There is a Turing functional $\Phi $ taking $A^\prime $ to a theory $T_A$ whose complexity is exactly that of the jump of A, and which has the property that $A \leq _T B$ if and only if $T_A \trianglelefteq T_B$ in Keisler’s order. In fact, by more elaborate means and related theories, we may keep the complexity at the level of A without using the jump.

This paper continues our study of computable point-free topological spaces and the metamathematical points in them. For us, a point is the intersection of a sequence of basic open sets with compact and nested closures. We call such a sequence a sharp filter. A function fF from points to points is generated by a function F from basic open sets to basic open sets such that sharp filters map to sharp filters. We restrict our study to functions that have at (...) least all computable points in their domains.We follow Turing's approach in stating that a point is computable if it is the limit of a computable sharp filter; we then define the Turing degree Deg of a general point x in an analogous way. Because of the vagaries of the definition, a result of J. Miller applies and we note that not all points in all our spaces have Turing degrees; but we also show a certain class of points do. We further show that in ℝn all points have Turing degrees and that these degrees are the same as the classical Turing degrees of points defined by other researchers.We also prove the following: For a point x that has a Turing degree and lies either on a computable tree T or in the domain of a computable function fF, there is a sharp filter on T or in dom converging to x and with the same Turing degree as x. Furthermore, all possible Turing degrees occur among the degrees of such points for a given computable function fF or a complete, computable, binary tree T. For each x ∈ dom for which x and fF have Turing degrees, Deg) ≤ Deg. Finally, the Turing degrees of the sharp filters convergent to a given x are closed upward in the partial order of all Turing degrees. (shrink)

A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let be a computable model and let R be an extra relation on the domain of . That is, R is not named in the language of . We define to be the set of Turing degrees of the images f under all isomorphisms f from to computable models. We investigate conditions on and R which are sufficient and necessary for (...) to contain every Turing degree. These conditions imply that if every Turing degree 0″ can be realized in via an isomorphism of the same Turing degree as its image of R, then contains every Turing degree. We also discuss an example of and R whose coincides with the Turing degrees which are 0′. (shrink)

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on (...) the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. (shrink)

We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of (...) Turing degrees. In particular, we show that every \(\Delta ^0_2\) -degree contains an NCR element. (shrink)

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number (...) of maximal sets whose many-one degrees are pairwise incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′. The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈, where M is a maximal set, and M is its complement. We showed that / 1 ≡ / 2 if, and only if, 1 = m 2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree. (shrink)

Let be an infinite computable structure, and let R be an additional computable relation on its domain A. The syntactic notion of formal hypersimplicity of R on , first introduced and studied by Hird, is analogous to the computability-theoretic notion of hypersimplicity of R on A, given the definability of certain effective sequences of relations on A. Assuming that R is formally hypersimple on , we give general sufficient conditions for the existence of a computable isomorphic copy of on whose (...) domain the image of R is hypersimple and of arbitrary nonzero computably enumerable Turing degree. (shrink)

This book presents new results in computability theory, a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field's connections with disparate areas of mathematical logic and mathematics more generally have grown deeper, and now have a variety of applications in topology, group theory, and other subfields. This monograph establishes new directions in the field, blending classic results with modern research areas such as algorithmic randomness. The significance of the book lies not only (...) in the depth of the results contained therein, but also in the fact that the notions the authors introduce allow them to unify results from several subfields of computability theory. (shrink)

If X0and X1are both generic, the theories of the degrees below X0and X1are the same. The same is true if both are random. We show that then-genericity orn-randomness of X do not suffice to guarantee that the degrees below X have these common theories. We also show that these two theories are different. These results answer questions of Jockusch as well as Barmpalias, Day and Lewis.

We construct a recursive model , a recursive subset R of its domain, and a Turing degree x 0 satisfying the following condition. The nonrecursive images of R under all isomorphisms from to other recursive models are of Turing degree x and cannot be recursively enumerable.

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 elementm $>$ 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. (shrink)

We study generalizations of shortest programs as they pertain to Schaefer’s problem. We identify sets of -minimal and -minimal indices and characterize their truth-table and Turing degrees. In particular, we show , , and that there exists a Kolmogorov numbering ψ satisfying both and . This Kolmogorov numbering also achieves maximal truth-table degree for other sets of minimal indices. Finally, we show that the set of shortest descriptions, , is 2-c.e. but not co-2-c.e. Some open problems are left (...) for the reader. (shrink)

Informal construction -- Formal construction -- Limiting results.

We study the problem of existence of maximal chains in the Turing degrees. We show that: 1. ZF+DC+"There exists no maximal chain in the Turing degrees" is equiconsistent with ZFC+"There exists an inaccessible cardinal"; 2. For all a ∈ 2ω.(ω₁)L[a] = ω₁ if and only if there exists a $\Pi _{1}^{1}[a]$ maximal chain in the Turing degrees. As a corollary, ZFC + "There exists an inaccessible cardinal" is equiconsistent with ZFC + "There is no (bold face) $\utilde{\Pi}{}_{1}^{1}$ (...) maximal chain of Turing degrees". (shrink)

We consider questions related to the rigidity of the structure R, the PTIME-Turing degrees of recursive sets of strings together with PTIME-Turing reducibility, pT, and related structures; do these structures have nontrivial automorphisms? We prove that there is a nontrivial automorphism of an ideal of R. This can be rephrased in terms of partial relativizations. We consider the sets which are PTIME-Turing computable from a set A, and call this class PTIMEA. Our result can be stated as (...) follows: There is an oracle, A, relative to which the PTIME-Turing degrees are not rigid . Furthermore, the automorphism can be made to preserve the complexity classes DTIMEA for all k 1, or to move any DTIMEA for n 2. From the existence of such an automorphism we conclude as a corollary that there is an oracle A relative to which the classes DTIME are not definable over R. We carry out the corresponding partial relativization for the many-one degrees to construct an oracle, A, relative to which the PTIMA-many-one degrees relative to A have a nontrivial automorphism, and one relative to which the lattice of sets in PTIMEA under inclusion have a nontrivial automorphism. The proof is phrased as a forcing argument; we construct the set A to meet a particular collection of dense sets in our notion of forcing. Roughly, the dense sets will guarantee that if A meets these sets and we split A into two pieces, A0 and A1, in a simple way, and then switching the roles of A0 and A1 in all computations from A will produce an automorphism of the ideal of PTIMA-degrees below A. We force A0 and A1 to have different PTIME-Turing degree; this will then make the automorphism nontrivial. An appropriately generic set A is constructed using a priority argument. (shrink)

Posner [6] has shown, by a nonuniform proof, that every ▵ 0 2 degree has a complement below 0'. We show that a 1-generic complement for each ▵ 0 2 set of degree between 0 and 0' can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above $\varnothing'$ . In the second half of the paper, we show (...) that the complementation of the degrees below 0' does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems. (shrink)

We define the bounded jump of $A$ by $A^{b}=\{x\in \omega \mid \exists i\leq x[\varphi_{i}\downarrow \wedge\Phi_{x}^{A\upharpoonright \!\!\!\upharpoonright \varphi_{i}}\downarrow ]\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact that it is strictly increasing and order-preserving on the bounded Turing degrees. We show that the bounded jump is related to the Ershov hierarchy. Indeed, for $n\geq2$ we have $X\leq_{bT}\emptyset ^{nb}\iff X$ is $\omega^{n}$-c.e. $\iff X\leq_{1}\emptyset ^{nb}$, extending the classical result that $X\leq_{bT}\emptyset (...) '\iff X$ is $\omega$-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every $X$ such that $\emptyset ^{b}\leq_{bT}X\leq_{bT}\emptyset ^{2b}$, there is a $Y\leq_{bT}\emptyset ^{b}$ such that $Y^{b}\equiv_{bT}X$. (shrink)

For [Formula: see text], the coarse similarity class of [Formula: see text], denoted by [Formula: see text], is the set of all [Formula: see text] such that the symmetric difference of [Formula: see text] and [Formula: see text] has asymptotic density [Formula: see text]. There is a natural metric [Formula: see text] on the space [Formula: see text] of coarse similarity classes defined by letting [Formula: see text] be the upper density of the symmetric difference of [Formula: see text] and (...) [Formula: see text]. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form [Formula: see text] where [Formula: see text] is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of [Formula: see text]. We then define a distance between Turing degrees based on Hausdorff distance in the metric space [Formula: see text]. We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly [Formula: see text], [Formula: see text], and [Formula: see text], and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree [Formula: see text] to be attractive if the class of all degrees at distance [Formula: see text] from [Formula: see text] has measure [Formula: see text], and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually [Formula: see text]-random, as well as a perfect set whose elements are mutually [Formula: see text]-generic. Finally, we study the completeness of [Formula: see text] from the perspectives of computability theory and reverse mathematics. (shrink)

We exhibit a finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees. Our method promises to lead to a full characterization of the finite lattices embeddable into the enumerable Turing degrees.

It is shown that every locally countable upper semi-lattice of cardinality the continuum can be embedded into the Turing degrees, assuming Martin’s Axiom.

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no prime ideal in the c.e. Turing degrees. This answers a question of Calhoun [2]. Every proper ideal in the c.e. Turing degrees has a low2 upper bound. Furthermore, the partial order of ideals under inclusion is dense.

Where $\underline{a}$ is a Turing degree and ξ is an ordinal $ , the result of performing ξ jumps on $\underline{a},\underline{a}^{(\xi)}$ , is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated.

TheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that (...) for all degreesa, bwhich are notK-trivial anda > d, b > dthere exists a degreevwhich is notK-trivial anda > v, b > v. This work sheds light to the question of the definability of theK-trivial degrees in the c.e. degrees. (shrink)

A computably enumerable degree a is called nonbounding, if it bounds no minimal pair, and plus cupping, if every nonzero c.e. degree x below a is cuppable. Let NB and PC be the sets of all nonbounding and plus cupping c.e. degrees, respectively. Both NB and PC are well understood, but it has not been possible so far to distinguish between the two classes. In the present paper, we investigate the relationship between the classes NB and PC, and (...) show that there exists a minimal pair which join to a plus cupping degree, so that PC ⊈ NB. This gives a first known difference between NB and PC. (shrink)

We show that the lattice L 20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (J). We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π 2 necessary and sufficient condition for embedding a finite pseudolattice into J.

We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.

Is there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial (...) automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more. (shrink)

Let I be a countable jump ideal in $\mathscr{D} = \langle \text{The Turing degrees}, \leq\rangle$ . The central theorem of this paper is: a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a (1) computes. We may replace "the join of an I-exact pair" in the above theorem by "a weak uniform upper bound on I". We also answer two minimality questions: the class of uniform upper bounds on (...) I never has a minimal member; if ∪ I = L α [ A] ∩ ω ω for α admissible or a limit of admissibles, the same holds for nice uniform upper bounds. The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces. (shrink)

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends (...) Sikora’s [28] investigation of orders on groups in topology and Solomon’s [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group Fn of rank n>1 has a bi-order in every Turing degree. (shrink)

Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is (...) based on results of Jensen and is detailed in [3] and [4]. In $\S1$ we review the basic definitions from [3] which are needed to state the general results. (shrink)

In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each (...) of its elements computable numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice. (shrink)

We show that the Π 4 -theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

We show that the $\Pi_4$-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

We say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c.e. degree x with $0 < x \leq a$ , there is a c. e. degree $y \not= 0'$ such that $x \vee y = 0/\'$ . We say that a is n-plus-cupping. if for every c. e. degree x, if $0 < x \leq a$ , then there is a $low_n$ c. e. degree 1 such that $x \vee l (...) = 0'$ . Let PC and $PC_n$ be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then $PC_{1} \subseteq PC_{2} \subseteq PC_3 = PC$ . In this paper we show that $PC_{1} \subset PC_2$ , so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [14] showing that $LC_{1} \subset LC_2$ , as well as extending the Harrington plus-cupping theorem [8]. (shrink)

We employ techniques related to Lempp and Lerman's “iterated trees of strategies” to directly measure a Σ5-predicate and use this in showing the index set of the cuppable r.e. sets to be Σ5-complete. We also show how certain technical devices arise naturally out of the iterated-trees context, in particular, links arise as manifestations of a generalized notion of “stage”.