The IFF proof procedure for abductive logic programming

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In this paper, we outline a proof procedure which combines reasoning with defined predicates together with reasoning with undefined, abducible, predicates. Defined predicates are defined in if-and-only-if form. Abducible predicates are constrained by means of integrity constraints. Given an initial query, the task of the proof procedure is to construct a definition of the abducible predicates and a substitution for the variables in the query, such that both the resulting instance of the query and the integrity constraints are implied by the extended set of definitions. The iff proof procedure can be regarded as a hybrid of the proof procedure of Console et al. and the SLDNFA procedure of Denecker and De Schreye. It consists of a number of inference rules which, starting from the initial query, rewrite a formula into an equivalent formula. These rules are: 1) unfolding, which replaces an atom by its definition; 2) propagation, which resolves an atom with an implication; 3) splitting, which uses distributivity to represent a goal as a disjunction of conjunctions; 4) case analysis for an equality X = t in the conditions of an implication, which considers the two cases X = t and X ≠ t; 5) factoring of two abducible atoms, which considers the two cases, where the atoms are identical and where they are different, 6) rewrite rules for equality, which simulate the unification algorithm; and 7) logical simplifications, such as A ∧ false ↔ false. The proof procedure is both sound and complete relative to the three-valued completion semantics. These soundness and completeness results improve previous results obtained for other proof procedures.