Bounding minimal pairs

Extract

A minimal pair of recursively enumerable (r.e.) degrees is a pair of degrees a, b of nonrecursive r.e. sets with the property that if c ≤ a and c ≤ b then c = 0. Lachlan [2] and Yates [4] independently proved the existence of minimal pairs. It was natural to ask whether for an arbitrary nonzero r.e. degree c there is a minimal pair a, b with a ≤ c and b ≤ c. In 1971 Lachlan and Ladner proved that a minimal pair below c cannot be obtained in a uniformly effective way from c for r.e. c ≠ 0. but the result was never published. More recently Cooper [1] showed that if c is r.e. and c′ = 0″ then there is a minimal pair below c.

In this paper we prove two results:

Theorem 1. There exists a nonzero r.e. degree with no minimal pair below it.

Theorem 2. There exists a nonzero r.e. degree c such that, if d is r.e. and 0 < d ≤ c, then there is a minimal pair below d.

The second theorem is a straightforward variation on the original minimal pair construction, but the proof of the first theorem has some novel features. After some preliminaries in §1, the first theorem is proved in §2 and the second in §3.

I am grateful to Richard Ladner who collaborated with me during the first phase of work on this paper as witnessed by our joint abstract [3]. The many discussions we had about the construction required in Theorem 1 were of great help to me.