Reduced ordered binary decision diagram with implied literals: a new knowledge compilation approach

Yong Lai, Dayou Liu, Sheng-Sheng Wang

Abstract: Reduced ordered binary decision diagram (ROBDD) is one of the most influential knowledge compilation languages. We generalize it by associating some implied literals with each node to propose a new language called ROBDD with implied literals (ROBDD-\(L\)) and show that ROBDD-\(L\) can meet most of the querying requirements involved in the knowledge compilation map. Then, we discuss a kind of subsets of ROBDD-\(L\) called ROBDD-\(L_i\) with precisely \(i\) implied literals \((0\le i\le \infty )\), where ROBDD-\(L_0\) is isomorphic to ROBDD. ROBDD-\(L_i\) has uniqueness over any given linear order of variables. We mainly focus on ROBDD-\(L_\infty \) and demonstrate that: (a) it is a canonical representation on any given variable order; (b) it is the most succinct subset in ROBDD-\(L\) and thus also meets most of the querying requirements; (c) given any logical operation ROBDD supports in polytime, ROBDD-\(L_\infty \) can also support it in time polynomial in the sizes of the equivalent ROBDDs. Moreover, we propose an ROBDD-\(L_i\) compilation algorithm for any \(i\) and an ROBDD-\(L_\infty \) compilation algorithm, and then we implement an ROBDD-\(L\) package called BDDjLu. Our preliminary experimental results indicate that: (a) the compilation results of ROBDD-\(L_\infty \) are significantly smaller than those of ROBDD; (b) the standard d-DNNF compiler c2d and our ROBDD-\(L_\infty \) compiler do not dominate the other, yet ROBDD-\(L_\infty \) has canonicity and supports more querying requirements and relatively efficient logical operations; and (c) the method that first compiles knowledge base into ROBDD-\(L_\infty \) and then converts ROBDD-\(L_\infty \) into ROBDD provides an efficient ROBDD compiler.