Search vs. Symbolic Techniques in Satisfiability Solving Guoqiang Pan ⋆ , Moshe Y. Vardi ⋆ Department of Computer Science, Rice University, Houston, TX gqpan,vardi@cs.rice.edu Abstract. Recent work has shown how to use OBDDs for satisfiability solving. The idea of this ap- proach, which we call symbolic quantifier elimination, is to view an instance of propositional satisfiabil- ity as an existentially quantified propositional formula. Satisfiability solving then amounts to quantifier elimination; once all quantifiers have been eliminated we are left with either 1 or 0. Our goal in this work is to study the effectiveness of symbolic quantifier elimination as an approach to satisfiability solving. To that end, we conduct a direct comparison with the DPLL-based ZChaff, as well as evaluate a variety of optimization techniques for the symbolic approach. In comparing the symbolic approach to ZChaff, we evaluate scalability across a variety of classes of formulas. We find that no approach domi- nates across all classes. While ZChaff dominates for many classes of formulas, the symbolic approach is superior for other classes of formulas. Once we have demonstrated the viability of the symbolic approach, we focus on optimization tech- niques for this approach. We study techniques from constraint satisfaction for finding a good plan for performing the symbolic operations of conjunction and of existential quantification. We also study var- ious variable-ordering heuristics, finding that while no heuristic seems to dominate across all classes of formulas, the maximum-cardinality search heuristic seems to offer the best overall performance. 1 Introduction Propositional-satisfiability solving has been an active area of research through out the last 40 years, starting from the resolution-based algorithm in [19] and the search-based algorithm in [18]. The latter approach, referred to as the DPLL approach, has since been the method of choice for satisfiability solving. In the last ten years, much progress have been made in developing highly optimized DPLL solvers, leading to efficient solvers such as ZChaff [46] and BerkMin [28], all of which use advanced heuristics in choosing variable splitting order, in performing efficient Boolean constraint propagation, and in conflict-driven learning to prune unnecessary search branches. These solvers are so effective that they are used as generic problem solvers, where problems such as bounded model checking [5], planning [32], scheduling [16], and many others are typically solved by reducing them to satisfiability problems. Another successful approach to propositional reasoning is that of decision diagrams, which are used to represent propositional functions. An instance of the approach is that of ordered Boolean decision diagrams (OBDDs) [8], which are used successfully in model checking [10] and planning [13]. The zero-suppressed variant (ZDDs) is used in prime implicants enumeration [36]. A decision-diagram representation also en- ables easy satisfiability checking, which amounts to deciding whether it is different than the empty OBDD [8]. Since decision diagrams usually represent the set of all satisfying truth assignments, they incur a sig- nificant overhead over search techniques that focus on finding a single satisfying assignment [15]. Thus, the only published comparison between search and OBDD techniques [44] used search to enumerate all satis- fying assignments. The conclusion of that comparison is that no approach dominates; for certain classes of formulas search is superior, and for other classes of formulas OBDDs are superior. Recent work has shown how to use OBDDs for satisfiability solving rather for enumeration [39]. The idea of this approach, which we call symbolic quantifier elimination, is to view an instance of proposi- tional satisfiability as an existentially quantified propositional formula. Satisfiability solving then amounts to quantifier elimination; once all quantifiers have been eliminated we are left with either 1 or 0. This en- ables us to apply ideas about existential quantifier elimination from model checking [38] and constraint satisfaction [21]. The focus in [39] is on expected behavior on random instances of 3-SAT rather than on efficiency. In particular, only a minimal effort is made to optimize the approach and no comparison to search methods is reported. Nevertheless, the results in [39] show that OBDD-based algoithms behave quite differently than search-based algorithms, which makes them worthy of further investigation. (Other recent approaches reported using decision diagrams in satisfiability solving [11, 24, 37]. We discuss these works in our concluding remarks). ⋆ Supported in part by NSF grants CCR-9988322, CCR-0124077, CCR-0311326, IIS-9908435, IIS-9978135, EIA- 0086264, ANI-0216467, and by BSF grant 9800096.