Decidability and Invariant Classes for Degree Structures

We present a decision procedure for the V3-theory of 3[0,0'], the Turing degrees below 0'. The two main ingredients are a new extension of embeddings result and a strengthening of the initial segments results below 0' of [Lei]. First, given any finite subuppersemilattice U ot 3[0,0'] with top element 0' and an isomorphism type V of a poset extending U consistently with its structure as an usl such that V and U have the same top element and V is an end extension of U — {0'}, we construct an extension of U inside S?[0,0'] isomorphic to V. Second, we obtain an initial segment W of 3[0,0'] which is isomorphic to U {0'} such that W U {0'} is a subusl of 3. The decision procedure follows easily from these results. As a corollary to the V3-decision procedure for 3, we show that no degree a > 0 is definable by any 3V-formula of degree theory. As a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any V or 3-formula. The analysis again uses the decision procedure for theV3-theory of 3. A similar analysis is carried out for the high/low hierarchy using the decision procedure for the V3-theory of 3[0,0']. (A jump class W is a-invariant if <r(a) holds for every a in W.) Introduction. This paper presents some new results dealing with decidability and definability within Th(.S'), the elementary theory of the poset of degrees of unsolvability. In the area of decidability, we show that V3fl T\i(3f [0,0']), theV3theory of the degrees below 0', is decidable. Lachlan [La] has shown that Tfh(2f) is undecidable, and Epstein [E] and Lerman [Lei] have obtained a similar result for Th^^O']). Schmerl (see [Lei]) has pulled both of these undecidability results down to the V3V level. In the other direction, results of Kleene and Post [KP] can be used to show that the 3 -theory of each of these structures is decidable, and Shore [Shi] and Lerman [Lei] have shown that the V3-theory of 2 is decidable. We present a decision procedure for the V3-theory of ^[0,0']. We first prove an extension of embeddings theorem which allows us to start with a finite subuppersemilattice U of ^[0,0'] with top element 0' and an isomorphism type V of a poset extending U consistently with its structure as an usl such that V and U have the same top element and V is an end extension of U — {0'}, and then to Received by the editors April 15, 1987 and, in revised form, August 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03D30; Secondary 03D55.