An And-Sum Circuit with Signed Edges That Is More Succinct than SDD

Knowledge compilation is a method of transforming knowledge into a compressed and tractable form for permitting more efficient operations. For Boolean functions, numerous representations have been proposed that enhance succinctness and tractability. In this paper, we introduce a new representation named structured Decomposable And-Sum Circuit (st-DASC), which employs AND and SUM nodes with signed edges, in place of the standard AND and OR nodes with unsigned edges. Notably, incorporating negative signs permits polytime logical negation. By following a knowledge compilation map, we show that st-DASCs are more succinct than Sentential Decision Diagrams (SDDs) while maintaining support for every operation on the knowledge compilation map that SDD supports. Furthermore, st-DASCs are even more succinct than structured d-DNNFs (st-d-DNNFs), which are more succinct than SDDs although they support fewer operations than SDDs. Accordingly, st-DASCs break the traditional trade-off between succinctness and tractability over SDDs and st-d-DNNFs.