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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.Keywords complexity theory · decomposable negation normal forms · knowledge compilation · ordered binary decision diagrams · sentential decision diagrams · storage access functions IntroductionKnowledge compilation is an area of research with a long tradition in artificial intelligence (see, e.g., [13]). An input formula is converted into a representation of the Boolean function that the formula defines from which some tasks can (hopefully) be done efficiently. Developing their knowledge compilation map Darwiche and Marquis identified sets of useful queries and transformations in the area of knowledge compilation and compared systematically different representation types w.r.t. their succinctness and efficient support of these operations [17]. One aim of their work was to decide whether representations can be transformed into equivalent ones of another representation type at the cost of increasing the representation size at most polynomially. Here we continue this part of their work. Sentential decision diagrams, or SDDs for short, introduced by Darwiche [16] are a promising representation type for propositional knowledge bases in artificial intelligence. Our main motivation in the paper is to characterize which kind of Boolean functions can be represented by SDDs of small size. Contribution and related workFor a representation type M let P(M) be the set of all Boolean functions representable by M in polynomial size w.r.t. the number of Boolean variables the functions are defined on. We call P(M) a complexity class. Our aim is to characterize the complexity class P(SDD) as precisely as possible. For the formal definitions of the following representation types see Section 2.OBDD....

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On the complexity of the hidden weighted bit function for various BDD models Informatique théorique et applications, tome 33, n o 2 (1999), p. 103-115 © AFCET, 1999, tous droits réservés. L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Theoretical Informaties and Applications