The class FDSet of while-programs equipped with finite sets as internal data type is considered in the context of generalized computability over abstract first order structures. This class is proved to be semi-universal, i.e. to be able to compute every computable function in every infinite finitely-generated structure. However, for this class the halting problem on finite structures is decidable within LinearSpace complexity (on the cardinality of structure), and it is proved that the dynamic logic of the class FDSet describes exactly the class LinearSpace computable global predicates on the class of all one-generated finite structures of a given signature. Some other classes of data complexity are also described in the paper in similar terms. Due to the semi-universality, this class FDSet is a good basis for developing generalized complexity theory.
