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The Turing Degrees: Global and Local Structure 1 The Turing Degrees: Global and Local Structure 1 Richard A. Shore c May 20, 2015 1THIS IS A ROUGH DRAFT. CORRECTIONS, COMMENTS AND SUGGGESTIONS ARE WELCOME. PLEASE SEND TO [email protected]. PLEASE IN- CLUDE THE DATE OF THE DRAFT. THANKS. ii Contents Introduction ix 0.1 Notation.................................... ix 0.2 Pairing and ordered sequences . ix 1 An Overview 1 1.1 History, intuition, undecidability, formalization, ... 1 1.2 Formal de…nitions, models of computation . 1 1.3 Relative computability: Turing machines with oracles . 1 1.4 Degrees, types of questions . 2 2 The Basics 3 2.1 Coding Turing machines with oracles . 3 3 The Turing Degrees 7 4 R.E. Sets and the Turing Jump 11 4.1 TheJumpOperator ............................. 11 4.2 Trees and König’sLemma . 12 4.3 Recursively Enumerable Sets . 16 4.4 Arithmetic Hierarchy . 21 4.5 The Hierarchy Theorem . 24 4.5.1 Indexsets ............................... 25 4.6 JumpHierarchies ............................... 25 5 Embeddings into the Turing Degrees 27 5.1 Embedding Partial Orders in ....................... 27 D 5.2 Extensions of embeddings . 37 5.3 Therangeofthejump ............................ 43 5.3.1 The Friedberg Jump Inversion Theorem . 44 5.3.2 The Shoen…eld Jump Inversion theorem . 45 5.4 Trees and sets of size the continuum . 46 iii iv CONTENTS 6 Forcing in Arithmetic and Recursion Theory 49 6.1 Notions of Forcing and Genericity . 49 6.2 The Forcing Language and Deciding Classes of Sentences . 55 6.3 Embedding Lattices . 63 6.4 E¤ective Successor Structures . 72 7 The Theories of and ( 00) 77 7.1 Interpreting ArithmeticD D . 77 7.2 Slaman-Woodin Forcing and T h( ) ..................... 79 D 7.3 T h(D 00) .................................. 84 8 Domination Properties 91 8.1 Introduction.................................. 91 0 8.2 R.E. and 2 degrees ............................. 92 8.3 High and GL2 degrees ............................ 96 8.4 De…nability and Biinterpretability in ( 00) ............... 105 8.5 Array Nonrecursive Degrees . .D . 112 9 Minimal Degrees and Their Jumps 117 9.1 Introduction.................................. 117 9.2 Perfect forcing and Spector minimal degrees . 117 9.3 Partial trees and Sacks minimal degrees . 124 9.4 Minimal degrees below degrees in H1 and GH1 .............. 126 9.5 Jumps of minimal degrees . 127 9.5.1 Narrow trees and GL1 minimal degrees . 127 9.5.2 Cooper’sjump inversion theorem . 128 9.6 The minimal degrees generate ....................... 131 D 10 Lattice Initial Segments of 137 10.1 Lattice Tables, trees and theD notion of forcing . 137 10.2 Initial segment conditions . 142 10.3 Constructing lattice tables . 147 10.4 Decidability of two quanti…er theory . 151 10.5 Undecidability of three quanti…er theory. 151 0 11 1 Classes 153 11.1Binarytrees .................................. 153 11.2 Finitely branching trees . 160 12 Pseudo-Jump Operators: De…ning 163 12.1 Completeness Theorems for REA operatorsA . 164 12.2 RE Operators and Minimal Covers . 171 12.3 The Join Theorem for !-r.e. Operators . 173 CONTENTS v 12.4 The De…nability of and the Failure of Homogeneity . 175 12.5 The Join Theorem forA REA operators . 176 13 Global Results 177 14 De…ning the Turing Jump 179 15 Appendices 181 15.1 Trees, Cantor and Baire space; topology; perfect sets . 181 15.2 Structures, Orders and Lattices . 181 15.3 Interpreting Structures and Theories . 182 16 Bibliography 183 vi CONTENTS preface Preface ....Thank previous students and especially Mia Minnes who took notes in TeX for the entire course in 2007 along with the other students that term who all took turns preparing lecture notes..... vii viii PREFACE intro Introduction The introduction Reference Style to Chapters Chapter n Sections §n.m?? notation 0.1 Notation N a; b; c; d; e; i; j; k; l; m; n; r; s; t; u; v; w; x; y; z N N f; g; h ! N 2 sets A; B; C; U; V; W; X; Y; Z f ambiguous! partial functions ; '; ::: Functionals ; ; ::: (continuous) 0.2 Pairing and ordered sequences Choice. Uniformity. 1 2 2 speci…c pairing polynomial 2 (x + 2xy + y + 3x + y) then n-tuples by recursion. then picture for listing the elements of a countable family of countable sets. Pairing functions: desiderata for x; y . x y 1 2 2 h i 2 3 ; 2 (x + 2xy + y + 3x + y) x; y; z = x; y; z etc. h i h h ii x1; : : : xn = n; x1; x2 ::: Uniformityh i overh lengthh h n. iii xi+1 p 1 for x1; : : : xn i h i pairing for functions f g; ifi Q Ai fi Ai :::i::: f j g strings ; ; ; ; ; ; String notation functional form if = x1; : : : xn then (i) = xi+1 (perhaps prefer h i x0; : : : xn 1 ); dom() = n = length of ; order by initial segments ; restriction h i j j for m , m and m = m. Apply to functions on all of N as well: f; j j j j f m. so strings as …nite sequences ix x INTRODUCTION Notation: set of all …nite sequences of elements of S denoted by S<! set of sequences of length n by Sn binary strings 0; 1 n 0; 1 <! Identify 0; 1 fwithg 2 fandg more generally 0; 1; : : : ; n 1 with n and so write 2n, 2<! 2N ... f g f g Also pairing for strings... Chapter 1 overview An Overview 1.1 History, intuition, undecidability, formalization, hist ... formal 1.2 Formal de…nitions, models of computation Turing machines (multitape with input, output and others) n-ary functions formally view as given by input an n-tuple x1; : : : ; xn coded on same input tape e.g. as B1x1 B1x2 :::B1xn B::: ??or in some nice way e.g. machine e; n ... or just realize can mimic such functions ...and use notation of n-ary inputs as shorthand.?h i Also allow multiple oracles f1; : : : fn ... Other notions: prim recursive + (search); register machines; equation calcu- lus?.. 1.3 Relative computability: Turing machines with relcomp oracles de…ne Turing machines, input (note how to deal with n-tuples as input coded on same on tape) above, output, computation also for oracles (functions) , Turing reducibility T de…ne formally do for explicitly for unary functions can do n-ary in same way or by coding n-tuples as numbers also oracles - could have tuple of oracles or view as code by function pairing ref speci…c coding procedures in . Church-Turing Thesis equality for partial functions = for if either are de…ned at x then both are de…ned and their values are equal. 1 2 CHAPTER 1. AN OVERVIEW use here example Proposition 1.3.1 Turing reducibility is re‡exive and transitive. Recursive. (continuous) Functionals; oracles as inputs degrees 1.4 Degrees, types of questions We have seen that Turing reducibility, T , is a re‡exive and transitive relation and so we can consider the equivalence classes for this relation, i.e. the classes of the form g g f & f g for any function f. These classes are called the Turing degrees, T T degset orf j simply the degrees. g As, by Exercise 1.4.1, every Turing degree contains a set (i.e. a characteristic function) and so we may, without any loss consider just the degrees of sets or the classes B B T A & A T B . We denote the collection of all degrees by . We typically denotef j the degree containing g f or A by f or a. The class is a partialD order D welldef under the induced relation de…ned by f g f T g. (See Exercise 1.4.2.) . , degset Exercise 1.4.1 For every function f NN there is a set A 2N of the same Turing 2 2 degree, i.e. f T A and A T f. welldef Exercise 1.4.2 The relation de…ned above is well de…ned, re‡exive and transitive on the degrees and so makes them into a partial order. N Notation 1.4.3 For functions f; g N , we write f T g to denote that f and g are 2 j Turing incomparable, i.e. f T g and g T f. Similarly, we write f g when f g and j g f. outline book algebraic, local... second order, global: de…nability, automorphisms, theory mention appendices Chapter 2 basics The Basics codeTM 2.1 Coding Turing machines with oracles To aid in our analysis of Turing machines with oracles and relative computability as relcomp described in 1.3, we introduce a master (universal) recursive function in two forms. First we have '(f; e; x; s) = y. Here the variables are f a function, e a number (index), x a number (input), s a number (steps) and the expression means that the Turing machine with index e and oracle j given input x and run for s many steps converges and outputs y. Secondly we have an approximation version '(; e; x; s) = y where the variables are a string (so an initial segment of a function), e a number (index), x a number (input), s a number (steps, use); and the expression means that the Turing machine with index e and oracle restricted to given input x and run for s many steps converges and outputs y. We say that '(f; e; x; s) or '(; e; x; s) converges, if there is such a y. The standard notation for this is '(f; e; x; s) or '(; e; x; s) . ??(In later chapters it will at times be convenient to# allow to be more# generally a …nite partial function from N to N rather than restricting its domain to be an initial segment of N.
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