Robert I. Soare

Recursively enumerable sets and degrees

TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Posts program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.

Computability and Recursion

We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “( Turing ) computability ” and “( general ) recursiveness ”. We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Churchs Thesis and Turings Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “ Computability Theory ” or simply Computability rather than “Recursive Function Theory.”

Automorphisms of the Lattice of Recursively Enumerable Sets Part I: Maximal Sets

Introduction . 80 ?1. Background material 82 S2. Automorphisms and maximal sets . 88 S3. Satisfying condition (2.2) and the hypotheses of the Extension Theorem . 92 S4. Proof of the Extension Theorem Part I: Motivation . ...... 97 S5. Proof of the Extension Theorem Part II: Covering ......... 105 S6. Proof of the Extension Theorem Part III: Mappings . ...... 114 S7. Conclusion 118

The d.r.e. degrees are not dense

Abstract By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n -r.e. degrees ( n ⩾2) and of the ω-r.e. degrees.

The Infinite Injury Priority Method

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {R,},ncw called requirements. If n s be undone for the sake of R, thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Posts problem and is characterized by the fact that each requirement is injured at most finitely often. Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity. The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own. The reader need have no prior knowledge of priority arguments for in ?1 we review the finite injury method using a version invented by Sacks for his Splitting Theorem [15]. In ?2 we discuss the two principal obstacles in extending the strategy to the infinite injury case. We show how the obvious and well-known solution to the first obstacle has automatically solved the second and more fundamental one. We then prove the two main lemmas upon which all of the theorems depend, and from these we prove the Thickness Lemma of Shoenfield [21, p. 83]. In ?3 we apply the method to derive the Yates Index Set Theorem, and results of

Degrees of orderings not isomorphic to recursive linear orderings

Abstract It is shown that for every nonzero r.e. degree c there is a linear ordering of degree c which is not isomorphic to any recursive linear ordering. It follows that there is a linear ordering of low degree which is not isomorphic to any recursive linear ordering. It is shown further that there is a linear ordering L such that L is not isomorphic to any recursive linear ordering, and L together with its ‘infinitely far apart’ relation is of low degree. Finally, an analogue of the recursion theorem for recursive linear orderings is refuted.

Computability theory and differential geometry

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature |K| ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machine Te, e ∈ ω, there is a sequence (uniformly effective in e) of homology n-spheres {P e k}k∈ω which are also hypersurfaces, such that P e k is diffeomorphic to the standard n-sphere S (denoted P e k ≈ diff S) iff Te halts on input k, and in this case the connected sum N k = M#P e k ≈ diff M , so N k ∈ Met(M), and N k is associated with a local minimum of the diameter

Recursively Enumerable Sets Modulo Iterated Jumps and Extensions of Arslanov's Completeness Criterion

Let W e be the e th recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleenes recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that W e = W f ( e ) . In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets. Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper. Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete ( i.e. A has degree 0 ′) iff there is a function f recursive in A with no fixed point .

Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory

We begin with the history of the discovery of computability in the 1930’s, the roles of Godel, Church, and Turing, and the formalisms of recursive functions and Turing automatic machines (a-machines). To whom did Godel credit the definition of a computable function? We present Turing’s notion [1939, §4] of an oracle machine (o-machine) and Post’s development of it in [1944, §11], [1948], and finally Kleene-Post [1954] into its present form. A number of topics arose from Turing functionals including continuous functionals on Cantor space and online computations. Almost all the results in theoretical computability use relative reducibility and o-machines rather than a-machines and most computing processes in the real world are potentially online or interactive. Therefore, we argue that Turing o-machines, relative computability, and online computing are the most important concepts in the subject, more so than Turing a-machines and standard computable functions since they are special cases of the former and are presented first only for pedagogical clarity to beginning students. At the end in §10–§13 we consider three displacements in computability theory, and the historical reasons they occurred. Several brief conclusions are drawn in §14.

The Δ₃⁰-automorphism method and noninvariant classes of degrees

A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = ({We}e∈ω ,⊆). Most previously known automorphisms Φ of the structure E of sets were effective (computable) in the sense that Φ has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is ∆3, and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of E. For example, we show that the orbit of every noncomputable (i.e., nonrecursive) c.e. set contains a set of high degree, and hence that for all n > 0 the well-known degree classes Ln (the lown c.e. degrees) and Hn = R −Hn (the complement of the highn c.e. degrees) are noninvariant classes. Department of Mathematics, University of California at Berkeley, Berkeley, California 94720 E-mail address: leo@math.berkeley.edu Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637-1538 E-mail address: soare@math.uchicago.edu World Wide Web address: http://www.Cs.uchicago.edu/∼soare