Automatic models of first order theories

Khoussainov and Nerode (2008) [14] posed various open questions on model-theoretic properties of automatic structures. In this work we answer some of these questions by showing the following results: (1) There is an uncountably categorical but not countably categorical theory for which only the prime model is automatic; (2) There are complete theories with exactly 3,4,5,… countable models, respectively, and every countable model is automatic; (3) There is a complete theory for which exactly 2 models have an automatic presentation; (4) If LOGSPACE=P then there is an uncountably categorical but not countably categorical theory for which the prime model does not have an automatic presentation but all the other countable models are automatic; (5) There is a complete theory with countably many countable models for which the saturated model has an automatic presentation but the prime model does not have one.