Abstract the Existential Theory of the Poset of R.e. Degrees with a Predicate for Single Jump Reducibility (original) (raw)

Definability in the Recursively Enumerable Degrees

Bulletin of Symbolic Logic, 1996

§1. Introduction. Natural sets that can be enumerated by a computable function (the recursively enumerable or r.e. sets) always seem to be either actually computable (recursive) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K? Let R be the r.e. degrees, i.e., the r.e. sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order (indeed, an uppersemilattice or usl) with least element 0, the degree (equivalence class) of the computable sets, and greatest element 1 or 0 ′ , the degree of K. Post's problem then asks if there are any other elements of R. The (positive) solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure R was in some way well behaved: Theorem 1.1 (Embedding theorem; Muchnik [1958], Sacks [1963]). Every countable partial ordering or even uppersemilattice can be embedded into R. Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degree a there are r.e. degrees b, c < a such that b ∨ c = a. Theorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degrees a < b there is an r.e. degree c such that a < c < b. These results led Shoenfield in 1963 to formulate the view that the structure was "nice" as the sweeping conjecture that the r.e. degrees, R, are a

On relative enumerability of Turing degrees

Archive for Mathematical Logic, 2000

Let d be a Turing degree, R[d] and Q[d] denote respectively classes of recursively enumerable (r.e.) and all degrees in which d is relatively enumerable. We proved in Ishmukhametov [1999] that there is a degree d containing differences of r.e.sets (briefly, d.r.e.degree) such that R[d] possess a least element m>0. Now we show the existence of a d.r.e. d such that R[ d] has no a least element. We prove also that for any REA-degree d below 0 the class Q[d] cannot have a least element and more generally is not bounded below by a non-zero degree, except in the trivial cases.

Computably enumerable sets and quasi-reducibility

Annals of Pure and Applied Logic, 1998

We consider the computably enumerable sets under the relation of Qreducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, RQ, ≤Q , under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of RQ, ≤Q is undecidable.

Reducibility orderings: Theories, definability and automorphisms

Annals of Mathematical Logic, 1980

introduced the notion of degree of unsolvability and the partial ordering ~ i on ~T, the set of such Tt, ring degrees, induced by Turing reducibility (Turing 13711, His paper with Kleene [ 1,4] contains the first serious analysis of this structure (~'r, ~x). They prove, for example, that all coantable partial orderings can be embedded in (ar, ~<~-). These embeddings show that the existential (it;st order~ theory of (~-r, ~r) is decidable, Next Spector [35], in a paper arising from Kleene's 1953 seminar, made an important inroad on the two quantifie, (i.e., VzI~ theory by showing that there is a minimal (Turing) degree. Sacks [31] extended these results and set forth some important conjectures on embeddings a~nd initial segments of ~'r. In particular he points out that one can prove the undecidability of the theory of (£ar, ~<v) by such results. This work inspired many papers by others eszabli,,hing better and better initial segment results. One milesto;~e was kachlan which showed that every countable distributive lattice can bc embedded as an initial segment of the Turing degrees. As the theory of distributive lattices was known to be undecidable, this sufficed to verify Sacks' conjezture that so is the theory of (@r, <~a-). (In fact it would have sufficed to embed all f-~nitc distributive lattices as was pointed out by Thomason [36] for hyperdegrees.) Two directions in which such results can be sharpened immediately come to mind, One is, where does the undecidability first arise in terms of quantifier complexity. The second is just how complicated is the full theory of (~, ~). (qhe results of Kleene and Post [ 14] showed only that the 3-theory was decidable while the coding of distributive lattices only showed that the full theory ha', degree at least 0'.) Further progress required further structural results. For the first question Lerman [20] supplied an essential ingredient by settling the full conjecture from Sacks , He showed that every finite lattice is embeddable ~, an initial segment of fib-. This can be combined with Kleene and Pc, st [14] to decide the 'q::l theory of

The Theory of the Metarecursively Enumerable Degrees

2000

Sacks (Sa1966a) asks if the metarecursively enumerable de- grees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O(!) or, equivalently, that of the truth set of L!CK 1 .

Decidability and Definability in the 0 -Enumeration Degrees

2005

Enumeration reducibility was introduced by Friedberg and Rogers in 1959 as a positive reducibility between sets. The enumeration degrees provide a wider context in which to view the Turing degrees by allowing us to use any set as an oracle instead of just total functions. However, in spite of the fact that there are several applications of enumeration reducibility in computable mathematics, until recently relatively little research had been done in this area. In Chapter 2 of my thesis, I show that the ∀∃∀-fragment of the first order theory of the Σ 0 2-enumeration degrees is undecidable. I then show how this result actually demonstrates that the ∀∃∀-theory of any substructure of the enumeration degrees which contains the ∆ 0 2-degrees is undecidable. In Chapter 3, I present current research that Andrea Sorbi and I are engaged in, in regards to classifying properties of non-splitting Σ 0 2-degrees. In particular I give proofs that there is a properly Σ 0 2-enumeration degree and that every ∆ 0 2-enumeration degree bounds a non-splitting ∆ 0 2-degree. Advisor: Prof. Steffen Lempp I am grateful to Steffen Lempp, my thesis advisor, for all the time, effort, and patience that he put in on my behalf. His insight and suggestions have been of great worth to me, both in and out of my research. I am especially grateful for his help in getting me back in school after my two-year leave of absence and for offering me a research assistantship so I could study for a year with him in Germany. I am also grateful to Andrea Sorbi for funding a visit to Siena, Italy that allowed me to do research with him, and for the friendship that has grown from our research contact. Hopefully we will be able to go running together in the mountains again. I would like to thank Todd Hammond for introducing me to mathematical logic, to Mirna Dzamonja for getting me excited about Computability Theory, and to Jerome Keisler, Ken Kunen, Arnie Miller, and Patrick Speissegger for teaching interesting logic classes. I would like to thank all of the wonderful teachers over the years who have encouraged my interest in mathematics, especially Patty Av3ery and Slade Skipper. Thanks also go to Eric Bach, Joel Robbin, and Mary Ellen Rudin for help they have given and for serving on my defense committee. I am very appreciative for my parents and sister, for the support and love they have given me over the past 31 years. The most appreciation, however, goes to my wonderful wife, Joy, for always being there for me. I could not have made it without her encouragement and unconditional love.

Decidability of the two-quantifier theory of the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices

The Journal of Symbolic Logic, 1996

We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theo...