Shapley Value Computation in Ontology-Mediated Query Answering

The Shapley value, originally introduced in cooperative game theory for wealth distribution, has found use in KR and databases for the purpose of assigning scores to formulas and database tuples based upon their contribution to obtaining a query result or inconsistency. In the present paper, we explore the use of Shapley values in ontology-mediated query answering (OMQA) and present a detailed complexity analysis of Shapley value computation (SVC) in the OMQA setting. In particular, we establish a PF/#P-hard dichotomy for SVC for ontology-mediated queries (T,q) composed of an ontology T formulated in the description logic ELHI_\bot and a connected constant-free homomorphism-closed query q. We further show that the #P-hardness side of the dichotomy can be strengthened to cover possibly disconnected queries with constants. Our results exploit recently discovered connections between SVC and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.