On the Relative Succinctness of Sentential Decision Diagrams

Abstract: Sentential decision diagrams (SDDs) introduced by Darwiche in 2011 are a promising representation type used in knowledge compilation. The relative succinctness of representation types is an important subject in this area. 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 types in polynomial size are investigated and SDDs are compared with representation types 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 these representation types. 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 a quasipolynomial simulation of SDDs by equivalent unambiguous nondeterministic OBDDs, a nondeterministic variant where there exists exactly one accepting computation for each satisfying input. As a side effect an open problem about the relative succinctness between SDDs and free binary decision diagrams (FBDDs) which are more general than OBDDs is answered.