This paper introduces two new compilation languages restricting weak decomposable negation normal form (wDNNF) circuits and integrates them into the knowledge compilation map. Positive (resp. negative) wDNNF circuits restrict wDNNF circuits so that each variable shared among the inputs of a conjunction node can only have positive (resp. negative) occurrences in that subcircuit. Unlike wDNNF circuits, pwDNNF (resp. nwDNNF) circuits satisfy the maximum (resp. minimum) cardinality query. We present a compiler for converting CNF formulae into pwDNNF and nwDNNF circuits by extending Bella - the state-of-the-art compiler for wDNNF circuits. We introduce a new caching scheme, called Cara, that exploits isomorphism. Using that scheme, we show a new compilation method based on copying subcircuits, which may significantly speed up compilations at the expense of increasing circuit sizes. Our experiments demonstrate that nwDNNF circuits are suitable for computing most probable explanations (MPEs) in two-layer Bayesian networks (BNs) with large domains.

This paper integrates weak decomposable negation normal form (wDNNF) circuits, introduced by Akshay et al. in 2018, into the knowledge compilation map. This circuit type generalises decomposable negation normal form (DNNF) circuits in such a way that they allow a restricted form of sharing variables among the inputs of a conjunction node. We show that wDNNF circuits have the same properties as DNNF circuits regarding the queries and transformations presented in the knowledge compilation map, whilst being strictly more succinct than DNNF circuits (that is, they can represent Boolean functions compactly). We also present and evaluate a knowledge compiler, called Bella, for converting CNF formulae into wDNNF circuits. Our experiments demonstrate that wDNNF circuits are suitable for configuration instances.