On the Number of Products to Represent Interval Functions by SOPs with Four-Valued Variables

Let A and B be integers such that A less than or equal to B. An n-variable interval function IN[n:A, B] is a mapping from{0,1}^n to {0,1}, where IN[n:A, B](X)=1 iff X is in the interval [A, B]. Such function is useful for packet classification in the internet, network intrusion detection system, etc. This paper considers the number of products to represent interval functions by sum-of-products expressions with two-valued and four-valued variables. It shows that to represent any interval function of n variables, an SOP with two-valued variables requires up to 2(n-2) products, while an SOP with four-valued variables requires at most n-1 products. These bounds are useful to estimate the size of a content addressable memory (CAM).