Transforming matings into natural deduction proofs
A procedure is given for transforming refutation matings into natural deduction proofs. Thus a theorem proving system which establishes the validity of a theorem by the general matings approach can apply this procedure to obtain a comprehensible proof of the theorem without further search. This illuminates the close relationship between matings and proofs, and serves as a step toward a synthesis between apparently quite different approaches to automated theorem proving.
From a refutation mating the system constructs a plan for a theorem, describing appropriate replications of quantifiers, substitutions, and matchings of atoms. Skolem functions play a useful role in refutation matings, but terms involving such functions are replaced by appropriate variables when plans are constructed. Once a plan has been constructed, the system constructs a proof outline, or fragmentary proof, on the basis of the structure of the theorem and general principles for constructing natural deduction proofs. In a proof outline certain lines (planned lines) are justified not by rules of inference, but by plans. The outline is filled in by applying transformation rules, which add additional lines to the proof, justify certain planned lines, and sometimes create new planned lines. The linkage between plans and the proof is maintained by keeping track of the ancestries of wffs and quantifiers in the proof. Using the plans and other information about the proof, the system controls the application of transformation rules. For example, ∀xA(x) is instantiated to A(t) in the proof if the plan requires that t be substituted for a variable corresponding to x. Special problems arise in constructing natural proofs of existentially complex theorems, so they are proved in alternative forms.
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