Decision Procedures for Guarded Logics

Abstract

An important class of decidable first-order logic fragments are those satisfying a guardedness condition, such as the guarded fragment (GF). Usually, decidability for these logics is closely linked to the tree-like model property - the fact that satisfying models can be taken to have tree-like form. Decision procedures for the guarded fragment based on the tree-like model property are difficult to implement. An alternative approach, based on restricting first-order resolution, has been proposed, and this shows more promise from the point of view of implementation. In this work, we connect the tree-like model property of the guarded fragment with the resolution-based approach. We derive efficient resolution-based rewriting algorithms that solve the Quantifier-Free Query Answering Problem under Guarded Tuple Generating Dependencies (GTGDs) and Disjunctive Guarded Tuple Generating Dependencies (DisGTGDs). The Query Answering Problem for these classes subsumes many cases of GF satisfiability. Our algorithms, in addition to making the connection to the tree-like model property clear, give a natural account of the selection and ordering strategies used by resolution procedures for the guarded fragment. We also believe that our rewriting algorithm for the special case of GTGDs may prove itself valuable in practice as it does not require any Skolemisation step and its theoretical runtime outperforms those of known GF resolution procedures in case of fixed dependencies. Moreover, we show a novel normalisation procedure for the widely used chase procedure in case of (disjunctive) GTGDs, which could be useful for future studies.