Herbrand Property, Finite Quasi-Herbrand Models, and a Chandra-Merlin Theorem for Quantified Conjunctive Queries

A structure enjoys the Herbrand property if, whenever it satisfies an equality between some terms, these terms are unifiable. On such structures the expressive power of equalities becomes trivial, as their semantic satisfiability is reduced to a purely syntactic check. In this work, we introduce the notion of Herbrand property and develop it in a finite model-theoretic perspective. We provide, indeed, a canonical realization of the new concept by what we call quasi-Herbrand models and observe that, in stark contrast with the naive implementation of the property via standard Herbrand models, their universe can be finite even in presence of functions in the vocabulary. We exploit this feature to decide and collapse the general and finite version of the satisfiability and entailment problems for previously unsettled fragments of first-order logic. We take advantage of the Herbrand property also to establish novel and tight complexity results for the aforementioned decision questions. In particular, we show that the finite containment problem for quantified conjunctive queries is NPTIME-complete, tightening along two dimensions the known 3EXPTIME upper bound for the general version of the problem (Chen, Madelaine, and Martin, LICS'08). We finally present an alternative view on this result by generalizing to such queries the classic characterization of conjunctive query containment via polynomial-time verifiable homomorphisms (Chandra and Merlin, STOC'77).