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.

notcurrently known that the elementary theory of a hyperbolic group is decidable. – HJRW Dec 1 '11 at 21:10