Solution Extraction from Long-distance Resolution Proofs

Much effort has been devoted in the past decade to the development of decision procedures for quantified Boolean formulas (QBFs) in order to improve their performance. Besides solving the decision problem of whether a closed QBF is true and returning yes or no, modern QBF solvers can produce clause or cube proofs to certify their answer. These proofs can be used to derive solutions to problems formulated as QBFs. For instance, if the QBF describes a synthesis problem, the solution to this problem can be generated from the proof. Examples for representations of such solutions are control circuits (manifested in Herbrand or Skolem functions) or control strategies. We review two approaches for solution generation below. We focus on false QBFs and clause Q-refutations. For true QBFs, both approaches work in a symmetric fashion. Balabanov and Jiang [1] propose to extract a Herbrand function for each universal (∀) variable from a Q-refutation of a false QBF. These functions are constructed in such a way that replacing each ∀ variable by its Herbrand function in the matrix and removing the prefix yields an unsatisfiable Boolean formula. Unsatisfiability can be checked by a common SAT solver if required. Their algorithm traverses the refutation and constructs a formula for each ∀ variable from the nodes of the refutation where the variable is reduced by universal reduction. The run-time of their extraction algorithm is polynomial in the size of the refutation. Goultiaeva et al. [3] employ a game-theoretic view to QBFs. They present an algorithm which executes a winning strategy for the universal player from the refutation. The existential (∃) player chooses a move, i.e., an assignment of truth values to all ∃ variables in the outermost quantifier block. The algorithm modifies the Q-refutation according to this move and removes the current quantifier block from the prefix. From the resulting refutation, the ∀ player computes her move, i.e., an assignment of truth values to all ∀ variables in the outermost quantifier block which is now universal. Modifications take place as above and the ∃ player continues. The run-time of this algorithm is also polynomial in the size of the refutation. For both methods, their runtime is directly related to the refutation size. It is therefore desirable to have short refutations. Since there are families of QBFs (like the one from Kleine Büning et al. [4] discussed below), for which any Q-refutation is exponential,