Craig Interpolation with Clausal First-Order Tableaux

Abstract

We show how variations of range-restriction and also the Horn property can be passed from inputs to outputs of Craig interpolation in first-order logic. The proof system is clausal tableaux, which stems from first-order ATP. Our results are induced by a restriction of the clausal tableau structure, which can be achieved in general by a proof transformation, also if the source proof is by resolution/paramodulation. Primarily addressed applications are query synthesis and reformulation with interpolation. Our methodical approach combines operations on proof structures with the immediate perspective of feasible implementation through incorporating highly optimized first-order provers.

Facets of the PIE Environment for Proving, Interpolating and Eliminating on the Basis of First-Order Logic

“…The implementation of the computation of Craig interpolants in PIE operates by a novel adaption of Smullyan's interpolation method [45,18] to clausal tableaux [57]. Suitable clausal tableaux can be constructed by the Prolog-based prover CM that is included in PIE .…”

“…The Skolem constant sk1 is converted to a quantified variable in a postprocessing operation. For a description of the interpolant extraction procedure, see [57]. Aside of the shown representation of quantified first-order formulas by Prolog ground terms, the system also supports a representation of clausal formulas as list of lists of terms (logic literals), with variables represented by Prolog variables.…”

“…Hypertableaux, either obtained from a hypertableau prover or obtained from a clausal tableau prover like CM by restructuring the tableau seem interesting as basis for interpolant extraction in query reformulation, as they allow to ensure that the interpolants are range restricted. Some related preliminary results are in[57].…”

PIE -- Proving, Interpolating and Eliminating on the Basis of First-Order Logic

“…Main applications in computer science are in verification [35] and query reformulation, based on its relationship to definability and construction of definientia in terms of a given vocabulary [43,4,3]. The PIE system supports the computation of Craig interpolants H from a closed clausal tableau that represents a proof that F entails G with a novel adaption of Smullyan's interpolation method [41,15] for clausal tableaux [51]. Suitable clausal tableaux can be constructed by the CM prover.…”

“…Hypertableaux, either obtained from a hypertableau prover or obtained from a clausal tableau prover like CM by restructuring the tableau seem interesting as basis for interpolant extraction in query reformulation, as they allow to ensure that the interpolants are range restricted. Some related preliminary results are in[51].…”