A discrete chain of degrees of index sets

Abstract

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

Precomplete Numberings

“…The finite levels of the difference hierarchy have some nice alternative descriptions in terms of the operations from the previous section. In the next theorem, item (1) follows from results of [61], while item (2) follows from results of [26] and the obvious equality…”

Fine hierarchies and m-reducibilities in theoretical computer science

“…[56,109,110,85,94]). Similar questions are also interesting for the more general case of ν-index k-partitions which are k-partitions of the form c • ν where c : S → k (in [110] they are called generalized index sets).…”

Discrete 𝜔-sequences of index sets

“…Moreover, it easily follows from the results in [3] The terminology and notation is that of [6]. Lemma 0.7 (Lemma 13 of [3]). If 8A < K x 8B and 0£ A, then 8A <8B.…”

“…That discrete «-sequences exist was proved in [3]; it was shown there that if \Z \ > 0 is the sequence of index sets of nonempty finite classes of finite sets (classified in [4] and, independently, in [2]), then \Z \ >Q is a discrete <y-sequence of index sets. Moreover, it easily follows from the results in [3] The terminology and notation is that of [6].…”