Watching Clauses in Quantified Boolean Formulae

The introduction of watched literals[1], a lazy data structure for satisfiability (SAT) search algorithms, has resulted in great improvements in the run-time of SAT solvers. Watched literals keeps track of two literals remaining in a clause so as to detect when a clause becomes unit or empty. Watched literals is non-trivial to implement in QBF search. Quantified Boolean Formulae (QBFs) are SAT formulae with some variables universally quantified. This changes the semantics of unit and false clauses. The issue of watching literals in QBF is addressed in [2]. In this paper, I show that the use of lazy data structures need not be restricted to literals in clauses. In SAT, the detection of pure literals and deleted variables appears to be unimportant[3], and so watched literals has so far been the only implemented lazy data structure. In search for QBF satisfiability, this is not the case; the detection of universal pure literals in particular is critical. This is because a universal pure literal is set false, and so removes the universal from clauses. These clauses could then lead to further unit propagation. To detect a pure literal, one must know the number of literals of each sign of a variable that exist in the problem. An easy way in which to do this is to keep a list of which clauses a variable occurs in an which sign the literal of the variable has in those clauses. If the positive and negative occurences are kept separate, detecting pure literals becomes trivial. The watched method improves upon this by keeping one positive and one negative watched “c-literal” per variable. If either is ever removed and no other can be found to take its place, the variable is detected as having a pure literal. In an experimental comparison, the watched clause method outperformed the standard method by up to 8825 times on some problems, and never performed worse. The watched clause method is not restricted to QBF and implementation in SAT may prove the effectiveness of pure literals in SAT.