  The class of threshold functions is known to be characterizable by functional
equations or, equivalently, by pairs of relations, which are called relational
constraints. It was shown by Hellerstein that this class cannot be
characterized by a finite number of such objects. In this paper, we investigate
classes of threshold functions which arise as intersections of the class of all
threshold functions with clones of Boolean functions, and provide a complete
classification of such intersections in respect to whether they have finite
characterizations. Moreover, we provide a characterizing set of relational
constraints for each class of threshold functions arising in this way.
