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Enumeration Reducibility and Polynomial Time Bounds Enumeration Reducibility and Polynomial Time Bounds Charles Milton Harris Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Pure Mathematics January 2006 The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. ii Acknowledgements I would like to thank Barry Cooper for his help, and for always managing to make time to talk and give advice, during my time as a Phd student under his supervision. I am extremely grateful and indebted to my father Roland and my step- mother Paddy for their unstinting support during my years as a student. I would also like to thank my brothers Paul and Simon, and my sisters Tessa and Jane, who have often looked after me during my time at university. Thankyou to Tina and Chris for the bed, the midnight feasts, and their friendship. Finally I would like to mention Aldo, Frantz, and Olivier (formerly) of the “Caf´edes Ar`enes” in Paris who put up with me and fed me almost every day for the best part of two years. iii iv ACKNOWLEDGEMENTS Abstract This thesis discusses enumeration reducibility and three of its variants. We begin with an appraisal of the basic properties of enumeration reducibility ( ≤e ). Underlying principles of the ≤e are discussed and six different definitions of the latter are introduced. Insights due to the different definitions are presented and particular attention is paid to Scott’s formulation of ≤e in terms of the λ-calculus. We introduce symmetric enumeration reducibility (≤se ) and we give a clas- sification of ≤se in terms of other standard reducibilities. We show that the natural embedding of the Turing degrees into the enumeration degrees (De) translates to an embedding ( ιse ) into Dse that preserves least element, suprema and infima. We define a weak and a strong jump and we observe that ιse preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers and diamond embeddings in Dse. We show that certain classes of symmetric enumeration de- grees are first order definable, in particular the classes of semirecursive, Σn ∪Πn, ∆n (for any n ∈ ω), and embedded Turing degrees. This last result allows us to conclude that the theory of Dse has the same 1-degree as the theory of Second Order Arithmetic. We present two polynomial time variants of ≤e , non deterministic poly- NP nomial time conjunctive reducibility (≤c ) and polynomial time enumeration reducibility (≤pe ). The degree structure induced by ≤pe over the computable sets (hRECpe, ≤i) is studied. It is shown that the latter is a non distributive v vi ABSTRACT upper semi-lattice that is not a lattice. We demonstrate the relativised ver- sion of the property that every non zero polynomial time enumeration (pe-) degree is part of a minimal pair. We prove that the pe-degrees are dense and moreover, that any countable distributive lattice can be embedded in any in- terval of the pe-degrees via embeddings that either preserve least or greatest element. We also explain why there exist similar embeddings of finite nowhere complemented distributive lattices that preserve least and greatest elements. SN Strong non deterministic polynomial time Turing reducibility ( ≤T ) is intro- SN duced. A canonical embedding of the degree structure induced by ≤T over the computable sets into hRECpe, ≤i is presented. We observe that this em- bedding preserves suprema and least element and that it is also essentially an NP embedding into the equivalent degree structure induced by ≤c . We prove the existence of relativised quasiminimal degrees in the case of the latter embed- ding. In conclusion we show that there exists sets A, B ∈ E such that A≤pe B P whereas AT B. Contents Acknowledgements iii Abstract v 1 Introduction 1 1.1 General Overview and Background . 1 1.2 Preliminaries. 5 1.2.1 Basic Notation. 5 1.2.2 Finite Sets and Functions. 6 1.2.3 Strings and Finite Initial Segments. 7 1.2.4 Partial Orders and Lattices. 7 2 Defining Enumeration Reducibility 9 2.1 Overview . 9 2.2 Turing Machines and Reductions . 10 2.2.1 Main Matter . 10 2.2.2 Other Reducibilities and Notation . 19 2.3 Constructive Enumeration Reducibility . 21 2.3.1 Main matter . 21 2.3.2 Other Facts and Notation . 27 2.4 Enumeration Reducibility and n-T-reducibility . 27 2.5 Non Constructive Enumeration Reducibility . 33 2.6 Enumeration reducibility and Lambda . 40 vii viii CONTENTS 2.6.1 The language Lambda and the C.E. Sets. 41 2.6.2 Enumeration Degrees and Lambda . 58 3 Symmetric Enumeration Reducibility 65 3.1 Overview . 65 3.2 Introduction to Symmetric Enumeration Reducibility . 66 3.3 Embedding the Turing Degrees . 70 3.4 Jump Operators . 73 3.5 The Symmetric Enumeration Hierarchy . 77 3.6 Basic Properties of Dse ........................ 82 3.7 CEA and co-CEA substructures of Dse . 89 3.8 Diamond embeddings and minimal covers . 92 3.9 Automorphisms and definability . 105 4 Polynomial Time Enumeration Reducibility 115 4.1 Overview . 115 4.2 Preliminaries . 116 4.2.1 Background Notation . 116 4.2.2 Coding Finite Sets and Pairs of Strings . 117 4.2.3 Turing Machines and Time Bounds . 118 4.2.4 Deterministic Turing Machines and Reductions . 119 4.2.5 Non Deterministic Turing machines and Reductions . 120 4.2.6 Complexity Classes . 126 4.3 Polynomial Time Enumeration Reducibility . 126 4.4 Basic Properties of hRECpe, ≤i . 150 4.5 Join and Meet Lemmas . 165 4.6 Lattice Embeddings in hRECpe, ≤i . 172 4.7 Embedding the sn-T-degrees . 181 P 4.8 Separating ≤T and ≤pe in E . 193 A Additional Proofs and Results 205 A.1 Proofs in Lambda . 205 CONTENTS ix A.2 Strong Definability in Lambda . 215 A.3 Complete sets, jumps and Lambda . 225 Bibliography 235 x CONTENTS Chapter 1 Introduction 1.1 General Overview and Background Enumeration reducibility (≤e ) is a relation between sets of numbers in which only positive information is processed. The first informal definition of ≤e appeared in an early draft of [Rog67]. According to this definition, a set A is enumeration reducible to a set B (A≤e B) if there is an effective procedure for getting an enumeration A from any enumeration of B. The formalised version1 of this definition appeared in [FR59]. Essentially it is this latter formulation that has been the basic tool in the ensuing research on enumeration reducibility and its degree structure. However two other equivalent definitions appeared in the literature in the 1970’s. The first of these was due to Selman [Sel71] who defined a relation G1 between sets such that AG1B iff, for any set X, if B is computably enumerable (c.e.) in X, then so is A. Selman proved in the same paper that (over non trivial sets), this relation is equivalent to ≤e despite the fact that the former is defined in a non constructive manner whereas the latter is defined with respect to a given effective procedure. The second equivalent definition appeared in Scott [Sco75]. In this paper Scott defined the language Lambda—an extension of the lambda calculus—and an interpretation in Pω 1In other words Definition 2.3.2 below. 1 2 CHAPTER 1. INTRODUCTION of the language according to which a set is c.e. iff it is the interpretation of a closed Lambda term. Moreover, for any A, B ⊆ ω, and supposing B to be a constant with interpretation B, A≤e B iff there exists a closed Lambda term u such that A is the interpretation of the (parametrised2) Lambda term (u) B. Thus an enumeration reduction corresponds to an application by a closed term in Scott’s language Lambda (whereas abstraction corresponds to the formation of an enumeration operator). In Chapter 2 we present these three different approaches to enumeration reducibility with a view to giving a more complete picture of the basic properties of the latter. In particular, by a careful analysis of oracle computations we reformulate ≤e in terms of oracle Turing machines (Definition 2.3.4). We also underline and extend previous work by Sasso, Cooper and McEvoy [McE84] on the relationship between deterministic (≤T) and non deterministic Turing reducibility, enumeration reducibility, and the relation “computably enumerable in” (c.e. in). A standard result that follows from the work reviewed in Chapter 2 is the fact that the Turing degrees (DT) canonically embed into the enumeration de- grees (De). In other words the Turing degrees can be regarded as a (partial order) substructure of the enumeration degrees. This leads to the question of the status of the former within the latter. In particular, are the Turing degrees first order definable in the enumeration degrees? A recent paper by Kalimullin [Kal03] addresses this question. Namely, Kalimullin shows that the enumera- tion jump operation is first order definable and, as a consequence of Friedberg’s completeness criterion, that the the class of embedded Turing degrees in the 0 cone above 0e is also first order definable within the structure. Kalimullin’s methods, which originate in earlier work on splitting properties of the embed- ded Turing degrees [AKC03] in De, underpin the main results in the work on the degree structure presented in Chapter 3. This work originated in a problem raised by McEvoy [McE84] of how one might refine the arithmetical hierarchy in such a way as to make it compatible with enumeration reducibility.
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