Translating Pseudo-Boolean Constraints into CNF

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Abstract

A Pseudo-Boolean constraint (PB-constraint) is a generalization of a clause. A PB-constraint is an inequality (equality) on a linear combination of Boolean literals \(\left(\sum_{i=1}^n a_il_i\ OP\ b\right)\) where a 1, ⋯ a n and b are constant integers, l 1, ⋯ ,l n are literals and OP is a comparison operator. The left-hand side of a PB-constraint under assignment \(\mathcal{A}\) is equal to the sum of the coefficients whose corresponding literals are mapped to true by \(\mathcal{A}\) . This kind of constraints has been widely used in expressing NP-complete problems. Several approaches have been proposed to translate a PB-constraint to CNF, [3], [2].