A b s t r a c t. We show that if T is the smallest discriminator clone on a set A, then the first order theory of finite powers of a finite algebra A with a distinguished subset closed under T is decidable. If A is a primal algebra and C is any discriminator clone on A, then the first order theory of finite algebras from V(A) with a distinguished subset closed under C is decidable. In particular, the first order theory of algebras from V(A) with a distinguished subalgebra is decidable.