A Lower Bound on DNNF Encodings of Pseudo-Boolean Constraints
This work establishes a fundamental theoretical limitation for the succinctness of DNNF encodings of pseudo-Boolean constraints, which is important for researchers and practitioners working on SAT/SMT solving and constraint programming.
This paper investigates the size of DNNF (decomposable negation normal form) encodings for pseudo-Boolean (PB) constraints. It proves that there exist PB-constraints for which all DNNF representations require exponential size, similar to previous findings for OBDDs.
Two major considerations when encoding pseudo-Boolean (PB) constraints into SAT are the size of the encoding and its propagation strength, that is, the guarantee that it has a good behaviour under unit propagation. Several encodings with propagation strength guarantees rely upon prior compilation of the constraints into DNNF (decomposable negation normal form), BDD (binary decision diagram), or some other sub-variants. However it has been shown that there exist PB-constraints whose ordered BDD (OBDD) representations, and thus the inferred CNF encodings, all have exponential size. Since DNNFs are more succinct than OBDDs, preferring encodings via DNNF to avoid size explosion seems a legitimate choice. Yet in this paper, we prove the existence of PB-constraints whose DNNFs all require exponential size.