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Building Herbrand Models for Sets of Guarded Clauses
Abstract
The guarded fragment of rst order logic, dened in [1], is interesting because many modal logics can be translated into it. Guarded clauses are a generalisation of clausal forms of guarded formulas, and sets of such clauses are decidable by ordered resolution [9]. We show that it is possible to transform a set S of guarded clauses (without equality) into a set of general Horn clauses G which has exactly one well-supported model M, and such that M is a model for S. Then G can be seen as a representation of M, since it is possible to evaluate ground atoms in M using G. By techniques presented in [4], G can further be transformed into a set of so-called primitive guarded clauses which represents M. An interpretation represented in this way allows the evaluation of guarded clauses and should be easy to understand for human users. The principle of our method is the processing of a set which is saturated under ordered resolution and factorisation. In order to determine a greate...
Given two inconsistent formulæ, a (reverse) interpolant is a formula implied by one, inconsistent with the other, and only containing symbols they share. Interpolation finds application in program analysis, verification, and synthesis, for example, towards invariant generation. An interpolation system takes a refutation of the inconsistent formulæ and extracts an interpolant by building it inductively from partial interpolants. Known interpolation systems for ground proofs use colors to track symbols. We show by examples that the color-based approach cannot handle non-ground refutations by resolution and paramodulation/superposition. We present a two-stage approach that works by tracking literals, computes a provisional interpolant, which may contain non-shared symbols, and applies lifting to replace non-shared constants by quantified variables. We obtain an interpolation system for non-ground refutations, and we prove that it is complete, if the only non-shared symbols in provisional interpolants are constants.
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