On Narrowing, Refutation Proofs and Constraints

We develop a proof technique for dealing with narrowing and refutational theorem proving in a uniform way, clarifying the exact relationship between the existing results in both elds and allowing us to obtain several new results. Reenements of narrowing (basic, LSE, etc.) are instances of the technique, but are also deened here for arbitrary (possibly ordering and/or equality constrained or not yet convergent or saturated) Horn clauses, and shown compatible with simpliication and other redundancy notions. By narrowing modulo equational theories like AC, compact representations of solutions, expressed by AC-equality constraints , can be obtained. Computing AC-uniiers is only needed at the end if one wants to "uncompress" such a constraint into its (doubly exponentially many) concrete substitutions.