A Complexity Assessment for Queries Involving Sufficient and Necessary Causes

In this work, we revisit a recently proposed multi-valued semantics for logic programs where each true atom in a stable model is associated with a set of expressions (or causal justifications) involving rule labels. For positive programs, these causal justifications correspond to the possible alternative proofs of the atom that further satisfy some kind of minimality or lack of redundancy. This information can be queried for different purposes such as debugging, program design, diagnosis or causal explanation. Unfortunately, in the worst case, the number of causal justifications for an atom can be exponential with respect to the program size, so that computing the complete causal model may become intractable in the general case. However, we may instead just be interested in querying whether some particular set of rules are involved in the atom derivation, either as a sufficient cause (they provide one of the alternative proofs) or as a necessary cause (they are mandatorily used in all proofs). In this paper, we formally define sufficient and necessary causation for this setting and provide precise complexity characterizations of the associated decision problems, showing that they remain within the first two levels of the polynomial hierarchy.