Recursively Enumerable Degrees Research Articles

The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

AbstractWhen attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of α-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the α-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of α-r.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the α-r.e. degrees is complicated, we get that for every admissible ordinal α, the α-r.e. degrees and the classical r.e. degrees are not elementarily equivalent.

Cupping the Recursively Enumerable Degrees by D.R.E. Degrees

We prove that there are two incomplete d.r.e. degrees (the Turing degrees of differences of two recursively enumerable sets) such that every non-zero recursively enumerable degree cups at least one of them to 0′, the greatest recursively enumerable (Turing) degree. 1991 Mathematics Subject Classification: primary 03D25, 03D30; secondary 03D35.

Interpretability and Definability in the Recursively Enumerable Degrees

We investigate definability in R, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. First we show that an SMA can be interpreted in R without parameters. Building on this, we prove that the recursively enumerable T-degrees satisfy a weak form of the bi-interpretability conjecture which implies that all jump classes Lown and Highn−1 n ⩾ 2 are definable in R without parameters and, more generally, that all relations on R that are definable in arithmetic and invariant under the double jump are actually definable in R. This partially answers Soare's Question 3.7 (R. Soare, Recursively enumerable sets and degrees (Springer, Berlin, 1987), Chapter XVI). 1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.

Definability in the Recursively Enumerable Degrees

§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 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 .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 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 .Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatb ∨ c = a.Theorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b.

The Decidability of the Existential Theory of the Poset of Recursively Enumerable Degrees with Jump Relations

We show that the existential theory of the recursively enumerable degrees in the language L containing predicates for order andn-jump comparability for alln, and constant symbols for least and greatest elements, is decidable. The decidability follows from our main theorem, where we show that any finite L-structure which is consistent with the order relation, the order-preserving property of the jump operator, and the property of the jump operator that the jump of an element is strictly greater than the element, can be embedded into the r.e. degrees.

Splittings of 0' into the Recursively Enumerable Degrees

AbstractLachlan [9] proved that there exists a non‐recursive recursively enumerable (r. e.) degree such that every non‐recursive r. e. degree below it bounds a minimal pair. In this paper we first prove the dual of this fact. Second, we answer a question of Jockusch by showing that there exists a pair of incomplete r. e. degrees a0, a1 such that for every non‐recursive r. e. degree w, there is a pair of incomparable r. e. degrees b0, b2 such that w = b0 V b1 and bi for i = 0, 1.Mathematics Subject Classification: 03D25, 03D30.

Discontinuity of Cappings in the Recursively Enumerable Degrees and Strongly Nonbranching Degrees

AbstractWe construct an r. e. degree a which possesses a greatest a‐minimal pair b0, b1, i.e., r. e. degrees b0 and b1 such that b0, b1 < a, b0 ∩ b1 = a, and, for any other pair c0, c1 with these properties, c0 ≤ bi and c1 ≤ b1‐i for some i ≤ 1. By extending this result, we show that there are strongly nonbranching degrees which are not strongly noncappable. Finally, by introducing a new genericity concept for r. e. sets, we prove a jump theorem for the strongly nonbranching and strongly noncappable r. e. degrees.Mathematics Subject Classification: 03D25.

RECURSIVELY ENUMERABLE SETS AND DEGREES A Study of Computably Generated Sets (Perspectives in Mathematical Logic)

By Robert I. Soare: pp. 437. DM.68.–. (Springer-Verlag, 1987)

On Pairs of Recursively Enumerable Degrees

On Pairs of Recursively Enumerable Degrees

The Quotient Semilattice of the Recursively Enumerable Degrees Modulo the Cappable Degrees

The Quotient Semilattice of the Recursively Enumerable Degrees Modulo the Cappable Degrees

An Algebraic Decomposition of the Recursively Enumerable Degrees and the Coincidence of Several Degree Classes with the Promptly Simple Degrees

An Algebraic Decomposition of the Recursively Enumerable Degrees and the Coincidence of Several Degree Classes with the Promptly Simple Degrees

Decomposition of Recursively Enumerable Degrees

It is shown that any nonzero recursively enumerable degree can be expressed as the join of two distinct such degrees having a greatest lower bound.

Minimal Upper Bounds for Sequences of Recursively Enumerable Degrees

Minimal Upper Bounds for Sequences of Recursively Enumerable Degrees

Jump Restricted Interpolation in the Recursively Enumerable Degrees

Sacks has shown [3, Theorem 3 of ? 6] that if b is r.e., f< 0' < u, f 4b and u is r.e. in 0' then there is an r.e. degree d such that b < d, f ;f d, and di = u. He also showed [3, Theorem 1 of ? 7] that if b, c are r.e. and b < c then there is an r.e. degree d with b < d < c and d' = c'. The methods which Sacks used to prove these theorems and, later, the Density Theorem [4] were combined by the author in [1]. There it is shown that if b, c are r.e. and b < c then there is an r.e. degree d such that b < d < c and d' = b'. In [6] Yates showed that if 0 < f < 0' then there is an r.e. degree d such that d I f and d' = 0. Yates based his proof on entirely different methods which earlier [5] had given the Density Theorem as a by-product. These theorems are all implied by the most general result of the present paper, Theorem 3, which states that if b and c are r.e., b ? c, {ff}iN < O' {f}i6N < c, f, S b and c : gi for all i, {g9}leN < u, b' < u, and u is r.e. in c then there is an r.e. degree d such that b < d < c, f, ;f d % gi for all i, and d' = u. Yates [6, page 260] has conjectured that there are degrees f and u such that 0 < f < O' < u, u is r.e. in 0', and there is no r.e. d such that d I f and di = u. In Corollary 3 Yates' conjecture is disproved by combining our main result with a splitting theorem from [2]. Corollary 3 implies that if b and c are r.e., b ? c, b < fi < c for all i < n, bY = O' < u, and u is r.e. in c then there is an r.e. degree d such that b < d < c, d I f, for all i < n, and d' = u.' The reader is assumed to be familiar with [2]. It should be noted in particular that any Greek letter other than X or [ denotes a special recursive function. In addition, the basic Lemmas 1, 2, and 3 are proved by means of observations on the jumps of sets constructed in the proofs of Theorems 1 and 2 and Lemma 6 of [2].

Interpolation and Embedding in the Recursively Enumerable Degrees

By a degree is meant a degree of recursive unsolvability. A degree is recursively enumerable (r.e.) just if it contains a recursively enumerable subset of N, the set of non-negative integers. Two infinite injury priority arguments are presented, in ? 2 and ? 3, which are generalizations of parts of Sacks' proof that the r.e. degrees are dense [9]. These results are in a form sufficiently flexible to admit a variety of applications. It is found, for example, that a degree which is a minimal r.e.-uniform upper bound for a sequence of degrees must be the join of a finite subsequence. By way of contrast, any non-recursive r.e. degree is a minimal upper bound for some strictly ascending sequence of r.e. degrees. It is also found that if a, b are r.e. and a < b, then any countable partial ordering is embeddable in the r.e. degrees between a and b with joins preserved whenever they exist. In ? 4 it is shown that if a' = 0' then there is always such an embedding which also preserves greatest and least elements when they exist. The proof of the latter result is a finite injury priority argument which is more closely related to two splitting theorems of Sacks [8, Th. 2 of ? 5 and Th. 2 of ? 61 than to the preceding results. Our basic notation follows Kleene [4], Kleene and Post [5], and Sacks [8]. A convenient property of Kleene's T-predicates is that if Tn(e, x1, * * * x ,nq y) or Tn(p e xi, ...,xny) holds then U(y) <y and xi<y for 1<? i n. As does Sacks we denote this fact by GND. It is convenient to have U(O) = 2, an inessential departure from Kleene. Another slight variation from Kleene's notation is that we put

Sublattices of the Recursively Enumerable Degrees

Sublattices of the Recursively Enumerable Degrees

Lower Bounds for Pairs of Recursively Enumerable Degrees

Lower Bounds for Pairs of Recursively Enumerable Degrees

The Recursively Enumerable Degrees are Dense

The Recursively Enumerable Degrees are Dense