Decidable models of small theories

Many counterexamples are known in the class of small theories due toGoncharov [3] and Millar [5]. The prime model of a decidable small theory is not necessarily decidable. The saturated model of a hereditarily decidable small theory is not necessarily decidable. A homogeneous model with uniformly decidable type spectra is not necessarily decidable. In this paper, I consider the question of what model theoretic properties are sufficient for the existence of such counterexamples. I introduce a subclass of the class of small theories, which I call AL theories, show the absence of Goncharov–Millar counterexamples in this class, and isolate amodel theoretic property that implies the existence of such anomalies among computable models.