In a computation using auxiliary informational inputs one can think of the external resource making itself available in different ways. One way is via an oracle as in Turing reducibility, where information is supplied on demand without any time delay. Alternatively the Scott graph model for lambda calculus suggests a situation where new information, only… (More)
We show that for any computably enumerable (c.e.) degree a and any low n–c.e. degree l (n ≥ 1), if l < a, then there are n–c.e. degrees a 0 , a 1 such that l < a 0 , a 1 < a and a 0 ∨ a 1 = a. In particular, there is no low maximal d.c.e. degree.
Although density fails in the d-recursively enumerable (d-r.e.) degrees, and more generally in the n-r.e. degrees (see ), we show below that the low 2 n-r.e. degrees are dense. This is achieved by combining density and splitting in the manner of Harrington's proof (see ) for the r.e. (=1-r.e.) case. As usual we use low 2-ness to eliminate infinite… (More)
This paper extends Post's programme to finite levels of the Ershov hierarchy of ∆2 sets. Our initial characterisation, in the spirit of Post , of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For… (More)