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Computability of Homogeneous Models

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable (CD) theory T in the spirit of Vaught [1961] combined with computability methods for (Turing) degrees d ≤ 0′. First we recast older results by Goncharov, Peretyatkin, and Millar in a more modern framework which we an apply. Then we survey recent results by Lange [ta1] which generalize the older results and which include positive results on when a certain homogeneous model A of T has an isomorphic copy of a given Turing degree. We then survey Lange [ta2] for negative results about when A does not, generalizing negative results by Goncharov, Peretyat’kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare [ta] about