Ordered Chaining Calculi for First-Order Theories of Binary Relations

We propose inference systems for binary relations with composition laws of the form S o T ~ U in the context of resolution-type theorem proving. Particulary interesting examples include transitivity, partial orderings, equality and the combination of equality with other transitive relations. Our inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and also prove their compatibility with the usual simplification techniques used in rewrite-based theorem provers. A key to the practicality of chaining techniques is the extent to which so-called variable chainings can be restricted. We demonstrate that rewrite techniques considerably restrict variable chaining, though we also show that they cannot be completely avoided in general. If a binary relation under consideration satisfies additional properties, such as symmetry, further restrictions are possible. In particular, we discuss orderings and partial congruence relations.