Craig Interpolation and Access Interpolation with Clausal First-Order Tableaux
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We show methods to extract Craig-Lyndon interpolants and access interpolants from clausal first-order tableaux as produced by automated first-order theorem provers based on model elimination, the connection method, the hyper tableau calculus and instance-based methods in general. Smullyan introduced an elegant method for interpolant extraction from "non-clausal" first-order tableaux. We transfer this to clausal tableaux where quantifier handling is based on prenexing and Skolemization. A lifting technique leads from ground interpolants of Herbrand expansions of Skolemized input formulas to quantified interpolants of the original input formulas. This is similar to a known interpolant lifting by Huang but based more straightforwardly on Herbrand's theorem instead of the auxiliary notion of relational interpolant. Access interpolation is a recent form of interpolation for formulas with relativized quantifiers targeted at applications in query reformulation and specified in the constructive framework of Smullyan's general tableaux. We transfer this here to clausal tableaux. Relativized quantification upon subformulas seems incompatible with lifting techniques that only introduce a global quantifier prefix. We thus follow a different approach for access interpolation: A structure preserving clausification leads to clausal ground tableaux that can be computed by automated first-order provers and, in a postprocessing step, can be restructured such that in essence the interpolant extraction from Smullyan's tableaux becomes applicable.
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