Are there standard examples of stable theories that are undecidable?

Question

What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality? For example, is there a natural example of an undecidable first order theory with nice stability properties?

Many standard examples, such as ACF and hyperbolic groups, seem to decidable. What about differentially closed fields ? Zariski geometries? What about theories obtained via Hrushovski fusion?

I should add that there artificially constructed examples of non-decidable stable theories: say add infinitely many algebraic types coding an undecidable set.

Answer 1

Let X be your favorite undecidable set of primes. In a language with one unary function symbol, say f^p(x) = x iff p is in X and also that there is a unique cycle of length p. Add that f is a bijection. Now the prime model contains cycles of exactly the lengths in X and the theory is undecidable. But it is categorical in all uncountable power and therefore omega stable.

Answer 2

I will refrain from discussing your general question, but since you specifically ask about differential fields, I can give the answer there. The theory of differentially closed fields of characteristic zero is decidable (lets say finitely many commuting derivations). There is in fact more that one can say.

Decidable theories have recursively presentable models. You can say more for differential fields. A recursive differential field has a recursively presentable differential closure. In model theoretic terms, the prime models over recursively presented structures are recursively presentable. This result is due to Harrington. I am not sure of the reference, as I have only read some notes on his proof.