A Complete Classification of the Complexity of Propositional Abduction

Abduction is the process of explaining a given query with respect to some background knowledge. For instance, p is an explanation for the query q given the knowledge $p\rightarrow q$. This problem is well known to have many applications, particularly in artificial intelligence (AI), and has been widely studied from both an AI and a complexity-theoretic point of view. In this paper we completely classify the complexity of propositional abduction in Schaefer's famous framework. We consider the case where knowledge bases are taken from a class of formulas in generalized conjunctive normal form. This means that the propositional formulas considered are conjunctions of constraints taken from a fixed finite language. We show that according to the properties of this language, deciding whether at least one explanation exists is either polynomial, NP-complete, or $\Sigma_2 {\mathrm{P}}$-complete. Our results are stated for a query consisting of a single, positive literal and for assumption-based solutions, i.e., the solutions must be formed upon a distinguished subset of the variables that is part of the input. We show, however, that our results can be interpreted "dually" for negative queries, and thus also for unrestricted (positive or negative) queries.