Relational Characterization of Multilevel Threshold Functions

It is well-known that the set of Boolean threshold functions can be characterized by pairs of relations called relational constraints. In this paper, we extend this characterization to encompass multilevel threshold functions using the framework of relational constraints. Specifically, we define a countably infinite set of pairs of relations that succinctly captures the set of multilevel threshold functions. For each pair of relations, our approach ensures that when a multilevel threshold function is applied to rows of any matrix, with its columns drawn from the first relation, the resulting output corresponds to an element from the second relation.