Turing Definability in the Ershov Hierarchy

We obtain the first nontrivial d.c.e. Turing approximation to the class of computably enumerable (c.e.) degrees. This depends on the following extension of the splitting theorem for the d.c.e. degrees: For any d.c.e. degree a, any c.e. degree b, if b < a, then there are d.c.e. degrees x0,x1 such that b < x0,x1 < a and a = x0 ∨ x1. The construction is unusual in that it is incompatible with upper cone avoidance. A basic and intractable problem of computability theory, closely related to other longstanding questions such as Downey’s conjecture, is: At what levels of the Ershov hierarchy does Turing definability of given lower levels of the hierarchy occur? In particular, are the computably enumerable (c.e.) degrees E definable within the class of d.c.e. (= 2-c.e.) degrees E 2? An important step towards a positive answer to this latter question is the establishment of natural d.c.e. Turing approximations to E . (A Turing approximation to a class S of degrees is a Turing definable class A for which A ⊆ S or S ⊆ A.) Arslanov [1985] showed that the class of cuppable d.c.e. degrees is a d.c.e. Turing approximation to E − {0}. However, by Lachlan’s observation that every nonzero d.c.e. degree has a nonzero c.e. predecessor, this class turns out to be E 2 − {0}. We provide below a first nontrivial d.c.e. Turing approximation to the c.e. degrees, by showing that every c.e. a has the property that each d.c.e. b > a is splittable in the d.c.e. cone above a. The nontriviality follows immediately from the fact that, for instance, there is a d.c.e. degree d maximal in E 2−{0′}, by Cooper, Harrington, Lachlan, Lempp and Soare [1991]. ∗This author was partially supported by an EPSRC Research Grant, “Turing Definability”, no. GR/M 91419, and an INTAS-RFBR Research Grant, “Computability and Models”, no. 97-0139. †This author was partially supported by an EPSRC Research Grant, “Turing Definability”, No. GR/M 91419 (UK) and by NSF Grant No. 69973048 and NSF Major Grant No. 19931020 ( P. R. CHINA). 1991 Mathematics Subject Classification. Primary 03D25, 03D30; Secondary 03D35.