1-genericity in the enumeration degrees

1988 ◽  
Vol 53 (3) ◽  
pp. 878-887 ◽  
Author(s):  
Kate Copestake
Keyword(s):  
Partial Function ◽  
Partial Ordering ◽  
Turing Degree ◽  
Turing Degrees ◽  
Original Formulation ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Generic Set

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

1984 ◽  
Vol 49 (2) ◽  
pp. 503-513 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Density Theorem ◽  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position ◽  
Density Problem

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

Partial degrees and the density problem

1982 ◽  
Vol 47 (4) ◽  
pp. 854-859 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Computation Tree ◽  
Partial Degree ◽  
Degree Structure ◽  
Turing Reducibility ◽  
Density Problem ◽  
Effective Computability ◽  
Degrees Of Unsolvability

A notion of relative reducibility for partial functions, which coincides with Turing reducibility on the total functions, was first given by S.C. Kleene in Introduction to metamathematics [4]. Following Myhill [7], this was made more explicit in Hartley Rogers, Jr., Theory of recursive functions and effective computability [8, pp. 146, 279], where some basic properties of the partial degrees or (equivalent, but notationally more convenient) the enumeration degrees, were derived. The question of density of this proper extension of the degrees of unsolvability was left open, although Medvedev's result [6] that there are quasi-minimal partial degrees (that is, nonrecursive partial degrees with no nonrecursive total predecessors) is proved.In 1971, Sasso [9] introduced a finer notion of partial degree, which also contained the Turing degrees as a proper substructure (intuitively, Sasso's notion of reducibility between partial functions differed from Rogers' in that computations terminated when the oracle was asked for an undefined value, whereas a Rogers computation could be thought of as proceeding simultaneously along a number of different branches of a ‘consistent’ computation tree—cf. Sasso [10]). His construction of minimal ‘partial degrees’ [11], while of interest in itself, left open the analogous problem for the more standard partial degree structure.

Nonisomorphism of lattices of recursively enumerable sets

1993 ◽  
Vol 58 (4) ◽  
pp. 1177-1188 ◽  
Author(s):  
John Todd Hammond
Keyword(s):  
Boolean Algebra ◽  
Isomorphism Class ◽  
Boolean Algebras ◽  
Turing Degree ◽  
Symmetric Difference ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Definition Of

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

Recursively enumerable generic sets

1982 ◽  
Vol 47 (4) ◽  
pp. 809-823 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Essential Feature ◽  
Turing Degree ◽  
New Class ◽  
Recursively Enumerable ◽  
Simple Set ◽  
Recursively Enumerable Sets

AbstractWe show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0′. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice ℰ of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions.

On degrees of unsolvability and complexity properties

1975 ◽  
Vol 40 (4) ◽  
pp. 529-540 ◽  
Author(s):  
Ivan Marques
Keyword(s):  
Turing Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Complete Sets ◽  
Turing Complete ◽  
Degrees Of Unsolvability

In this paper we present two theorems concerning relationships between degrees of unsolvability of recursively enumerable sets and their complexity properties.The first theorem asserts that there are nonspeedable recursively enumerable sets in every recursively enumerable Turing degree. This theorem disproves the conjecture that all Turing complete sets are speedable, which arose from the fact that a rather inclusive subclass of the Turing complete sets, namely, the subcreative sets, consists solely of effectively speedable sets [2]. Furthermore, the natural construction to produce a nonspeedable set seems to lower the degree of the resulting set.The second theorem says that every speedable set has jump strictly above the jump of the recursive sets. This theorem is an expected one in view of the fact that all sets which are known to be speedable jump to the double jump of the recursive sets [4].After this paper was written, R. Soare [8] found a very useful characterization of the speedable sets which greatly facilitated the proofs of the results presented here. In addition his characterization implies that an r.e. degree a contains a speed-able set iff a′ > 0′.

Jump embeddings in the Turing degrees

1991 ◽  
Vol 56 (2) ◽  
pp. 563-591 ◽  
Author(s):  
Peter G. Hinman ◽  
Theodore A. Slaman
Keyword(s):  
Partial Ordering ◽  
Turing Degrees ◽  
First Order ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Locally Countable ◽  
Image Set ◽  
Turing Reducibility ◽  
Image Position ◽  
Order Structures

Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embeddingwhere D is the set of all Turing degrees and <T is Turing reducibility. If (P, ≤P) is a countable partial ordering, then the image of the embedding may be taken to be a subset of R, the set of recursively enumerable degrees. Without attempting to make the notion completely precise, we shall call embeddings of the first sort global, in contrast to local embeddings which impose some restrictions on the image set.

The Sacks density theorem and Σ2-bounding

1996 ◽  
Vol 61 (2) ◽  
pp. 450-467 ◽  
Author(s):  
Marcia J. Groszek ◽  
Michael E. Mytilinaios ◽  
Theodore A. Slaman
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Models Of Arithmetic

AbstractThe Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P− + BΣ2. The proof has two components: a lemma that in any model of P− + BΣ2, if B is recursively enumerable and incomplete then IΣ1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

A limit on relative genericity in the recursively enumerable sets

1989 ◽  
Vol 54 (2) ◽  
pp. 376-395 ◽  
Author(s):  
Steffen Lempp ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degrees ◽  
Recursively Enumerable Set ◽  
Recursively Enumerable ◽  
Low Degree ◽  
Recursively Enumerable Sets ◽  
Nonzero Degree

AbstractWork in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of A ⊕ W is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W ⊕ A)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.

Jumps of quasi-minimal enumeration degrees

1985 ◽  
Vol 50 (3) ◽  
pp. 839-848 ◽  
Author(s):  
Kevin McEvoy
Keyword(s):  
Turing Degree ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Enumeration Reducibility ◽  
Degree Structure ◽  
Usual Topology ◽  
Definition Of ◽  
Second Category ◽  
Degrees Of Unsolvability

Enumeration reducibility is a reducibility between sets of natural numbers defined as follows: A is enumeration reducible to B if there is some effective operation on enumerations which when given any enumeration of B will produce an enumeration of A. One reason for interest in this reducibility is that it presents us with a natural reducibility between partial functions whose degree structure can be seen to extend the structure of the Turing degrees of unsolvability. In [7] Friedberg and Rogers gave a precise definition of enumeration reducibility, and in [12] Rogers presented a theorem of Medvedev [10] on the existence of what Case [1] was to call quasi-minimal degrees. Myhill [11] also defined this reducibility and proved that the class of quasi-minimal degrees is of second category in the usual topology. As Gutteridge [8] has shown that there are no minimal enumeration degrees (see Cooper [3]), the quasi-minimal degrees are very much of interest in the study of the structure of the enumeration degrees. In this paper we define a jump operator on the enumeration degrees which was introduced by Cooper [4], and show that every complete enumeration degree is the jump of a quasi-minimal degree. We also extend the notion of a high Turing degree to the enumeration degrees and construct a “high” quasi-minimal enumeration degree—a result which contrasts with Cooper's result in [2] that a high Turing degree cannot be minimal. Finally, we use the Sacks' Jump Theorem to characterise the jumps of the co-r.e. enumeration degrees.

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