Shapley Value Computation in Ontology-Mediated Query Answering

TL;DR

This paper analyzes the computational complexity of calculating the drastic Shapley value in ontology-mediated query answering, establishing a clear dichotomy between tractable and #P-hard cases for various query types.

Contribution

It provides a detailed complexity analysis and a dichotomy result for the drastic Shapley value computation in ontology-mediated query answering, extending understanding of its computational boundaries.

Findings

SVC^{dr} is either in FP or #P-hard for all relevant queries.

Complexity depends on the structure of the ontology and query.

Results connect Shapley value computation with probabilistic query evaluation.

Abstract

The Shapley value was originally introduced in cooperative game theory as a wealth distribution mechanism. It has since found use in knowledge representation and databases for the purpose of assigning scores to formulas and database tuples based upon their contribution to obtaining a query result or inconsistency. The application of the Shapley value outside of its original setting relies upon defining a numeric wealth function that captures the phenomenon of interest. In the case of database queries, recent work has focused on the so-called drastic Shapley value, obtained by translating a Boolean query into a 0/1 function based upon whether the query is satisfied or not. The present paper explores the use of the drastic Shapley value in the context of ontology-mediated query answering (OMQA). We present a detailed complexity analysis of the drastic Shapley value computation (SVC$^{dr}$) problem in the OMQA setting. In particular, we establish a dichotomy result that shows that for every ontology-mediated query (T,q) composed of an ontology T formulated in the description logic $\mathcal{ELHI}_\bot$ and a connected constant-free homomorphism-closed query q the corresponding SVC$^{dr}$ problem is either tractable (in FP) or #P-hard. We further show how 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$^{dr}$ and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.