  Many counterexamples are known in the class of small theories due to
Goncharov and Millar. 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 questions 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 a model theoretic property that
implies the existence of such anomalies among computable models.
