Querying the Guarded Fragment

Abstract

Evaluating a boolean conjunctive query q over a guarded first-order theory T is equivalent to checking whether (T & not q) is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories T and (unions of) conjunctive queries q that (i) T implies q iff T implies q over finite models, that is, iff q is true in each finite model of T and (ii) determining whether T implies q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded-bisimulation-invariant PTIME.