Inference for Annotated Logics over Distributive Lattices

The inference rule ✵-resolution was introduced in [18] as a technique for developing an SLD-style query answering procedure for the logic programming subset of annotated logic. This paper explores the properties of ✵-resolution in the general theorem proving setting. In that setting, it is shown to be complete and to admit a linear restriction. Thus ✵-resolution is amenable to depth-first control strategies that require little memory. An ordering restriction is also described and shown to be complete, providing a promising saturation-based procedure for annotated logic. The inference rule essentially requires that the lattice of truth values be ordinary. Earlier investigations left open the question of whether all distributive lattices are ordinary; this is answered in the affirmative here.