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Some Results on Enumeration Reductibility

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Some Results on Enumeration Reductibility SOME RESULTS ENUMERATION REDUCIBILITY by LANCE GUTTERIDGE B.Sc., University of British Columbia, 1967 M.Sc., University of British Columbia, 1969 A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREZWNTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics LANCE GUTTERIDGE 1971 SIMON FR9SER 'VNIVERS ITY AUGUST 1971 APPROVAL Name : Lance Gutteridge Degree : Doctor of Philosophy Title of Dissertation: Some results on enumeration reducibility 1 Examining Committee: 7 , --L-- - I ' r , Chairman: N. R. Reilly ,' A. H. Lachlan Senior Supervisor A. R. Freedman - .- - G. E. Sacks External Examiner, Professor, Massachusetts Institute of Technology Cambridge, Mass. Date ~pproved: September 24, 1971 ABSTRACT Enumeration reducibility was defined by Friedberg and Rogers in 1959. Medvedev showed that there are partial degrees which are not total. Rogers in his book Theory of Recursive Functions and Effective Computability gives all the basic results and definitions concerning enumeration reducibility and the partial degrees. He mentions in this book that the existence of a minimal partial degree is an open problem. In this thesis it is shown that there are no minimal partial degrees. This leads naturally to the conjecture that the partial degrees are dense. This thesis leaves this question unanswered, but it is shown that there are no degrees minimal above a total degree, and there are at most countably many degrees minimal above a non-total degree. J. W. Case has proved several results about the partial degrees. He conjectured that there is no set in a total degree whose complement is in a non-total degree. In this thesis that conjecture is disproved. Finally, Case's result that there is a minimal pair of partial degrees is strengthened to show that there is a minimal pair of partial degrees which are total and form a minimal pair of r.e. degrees. The content of this thesis has been vastly influenced by Professor A. H. Lachlan. The author appreciates the mathematical assistance and personal encouragement which he gave at all hours and through all crises. The finished form of this thesis owes its neatness and presenta- bility to the typing skill of Mrs. A. Gerencser. Without her co- operation the author could never have met his deadlines. The author is also indebted to the National Research Council of Canada without whose financial aid in the form of a bursary the work on this thesis could not have been continued. TABLE OF CONTENTS PAGE CHAPTER I INTRODUCTION AND TERMINOLOGY 1 51.1 INTrnDUCTION f 51.2 TERMINOLOGY 1 1.3 ENUMERATION REDUCIBILITY 4 51.4 INFINITE GAMES 7 CHAPTER 11 A SET IN A TOTAL DEGREE WHOSE COMPLEMENT IS 10 IN A NON-TOTAL DEGREE 2.1 INTRODUCTION 52.2 DESCRIPTION OF THE GAME 52.3 AN EFFECTIVE WINNING STPATEGY 52.4 A PROOF THAT THE STRATEGY IS A WINNING STRATEGY 52.5 CONCLUSIONS CHAPTER 111 MINIMAL PAIRS OF PARTIAL DEGREES INTRODUCTION 53.2 DESCRIPTION OF THE GAME 53.3 BASIC IDEA EEHIND THE STRATEGY 53.4 A WINNING STRATEGY FOR THE GAME 53.5 PROOF THAT THE STRATEGY IS A WINNING ONE 53.6 CONCLUSIONS CHAPTER IV THERE ARE NO MINIMAL PARTIAL DEGREES 4. INTRODUCTION 54.3 STRATEGIES FOR TEGAMES Cv PAGE CHAPTER V DEGREES MINIMAL ABOVE A NON-TOTAL DEGREE 5 3 9 5.1 INTrnDUCTION 5 5.2 EXTENDED ENUMERATION OPERATORS 55.3 AN ENUMERATION STRATEGY 55.4 CONCLUSIONS BIBLIOGRAPHY CHAPTER I INTRODUCTION AND TERMINOLOGY INTRODUCTION In this thesis we consider the partial (enumeration) degrees as defined by Friedberg and Rogers [2]. In Chapter I1 we disprove a conjecture by Case [l] that there is no set A in a total degree whose complement is in a non-total degree. Case [l] has shown that there is a minimal pair of partial degrees. In Chapter I11 we strengthen this result to show that there is a pair of co-r.e. sets whose partial degrees form a minimal pair of partial degrees; and hence their Turing degrees form a minimal pair of r.e. degrees. In Chapter IV we show that there are no minimal partial degrees and demonstrate that this result relativizes to show that there are no degrees minimal above a total degree. This still leaves open the question as to whether the partial degrees are dense. In Chapter V we move closer to a solution of the density problem by showing that given any partial degree -a there can be at most a countable number of partial degrees minimal above .a.- In this section we intend to outline all the basic notation and definitions used in this thesis. For a detailed explanation and study of recursion theory the reader is referred to Rogers [5]. We will denote subsets of N by upper case letters, with D, E, F being reserved for finite subsets. Members of N will be represented by lower case letters except for f, g, h which are reserved for total functions. Partial functions (that is, functions whose domain is a subset of N) will be denoted by $ and $. A will denote the complement of A, will be used for the C~ characteristic function of A and A A B will denote the symmetric difference of A and B. For notational convenience we will sometimes use ~[n]to denote CX(n). We will call a finite initial segment of a characteristic function an initial function. A join B is the set We define the binary function T by, r(x1 y) = f(x2 + 2xy + y2 + 3x+ y), and T is a recursive, one to one mapping of N x N onto N (see k Rogers [ 5, p. 641 1 . For each k we define the k-ary function T as follows : k Me abbreviate T (xl, ..., 5) by <XI, . -.f \>. For each k we k define the projection functions . , 1 5 i I kt by 1 k We will usually drop the superscript on n when its value is clear i from the context. A set B is single-valued if <n, j>€B and <n, k>€B+ j=k. A set B is total if for all n there is a m such that <n, m> € B. If D is finite. i.e. D = {xl. ..., x < x < ... < XI then -- +. 1-- 2 Al "k the canonical index of D is 2 + ... + 2 , if D= p) we let its canonical index be 0. We denote by Di the finite set whose canonical index is i. We will often make no distinction between a finite set and its canonical index, as in the use of <E, x> for <itx>, where Di = E. We will assume that the reader is familiar with the recursive partial functionals, (see [6. p.150]) . we let <o~>:=~ be an effective enumeration of the recursive partial th functionals. For each e we denote the e -partial recursive function An[@ (C n)] by Oe. The function f is recursive if for some e. e B' 4e is total and f(n) = Oe(n) for all n. A set is recursive just if its characteristic function is recursive. A set is recursively enumerable if it is the range of some parkial recursive function. We denote the range of Oe by We. We say f is Turing-reducible to g (f 5T g) if f = hn[Oe(g, n) ] for some e. We extend this reducibility to sets by defining A 5 B if C < Two functions f and g T A -T C~. are Turing equivalent (f 9) if f5 gandg5 f, It canbeshown -, T T that E is an equivalence relation (see (5, p.1371). The equivalence T classes under = of total functions (or sets by considering their -T characteristic function) are called the Turing-degrees. A Turing-degree that contains an r.e. set is calle6 an r.e. degree. We will use small underlined letters for the Turing-degrees, for example 2, b, 2. The jump of a set A, denoted A', is the set Ie : Oe(CAf e) is defined}. The jump operator is well defined for the degrees and if -a is the degree of A then we denote by 2' the degree of A'. We use 2 to denote the smallest degree, that is the degree of all recursive functions. We reserve the symbol K for the set {e : e f we} and the symbol Kg for the set {<n, e> : n € we}. Both K and K are of degree 2'. 0 1.3 ENUMERATION REDUCIBILITY Enumeration reducibility is defined in Friedberg and Rogers [2] and discussed in Rogers [ 5, p. 1461 . Intuitively, a set A is enumera- tion reducible to B is there is an algorithm that will work on any enumeration of B (the input enumeration) and produce an enumeration of A (the output enumeration). The formal definitions are as follows: 1.3.1 W(A1 = {x : for some D, D c A and <D, x> € w}. 1.3.2 A is enumeration reducible to B (A Se B) if there is an r.e. set W such that A = W(B). 1.3.3 A= B if A5 B and B5 A. e e e 1.3.4 A< B if A 5 B and B $ A. e e e ~USto every r.e. set W there is associated a function e 'e N from 2* to 2 . We call this function an enumeration operator. In this thesis we shall not distinguish between an r.e. set V and its e associated operator me, for example if @ is the enumeration operator associated with the r. e. set W we will write W (A) for @ (A) .
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