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

Let A and B be integers such that A ≤ B. An n-variable interval function is a mapping IN[n : A, B] : {0,1}ⁿ → {0,1}, where IN[n : A, B](X) = 1 iff A ≤ X ≤ 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).