First-order specifications of programmable data types

We consider first-order specifications together with the restriction to acc ept only programmable algebras as models. We provide a criterion which link s this approach with the "generation principle": all programmable models of any specification SP that meets this criterion are reachable. We also show an example of a specification which does not satisfy the criterion and adm its a programmable yet nonreachable model. Moreover, a general method of sh owing the existence of programmable but nonreachable models for a class of first-order specifications is given.