Both SAT and (number sign)SAT can represent difficult problems in seemingly dissimilar areas such as planning, verification, and probabilistic inference. Here, we examine an expressive new language, (number sign)-SAT, that generalizes both of these languages. (nubmer sign)-SAT problems require counting the number of satisfiable formulas in a concisely-describable set of existentially quantified, propositional formulas. We characterize the expressiveness and worst-case difficulty of (number sign)-SAT by proving it is complete for the complexity class (PNP 1), and relating this class to more familiar complexity classes. We also experiment with three new general-purpose (number sign)-SAT solvers on a battery of problem distributions including a simple logistics domain. Our experiments show that, despite the formidable worst-case complexity of (PNP 1), many of the instances can be solved efficiently by noticing and exploiting a particular type of frequent structure.
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