On minimal wtt-degrees and computably enumerable Turing degrees

The Bulletin of Symbolic Logic, 2018

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of ${\rm{\Delta }}_2^0$ functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

2014

We show the decidability of the existential theory of the recursively enumerable degrees in the language of Turing reducibility, Turing reducibility of the Turing jumps, and least and greatest element. The Existential Theory of the Poset of R.E. Degrees with a Predicate for Single Jump Reducibility

Transactions of the American Mathematical Society, 1988

Notre Dame Journal of Formal Logic, 1997

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.

Proceedings of Computational Complexity. Twelfth Annual IEEE Conference, 1997

We prove that the theory of EXPTIME degrees with respect to polynomial time Turing and many-one reducibility is undecidable. To do so we use a coding method based on ideal lattices of Boolean algebras which w as introduced in [7]. The method can be applied in fact to all hyper-polynomial time classes.

Logical Methods, 1993

In the setting of the parameterized reducibilities introduced by the second author and Mike fellows, we prove a number of decidability and definability results. In particular the undecidability of the relevant m-degree structures is proven. The relationship with classical notions is analysed, and this leads to a number of observations about classical constructions in the P T IM E degrees. Methods include 0 and 0 priority arguments combined with speedup type arguments. From among Anil Nerode's many superb contributions, one recurrent theme is the analysis of definability and decibility results in the structures of recursion theory. For instance the papers [MaN], [MN], [NSm], [NSh] and [NR1,2] are clearly of this ilk. In the present paper we wish to analyse the structures introduced by the first author and Mike Fellows[DF1-5], in the same spirit. This seems particularly apt in view of Anil's interest in polynomial time and polynomialy graded structures [NR3]. In these structures we plan to prove a number of decidability and definability results, as well as examining relationships with classical notions. Before we state specific results it is perhaps appropriate to include a brief recap of the Downey-Fellows setting, and its motivations as it is still rather novel. While the N P completeness phenomenom is a good tool to explain the apparent intractability of many combinatorial problems, it is really a fairly coarse measure in the sense that from a practical viewpoint many N P complete problems can behave quite differently with to respect to the spectrum of solutions. Furthermore N P completeness does not say much about intractability in P . To be specific, many combinatorial problems have the prop-

2017

The relationship between enumeration degrees and abstract models of computability inspires a new direction in the field of computable structure theory. Computable structure theory uses the notions and methods of computability theory in order to find the effective contents of some mathematical problems and constructions. The paper is a survey on the computable structure theory from the point of view of enumeration reducibility.

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

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 Journal of Symbolic Logic, 2003

Given two incomparable c.e. Turing degrees a and b , we show that there exists a c.e. degree c such that c = ( a ∪ c ) ∩ ( b ∪ c ), a ∪ c ∣ b ∪ c , and c &lt; a ∪ b .

Theoretical Computer Science, 1992

Shore, R.A. and T.A. Slaman, The p-T degrees of the recursive sets: lattice embeddings, extensions of embeddings and the two-quantifier theory, Theoretical Computer Science 97 (1992) 2633284. Ambos-Spies (1984a) showed that the two basic nondistributive lattices can be embedded in R,.,, the polynomial-time Turing degrees of the recursive sets. We introduce more general techniques to extend his results to show that every recursive lattice can be embedded in R,,. In addition to lattice-theoretic representation theorems, we use the scheme of priority style arguments coupled with "looking back" techniques presented in Shinoda and Slaman (1988, 1990). We also generalize the density type results of Ladner (1975) and many others to settle the full extension of the embedding problem for R,.,. Combined with the logical analysis of sentences with one alternation of quantifiers (Shore 1978, Lerman 1983), these results suffice to decide the full U-theory of R,,. They also give a strong nonhomogeneity result: the p-time degrees of the sets recursive in (and, if desired, p-time above) two distinct sets A and E are almost never isomorphic. The situation for the p-time many-one degrees is quite different. We decide the extension of the embedding problem (differently than for R,,) but not the t/3-theory. A notion of reducibility 6, between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is reducible to a set B, A <, B, if *Research partially supported by NSF Grants DMS-8601048 and DMS-8902797.

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...

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.

Journal of Symbolic Logic, 1970