The Complexity of Abduction for Equality Constraint Languages

Abduction is a form of nonmonotonic reasoning that looks for an explanation for an observed manifestation according to some knowledge base. One form of the abduction problem studied in the literature is the propositional abduction problem parameterized by a structure \Gamma over the two-element domain. In that case, the knowledge base is a set of constraints over \Gamma, the manifestation and explanation are propositional formulas. In this paper, we follow a similar route. Yet, we consider abduction over infinite domain. We study the equality abduction problem parameterized by a relational first-order structure \Gamma over the natural numbers such that every relation in \Gamma is definable by a Boolean combination of equalities, a manifestation is a literal of the form (x = y) or (x != y), and an explanation is a set of such literals. Our main contribution is a complete complexity characterization of the equality abduction problem. We prove that depending on \Gamma, it is \Sigma^P_2-complete, or NP-complete, or in P.