We study pseudo-Boolean constraints (PBC) and their special case cardinality constraints (CARD) from the perspective of knowledge representation. To this end, the succinctness of PBC and CARD is compared to that of many standard propositional languages. Moreover, we determine which queries and transformations are feasible in polynomial time when knowledge is represented by PBC or CARD, and which are not (unconditionally or unless P = NP). In particular, the advantages and disadvantages compared to CNF are discussed.

We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNFformulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.

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