The Complexity of Logic-Based Abduction

Thomas Eiter, Georg Gottlob

Abstract: Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logic- based abduction. Candidates for abductive explanations are usually subjected to minimality criteria such as subset-minimality, minimal cardinality, minimal weight, or minimality under prioritization of individual hypotheses. This paper presents a comprehensive complexity analysis of relevant problems related to abduction on propositional theories. They show that the different variations of abduction provide a rich collection of natural problems populating all major complexity classes between P and Σ P3 , Π P3 in the refined polynomial hierarchy. More precisely, besides polynomial, NP-complete and co-NP-complete abduction problems, abduction tasks that are complete for the classes Δ Pi , Δ Pi [O(logn), Σ Pi , and Π Pi , for i=2,3, are identified.