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IntroductionTwo central areas of the study of computability theory, also called recursion theory, are recursive enumerability and the (Turing) degrees of unsolvability, and in particular, the degrees <0 H . One of the characterizations of the recursively enumerable (r.e.) sets is as those sets A that can be effectively approximated with at most one change in the approximation. We begin by guessing that x is not in A and we may change our mind at most once to put x in A (when it is enumerated into A according to the usual de®nition of an r.e. set). The natural generalization of this property is to allow the approximation to change more often. We know that the sets with degree <0 H can be recursively approximated, but we could not in general recursively ®nd the number of changes made in the approximation for a given number. A set A is called d:r:e: if there are two r.e. sets B, C such that A B À C. We begin by guessing that x is not in A and we change our mind at most twice to put x in A (put it in B) and then may remove x from A (put x in C). We begin with a review of some results related to the questions of cupping (that is, join) and complementation.After Sacks [16] proved the Density Theorem, Shoen®eld [18] conjectured that for all ®nite partial orderings P Í Q with join and least and greatest elements, any embedding of P into the upper semi-lattice R (the Turing degrees of recursively enumerable sets (r.e. degrees)) can be extended to an embedding of Q into the same. In particular, (C1) if a, b are incomparable then they have no in®mum in R; and (C2) given r.e. degrees 0 < b < a there exists an r.e. degree c < a such that a b _ c. Both of these consequences are false. Lachlan [10] and independently Yates [23] refuted Schoen®eld's conjecture by showing that there exist non-zero r.e. degrees a, b such that a^b 0. Lachlan [9] also disproved (C2). Yates, Cooper and Harrington (see Miller [13]) showed that there is a non-zero recursively enumerable degree that is not half of a pair of incomplete recursively enumerable degrees with join 0 H . Harrington and Shelah [8] showed the undecidability of the ®rst order theory of R by using cupping properties of R.The situation is very different for D<0 H . Posner and Robinson [15] ®rst showed that every non-zero element of D<0 H can be cupped to 0 H by an element which is less than 0 H . In further work, Posner [14] showed (through a non-uniform proof) that every degree in D<0 H is one of a pair of degrees with join 0 H and meet 0. We say that two such degrees are complements (in the D 0 2 -degrees). In [21], Slaman and Steel gave a different proof of Posner's theorem in which the complement A for a set X is in fact produced uniformly from X. Sasso [17] showed that there is a minimal degree less than or equal to 0 H incomparable with each intermediate r.e. degree. As a dual of this result, Li [12] and Cooper and Seetapun (in preparation) have proved that there is a degree < 0 H which cups every non-zero r.e. degree to 0 H . Very recently, Slaman and Soare [20] ®nally solve...

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