Type structure complexity and decidability

We prove that for every countable homogeneous model C6 such that the set of recursive types of Th(d) is E?, C, is decidable iff the set of types realized in CT is a E? set of recursive types. As a corollary to a lemma, we show that if a complete theory T has a recursively saturated model that is decidable in the degree of T, then T has a prime model. In this paper all models mentioned will be assumed countable. If 6f is homogeneous and realizes either no nonprincipal types [1] or all recursive types [2] then 6 is decidable iff the set of types realized by 6f is an r.e. set of recursive types. An examination of the proofs involved leads naturally to the conjecture that the techniques can be combined to prove (*) for those 6f that are simply homogeneous. Unfortunately this is false in general [5]. However, if the structure of recursive types of the theory of 6f is not pathological, then (*) can be proved for those 6f which are simply homogeneous. Specifically, the principal result of this paper is to prove that if 6f is homogeneous and the set of recursive types of the theory of 6f is Y , then 6f is decidable iff the set of types realized by 6f is a Y 2 set of recursive types. Since every complete theory's set of recursive types is fl0, the result is the best possible, in light of [5]. Notations and conventions. All types in this paper are assumed complete. A specific effective first order language L is assumed fixed, as well as an effective enumeration {a1 I i < c of all formulas of the language. An n-type F is recursive if the set {i I ai E r(x0. . . xn-I)} is recursive. {f l i < () is an effective enumeration of all partial recursive functions y: X -3 2. An index e for a recursive type F is a natural number such that Me is the characteristic function for F relative to {ai I i < WI). A set of recursive types is Y:1 (Y2) if there is a Y:1 (E ) set of indices for the types in that set. We will say {F1 I i < co is an effective enumeration of types if there is some recursive f such that f(i) is an index for F1, i < c. FS will denote the first s formulas (order determined by index) of r n {ai I i < }o .k 0= if k 0, and 70 if k = 1. {ci I i < () will be distinct constant symbols not in L and {' I i < WI an effective enumeration of all sentences in L U {cl I i < w) such that each sentence occurs Received by the editors March 14, 1980. 1980 Mathematics Subject Classification. Primary 03F50; Secondary 03H1 3. 'The preparation of this paper was partially supported by Grant NSF-MCS-7900824. ? 1982 American Mathematical Society 0002-9947/81/0000-1054/$03.CO-