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Abstract:
We propose inference systems for dealing with transitive relations
in the context of resolution-type theorem proving.
These 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 for transitive relations in general.
If the given relation satisfies additional properties,
such as symmetry, further restrictions are possible.
In particular,
we discuss (partial) equivalence relations and congruence relations.
