论文信息 - Semirecursive sets and positive reducibility

Semirecursive sets and positive reducibility

1. Introduction. In this paper the notions of " semirecursive set" and "positive reducibility" introduced in [8] are studied and applied to problems in recursive function theory. It will be shown that every r.e. set with regressive complement is semirecursive and that every r.e. truth-table degree contains an r.e. semirecursive set. It will follow that there are hypersimple sets A with A x A <mA and that there are sets which are truth-table complete but not p-complete. It will be shown that every degree contains a semirecursive set but that the degrees of immune semirecursive sets are precisely the nonrecursive degrees which are r.e. in 0'. From the latter result it follows at once that every nonrecursive degree which is r.e. in O' contains a hyperimmune set, for immune semirecursive sets are hyperimmune. Finally it will be shown that there is a simple semirecursive set with a nonregressive complement. As McLaughlin has pointed out, the construction also shows that the intersection of an r.e. set with a regressive set need not be regressive.

C. Jockusch

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