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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 $\mathcal{C}$-BDMC sentence is a monotone circuitwhich satisfies decomposability property (such as in DNNF) in which the inputs(or leaves) are associated with CNF encodings from a given base class$\mathcal{C}$. We consider the class of propagation complete (PC) encodings asa base class and we show that PC-BDMCs are polynomially equivalent to PCencodings. Additionally, we use this to determine the properties of PC-BDMCsand PC encodings with respect to the knowledge compilation map including thelist of efficient operations on the languages.

Backdoor Decomposable Monotone Circuits and their Propagation Complete Encodings