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We investigate the size of first-order rewritings of conjunctive queries overOWL 2 QL ontologies of depth 1 and 2 by means of hypergraph programs computingBoolean functions. Both positive and negative results are obtained. Conjunctivequeries over ontologies of depth 1 have polynomial-size nonrecursive datalogrewritings; tree-shaped queries have polynomial positive existentialrewritings; however, in the worst case, positive existential rewritings canonly be of superpolynomial size. Positive existential and nonrecursive datalogrewritings of queries over ontologies of depth 2 suffer an exponential blowupin the worst case, while first-order rewritings are superpolynomial unless$\text{NP} \subseteq \text{P}/\text{poly}$. We also analyse rewritings oftree-shaped queries over arbitrary ontologies and observe that the queryentailment problem for such queries is fixed-parameter tractable.

On the Succinctness of Query Rewriting over OWL 2 QL Ontologies with Shallow Chases