UNIVERSAL POSITIVE PREORDERS

In this paper, we investigate universal objects in the class of positive preorders with respect to computable reducibility, we constructed a computable numbering of this class and proved theorems on the existence of a universal positive lattice and a universal weakly precomplete.

Complexity for partial computable functions over computable Polish spaces

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

Computable elements and functions in effectively enumerable topological spaces

This paper is a part of the ongoing program of analysing the complexity of various problems in computable analysis in terms of the complexity of the associated index sets. In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective ω–continuous domains and the natural numbers with the discrete topology satisfy this condition. We prove weak and strong analogues of Rice's theorem for computable elements. Then, we construct principal computable numberings of partial majorant-computable real-valued functions and co-effectively closed sets and calculate the complexity of index sets for important problems such as root verification and function equality. For example, we show that, for partial majorant-computable real functions, the equality problem is Π11-complete.

A decomposition of the Rogers semilattice of a family of d.c.e. sets

Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up to equivalence) exactly two computable Friedberg numberings μ and ν, and μ reduces to any computable numbering not equivalent to ν. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open.

Diagonals and -maximal sets

In analogy to the definition of the halting problem K, an r.e. set A is called a diagonal iff there is a computable numbering ψ of the class of all partial recursive functions such that A = {i ∈ ω: ψi(i)↓} (in that case we say that A is the diagonal of ψ). This notion has been introduced in [10]. It captures all r.e. sets that can be constructed by diagonalization.It was shown that any nonrecursive r.e. T-degree contains a diagonal, that for any diagonal A there is an r.e. nonrecursive nondiagonal B ≤TA, and that there are r.e. degrees a such that any r.e. set from a is a diagonal.In §2 of the present paper we show that the property “A is a diagonal” is elementary lattice theoretic (e.l.t.). This result complements and generalizes previous results of Harrington and Lachlan, respectively. Harrington (see [16, XV. 1]) proved that the property of being the diagonal of some Gödelnumbering (i.e., of being creative) is e.l.t., and Lachlan [12] proved that the property of being a simple diagonal (i.e., of being simple and not hh-simple [10]) is e.l.t.In §§3 and 4 we study the position of diagonals and nondiagonals inside the lattice of r.e. sets. We concentrate on an important class of nondiagonals, generalizing the maximal and hemimaximal sets: the -maximal sets. Using the results from [6] we are able to classify the -maximal sets that can be obtained as halfs of splittings of hh-simple sets.

Diagonals and semihyperhypersimple sets

The most basic construction of an r.e. nonrecursive set—e.g. of the halting problem—proceeds by taking the diagonal of a recursive enumeration of all r.e. sets. We will answer the question of which r.e. sets can be constructed in this manner.If ψ is a computable numbering of some class of partial recursive functions, we define the diagonal of ψ to be the set Kψ ≔ {i ∈ ω ∣ ψi(i)↓}- It is well known that Kφ is creative if φ is a Gödelnumbering, and that for each creative set K there exists a Gödelnumbering φ such that K = Kφ. That is to say, the class of diagonals of Gödelnumberings is characterized as the class of creative sets. This class was shown to be elementary lattice theoretic (e.l.t.) by Harrington (see [So87, XV. 1.1]).We give a characterization of diagonals of arbitrary computable numberings of the class P1 of all partial recursive functions. To this end we introduce the notion of a semihyperhypersimple (shhs) set, which generalizes the notion of hyperhypersimplicity to nonsimple sets. It is shown that the diagonals of numberings of P1 are exactly the non-shhs sets. Then, properties of shhs sets are discussed. For example, for each nonrecursive r.e. set A there exists a nonrecursive shhs set B ≤TA, but not every r.e. T-degree contains a shhs set. These results build upon previous work by Downey and Stob [DSta].The question whether the property “shhs” is (elementary) lattice theoretic remains open. A positive answer would give both an analog of Harrington's result mentioned above, and a generalization of the well-known fact, due to Lachlan [La68], that hyperhypersimplicity is e.l.t. Therefore, we suspect that shhs sets turn out to be useful in the study of the lattice of r.e. sets.Previously, for several constructions from recursion theory the role of the underlying numbering of P1 was investigated; see Martin ([Ma66a] or [So87, V.4.1]) and Lachlan ([La75] or [Od89, III.9.2]) for Post's simple set, and Jockusch and Soare ([JS73]; cf. also [So87, XII.3.6, 3.7]) for Post's hypersimple set. However, only Gödelnumberings were considered. An explanation for the greater variety which arises when arbitrary numberings of P1 are admitted is provided by the fact that the index set of Gödelnumberings is less complex than the index set of all numberings of P1. The former is Σ1-complete; the latter is Π4-complete.