Augmenting Clause Learning with Implied Literals

There exist various approaches in SAT solvers that aim at extending inference based on unit propagation. For instance, probing [5] simply applies unit propagation of literals at the root node in order to detect failed literals [3] or to populate literal implication lists. The latter information can then, for instance, be used to shrink clauses by hidden literal elimination (e.g., if a → b then (a ∨ b ∨ c) can be reduced to (b ∨ c); cf. [4]). Here we propose to strengthen clause learning by dynamically inferring lit-erals that the newly learned clause entails. We say that a literal l is an implied literal for a clause C if all literals of C entail l. For instance, if a → d, ¬b → d, and c → d, then (a ∨ ¬b ∨ c) entails d. While this insight has already been exploited in several methods (e.g., variations of hyper binary resolution and hidden literal elimination), we apply it to clause learning: when the SAT solver derives a new conflict clause c, we check if the literals in c imply a single or multiple literals which can then be propagated as new unit literals. In order to employ this technique we first need to generate implication lists L(l) = UnitPropagation(l) for each literal l. This is done at the root node of the search tree before the solving process starts and then periodically during search. During this computation, we also add not yet existing binary clauses corresponding to ∀l ∈ L(p) : ¬l → ¬p. As one might expect, we detect failed literals as well and add and propagate their negations as new unit literals; we do the same for all literals in the intersection of L(p) and L(¬p) for all p ∈ F. Once the implied literal lists for each literal are computed, we iterate over the clauses in the original theory F and propagate all implied literals as new unit clauses. Ideally we would like to augment these lists with new implications whenever new clauses have been learned by the solver. However, this operation can be computationally expensive and we must control how frequently it is performed. While learned clauses are usually 'forgotten' over time, we hold on to the implication lists and only extend them, when possible. Note that this can enable inference that might not be explicitly captured in the current clausal theory. Whenever …