Canonical Inference for Implicational Systems

Completion is a general paradigm for applying inferences to generate a canonical presentation of a logical theory, or to semi-decide the validity of theorems, or to answer queries. We investigate what canonicity means for implicational systems that are axiomatizations of Moore families – or, equivalently, of propositional Horn theories. We build a correspondence between implicational systems and associative-commutative rewrite systems, give deduction mechanisms for both, and show how their respective inferences correspond. Thus, we exhibit completion procedures designed to generate canonical systems that are “optimal” for forward chaining, to compute minimal models, and to generate canonical systems that are rewrite-optimal. Rewrite-optimality is a new notion of “optimality” for implicational systems, one that takes contraction by simplification into account.