Augmenting Ordered Binary Decision Diagrams with Conjunctive Decomposition

This paper augments OBDD with conjunctive decomposition to propose a generalization called OBDD[∧]. By imposing reducedness and the finest ∧-decomposition bounded by integer i (∧î-decomposition) on OBDD[∧], we identify a family of canonical languages called ROBDD[∧î], where ROBDD[∧0̂] is equivalent to ROBDD. We show that the succinctness of ROBDD[∧î] is strictly increasing when i increases. We introduce a new time-efficiency criterion called rapidity which reflects that exponential operations may be preferable if the language can be exponentially more succinct, and show that the rapidity of each operation on ROBDD[∧î] is increasing when i increases; particularly, the rapidity of some operations (e.g., conjoining) is strictly increasing. Finally, our empirical results show that: a) the size of ROBDD[∧î] is normally not larger than that of its equivalent ROBDD[∧ î+1 ]; b) conjoining two ROBDD[∧1̂]s is more efficient than conjoining two ROBDD[∧0̂]s in most cases, where the former is NP-hard but the latter is in P; and c) the space-efficiency of ROBDD[∧∞̂] is comparable with that of d-DNNF and that of another canonical generalization of ROBDD called SDD.