Nested antichains for WS1S

We propose a novel approach for coping with alternating quantification as the main source of nonelementary complexity of deciding WS1S formulae. Our approach is applicable within the state-of-the-art automata-based WS1S decision procedure implemented e.g. in Mona . The way in which the standard decision procedure processes quantifiers involves determinization, with its worst case exponential complexity, for every quantifier alternation in the prefix of a formula. Our algorithm avoids building the deterministic automata—instead, it constructs only those of their states needed for (dis)proving validity of the formula. It uses a symbolic representation of the states, which have a deeply nested structure stemming from the repeated implicit subset construction, and prunes the search space by a nested subsumption relation, a generalization of the one used by the so-called antichain algorithms for handling nondeterministic automata. We have obtained encouraging experimental results, in some cases outperforming Mona , and some of the other recently proposed approaches, by several orders of magnitude.