Relations of Threshold and k-Interval Boolean Functions

Every k-interval Boolean function f can be represented by at most k intervals of integers such that vector x is a truepoint of f if and only if the integer represented by x belongs to one of these k (disjoint) intervals. Since the correspondence of Boolean vectors and integers depends on the order of bits an interval representation is also specified with respect to an order of variables of the represented function. Interval representation can be useful as an efficient representation for special classes of Boolean functions which can be represented by a small number of intervals. In this paper we study inclusion relations of threshold and k-interval Boolean functions. We show that positive 2-interval functions constitute a (proper) subclass of positive threshold functions and that such inclusion does not hold for any k > 2. We also prove that threshold functions do not constitute a subclass of k-interval functions, for any k. Acknowledgements: I would like to thank my supervisor Ondřej Čepek for helping me to improve readability and overall quality of this paper. The research was supported by Czech Science Foundation (GACR) under the contract no. 201/05/H014.