Let A 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 A, first introduced and studied by G. 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 hypersimp...

The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y . Here X is strongly hyperimmune-free relative to Y if and only if for every partial X-recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a) < p(a), that is, p is majorised by q. For X being hyperi...

The intuitionistic linguistic set (ILS) is an extension of linguisitc variable. To overcome the drawback of using single real number to represent membership degree and non-membership degree for ILS, the concept of interval-valued intuitionistic linguistic set (IVILS) is introduced through representing the membership degree and non-membership degree with intervals for ILS in this paper. The oper...

We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeld...

A set A ⊆ ω is called 2-computably enumerable (2-c.e.), if there are computably enumerable sets A1 and A2 such that A1 −A2 = A. A set A ⊆ ω is quasi-reducible to a set B ⊆ ω (A ≤Q B), if there is a computable function g such that for all x ∈ ω we have x ∈ A if and only if Wg(x) ⊆ B. This reducibility was introduced by Tennenbaum (see [1], p.207) as an example of a reducibility which differs fro...

We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the so-called join-semidistributive...

R. Shore proved that every recursively enumerable (r.e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice E of all r.e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r.e. sets which can be sp...

We study low level nondefinability in the Turing degrees. We prove a variety of results, including for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤, REA) where REA stands for the “r.e. in and above” predicate. In contrast, this property is definable by a Π2 formula in this language. We also show that the Σ1-theory of (D,≤, REA) is decidable.

A set S of vertices of a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γc(G) is the minimum size of a connected dominating set of G. In this paper we prove that γc(G) + γc(G) ≤ min{δ(G), δ(G)} + 4 for every n-vertex graph G such that G and G have diameter 2 and show that ...