On the Relative Succinctness of Sentential Decision Diagrams

Abstract

Sentential decision diagrams (SDDs) introduced by Darwiche in 2011 are a promising representation language for propositional knowledge bases. The relative succinctness of representation languages is an important subject in knowledge compilation. The aim of the paper is to identify which kind of Boolean functions can be represented by SDDs of small size with respect to the number of variables the functions are defined on. For this reason the sets of Boolean functions representable by different representation languages in polynomial size are investigated and SDDs are compared with representation languages from the classical knowledge compilation map of Darwiche and Marquis. Ordered binary decision diagrams (OBDDs) which are a popular data structure for Boolean functions are one of them. SDDs are more general than OBDDs by definition but only recently, a Boolean function was presented with polynomial SDD size but exponential OBDD size. This result is strengthened in several ways. The main result is that a function can be represented by SDDs of small size if the function and its negation have small restricted nondeterministic OBDD representations. Moreover, for important Boolean functions called storage access function polynomial-size SDDs are presented. As a side effect an open problem about the relative succinctness between SDDs and free binary decision diagrams which are more general than OBDDs is answered.