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Shapley Value Computation in Ontology-Mediated Query Answering

Meghyn Bienvenu, Diego Figueira, Pierre Lafourcade

TL;DR

A FP / #P-hard dichotomy for SVC for SVC for ontology-mediated queries (T, q) composed of an ontology T is formulated in the description logic ELHIbot and a connected constant-free homomorphism-closed query q is established.

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.

Shapley Value Computation in Ontology-Mediated Query Answering

TL;DR

A FP / #P-hard dichotomy for SVC for SVC for ontology-mediated queries (T, q) composed of an ontology T is formulated in the description logic ELHIbot and a connected constant-free homomorphism-closed query q is established.

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) 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 and a connected constant-free homomorphism-closed query q the corresponding SVC 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 and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.
Paper Structure (24 sections, 14 theorems, 32 equations, 7 figures)

This paper contains 24 sections, 14 theorems, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Databases
  4. Queries
  5. Description Logic Knowledge Bases
  6. Querying Description Logic KBs
  7. Canonical Models
  8. Complexity
  9. Shapley Value: Definition & Basic Results
  10. Definition of Shapley Value
  11. Applying the Shapley Value to Databases
  12. Drastic Shapley Value Computation in the Database Setting
  13. Hardness Result for Reachability
  14. Drastic Shapley Value Computation in the Ontology Setting: Definitions and First Intractability Results
  15. Axiom Entailment
  16. ...and 9 more sections

Key Result

proposition 1

We have slightly modified the formulation to suit our setting, dropping mention of ABox signatures and functional roles, not considered in this work, and rephrasing in terms of Boolean queries with constants, rather than queries with answer variables. Let $Q=\withT{q}$ be a "Boolean" "OMQ" from $(\m

Figures (7)

  • Figure 1: Illustration of the graph $G_i$.
  • Figure 2: An example "KB", with data and knowledge about a recipe from escoffierGuideCulinaireAidememoire1903. The arrows represent "role assertions" and labels on top of boxes (e.g. Meat) represent "concept assertions".
  • Figure 3: Variants of the graph $G$ for the "reduction". Adapted from livshitsShapleyValueTuples2021.
  • Figure 4: Encoding $\intro*\PG$ of $G$ in $\P$. The "endogenous" "assertions" are indicated by thick lines.
  • Figure 5: Illustration of why $\Right(a_\chi)$ contains $\Below(a_\lambda)$ and $\Right(a_\lambda)$.
  • ...and 2 more figures

Theorems & Definitions (35)

  • proof
  • proposition 1: bienvenuFirstOrderRewritabilityContainment2020
  • theorem 1: bienvenuOntologyMediatedQueryAnswering2015
  • proof
  • remark 1
  • theorem 2: ourpods24
  • proposition 2
  • proof
  • proposition 3
  • proof
  • ...and 25 more