numerous research colleagues that I thank for their hospitality, friendship, and working with me: Peter Andrews, Jeremy Avigad, Ulrich Berger, Lars Birkedal, Jens Blanck, Vasco Brattka, Chad Brown, Stephen Brookes, Aurelio Carboni, Douglas Cenzer, Edmund Clarke, Yuri Ershov, Mart´ın Escard´o,Marcelo Fiore, Peter Freyd, Harvey Friedman, Robert Harper, Reinhold Heckmann, Jeff Helzner, Peter Hertling, Jesse Hughes, Martin Hyland, Achim Jung, Elham Kashefi, Mathias Kegelmann, Klaus Keimel, Michal Konecny, Marko Krznari´cJohn Langford, David Lester, Peter Lietz, F. William Lawvere, John Longley, Keye Martin, Mat´ıasMenni, Mike Mislove, Aleks Nanevski, Dag Normann, Jaap van Oosten, Marko Petkovˇsek,TomaˇzPisanski, Martin Raiˇc,Mike Reed, DuˇsanRepovˇs,John Reynolds, Edmund Robinson, Alexander Rohr, Giuseppe Rosolini, Steven Rudich, Matthias Schr¨oder,Alex Simpson, Dieter Spreen, Chris Stone, Thomas Streicher, Matthew Szudzik, AleˇsVavpetiˇc,Klaus Weihrauch, Kevin Watkins, Stephen Wolfram, Xudong Zhao, and anyone else I might have forgotten. My writing a dissertation was as much of an endeavor for me as it was for my friends who stood by me. I am most grateful to my good friend and companion Vandi, my friend and officemate Elly, my housemates, all my other friends in Pittsburgh, my soon-to-be-rich friends in Slovenia, and my family. Contents Introduction 11 1 Categories of Modest Sets 21 1.1 Partial Combinatory Algebras . 21 1.1.1 The First Kleene Algebra N ............................ 23 1.1.2 Reflexive Continuous Posets . 23 1.1.3 The Graph Model P ................................ 24 1.1.4 The Universal Domain U ............................. 27 1.1.5 The Partial Universal Domain V ......................... 28 1.1.6 The Second Kleene Algebra B ........................... 30 1.1.7 PCA over a First-order Structure . 32 1.2 Modest Sets . 35 1.2.1 Modest Sets as Partial Equivalence Relations . 36 1.2.2 Modest Sets as Representations . 37 1.3 Properties of Modest Sets . 37 1.3.1 Finite Limits and Colimits . 38 1.3.2 The Locally Cartesian Closed Structure . 40 1.3.3 The Regular Structure . 42 1.3.4 Projective Modest Sets . 43 1.3.5 Factorization of Morphisms . 45 1.3.6 Inductive and Coinductive Types . 45 1.3.7 The Computability Operator . 51 1.4 Applicative Morphisms . 51 1.4.1 Applicative Adjunction between N and P] .................... 55 1.4.2 Applicative Retraction from (P; P]) to (B; B]).................. 55 1.4.3 Applicative Inclusion from (P; P]) to (U; U])................... 56 1.4.4 Equivalence of (P; P]) and (V; V])......................... 57 1.4.5 Equivalence of Reflexive Continuous Lattices . 58 2 A Logic for Modest Sets 61 2.1 The Logic of Simple Types . 62 2.1.1 First-order Predicate Logic . 62 2.1.2 Maps and Function Spaces . 63 2.1.3 Products . 65 2.1.4 Disjoint Sums . 66 8 2.1.5 The Empty and the Unit Spaces . 67 2.1.6 Subspaces . 68 2.1.7 Quotient Spaces . 70 2.1.8 Factorization of Maps . 72 2.2 The Logic of Complex Types . 74 2.2.1 Dependent Sums and Products . 74 2.2.2 Inductive Spaces . 75 2.2.3 Coinductive Spaces . 77 2.3 Computability, Decidability, and Choice . 80 2.3.1 Computability . 80 2.3.2 Decidable Spaces . 84 2.3.3 Choice Principles . 85 3 The Realizability Interpretation of the Logic in Modest Sets 89 3.1 The Interpretation of Logic . 89 3.2 Simple Types . 93 3.2.1 Function Spaces . 93 3.2.2 Products . 93 3.2.3 Disjoint Sums . 94 3.2.4 The Empty and the Unit Spaces . 94 3.2.5 Subspaces . 94 3.2.6 Quotient Spaces . 95 3.3 Complex Types . 96 3.3.1 Dependent Sums and Products . 96 3.3.2 Inductive and Coinductive Spaces . 96 3.4 The Computability Predicate . 97 3.5 Choice Principles . 98 3.5.1 Markov's Principle . 98 3.5.2 Choice Principles . 98 3.6 The Realizability Operator . 98 4 Equilogical Spaces and Related Categories 105 4.1 Equilogical Spaces . 105 4.1.1 Equilogical Spaces . 106 4.1.2 Effective Equilogical Spaces . 111 4.1.3 Effectively Presented Domains as a Subcategory of Equeff . 117 4.1.4 Domains with Totality as a Subcategory of Equilogical Spaces . 118 4.1.5 Equilogical Spaces as a Subcategory of Domain Representations . 133 4.2 Equilogical Spaces and Type Two Effectivity . 136 4.2.1 Mod(B; B]) as a Subcategory of Effective Equilogical Spaces . 137 4.2.2 Sequential Spaces . 139 4.2.3 Admissible Representations . 141 4.2.4 A Common Subcategory of Equilogical Spaces and Mod(B) . 144 4.3 Sheaves on Partial Combinatory Algebras . 148 4.3.1 Sheaves over a PCA . 148 Contents 9 4.3.2 Functors Induced by Applicative Morphisms . 150 4.3.3 Applicative Retractions Induce Local Maps of Toposes . 151 4.3.4 A Forcing Semantics for Realizability . 151 4.3.5 A Transfer Principle for Modest Sets . 152 5 Computable Topology and Analysis 155 5.1 Countable Spaces . 156 5.2 The Generic Convergent Sequence . 158 5.3 Semidecidable Predicates . 161 5.3.1 Σ-partial Maps and Lifting . 172 5.4 Countably Based Spaces . 175 5.4.1 Countably Based Spaces in Mod(N) . 180 5.5 Real Numbers . 187 5.5.1 Integers and Rational Numbers . 187 5.5.2 The Construction of Cauchy Reals . 188 5.5.3 The Algebraic Structure of Reals . 192 5.5.4 Discontinuity of Real Maps . 201 5.6 Metric Spaces . ..
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