Backdoor Decomposable Monotone Circuits and their Propagation Complete Encodings

We describe a compilation language of backdoor decomposable monotone circuits (BDMCs) which generalizes several concepts appearing in the literature, e.g. DNNFs and backdoor trees. A BDMC sentence is a monotone circuit which satisfies decomposability property (such as in DNNF) in which the inputs (or leaves) are associated with CNF encodings of some functions required to be propagation complete (PC) or at least unit refutation complete (URC). BDMCs are strictly more succinct than both DNNF and backdoor trees. On the other hand, we show that a representation of a boolean function with a BDMC can be compiled into a PC encoding of the same function whose size is polynomial in the size of the input BDMC sentence. As a consequence, BDMCs are equally succinct as PC encodings, however, their structure allows to incorporate parts equivalent to a DNNF. This makes BDMCs suitable for applications, where it is beneficial to combine DNNF with tractable classes of CNF formulas like 2-CNF or renamable Horn formulas.