Constraint satisfaction problems in clausal form: Autarkies, deficiency and minimal unsatisfiability by

Abstract. In this paper, we present a new way to preprocess Boolean formulae in Conjunctive Normal Form (CNF). In contrast to most of the current pre-processing techniques, our approach aims at improving the filtering power of the original clauses while producing a small number of additional and relevant clauses. More precisely, an incomplete redundancy check is performed on each original clauses through unit propagation, leading to either a sub-clause or to a new relevant one generated by the clause learning scheme. This preprocessor is empirically compared to the best existing one in terms of size reduction and the ability to improve a state-of-the-art satisfiability solver. 1

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The logical and algorithmic properties of stable conditional independence (CI) as an alternative structural representation of conditional independence information are investigated. We utilize recent results concerning a complete axiomatization of stable conditional independence relative to discrete probability measures to derive perfect model properties of stable conditional independence structures. We show that stable CI can be interpreted as a generalization of Markov networks and establish a connection between sets of stable CI statements and propositional formulas in conjunctive normal form. Consequently, we derive that the implication problem for stable CI is coNP-complete. Finally, we show that Boolean satisfiability (SAT) solvers can be employed to efficiently decide the implication problem and to compute concise, non-redundant representations of stable CI, even for instances involving hundreds of random variables. Key words: conditional independence, graphical models, stable conditional independence, computational complexity, concise representation 1