On the limitations of ordered representations of functions

Abstract

We demonstrate the limitations of various ordered representations that have been considered in the literature for symbolic model checking including BDDs [3], +-BMDs [6], HDDs [15], MTBDDs [13] and EVBDDs [25]. We introduce a lower bound technique that applies to a broad spectrum of such functional representations. Using an abstraction that encompasses all these representations, we apply this technique to show exponential size bounds for a wide range of integer and boolean functions that arise in symbolic model checking in the definition and implicit exploration of the state spaces. We give the first examples of integer functions including integer division, remainder, high/low-order words of multiplication, square root and reciprocal that require exponential size in all these representations. Finally, we show that there is a simple regular language that requires exponential size to be represented by any +-BMD, even though BDDs can represent any regular language in linear size.