Expected Shapley-Like Scores of Boolean Functions: Complexity and Applications to Probabilistic Databases

TL;DR

This paper extends Shapley-like scores to probabilistic databases, establishing their computational complexity, designing efficient algorithms for certain cases, and demonstrating practical applications with experimental validation.

Contribution

It introduces a polynomial-time algorithm for computing expected Shapley values for Boolean functions represented as decomposable circuits in probabilistic databases.

Findings

Expected Shapley values and Boolean function expectations are polynomial-time interreducible.

A polynomial-time algorithm is developed for decomposable circuit representations.

Experimental validation confirms the algorithm's feasibility on standard benchmarks.

Abstract

Shapley values, originating in game theory and increasingly prominent in explainable AI, have been proposed to assess the contribution of facts in query answering over databases, along with other similar power indices such as Banzhaf values. In this work we adapt these Shapley-like scores to probabilistic settings, the objective being to compute their expected value. We show that the computations of expected Shapley values and of the expected values of Boolean functions are interreducible in polynomial time, thus obtaining the same tractability landscape. We investigate the specific tractable case where Boolean functions are represented as deterministic decomposable circuits, designing a polynomial-time algorithm for this setting. We present applications to probabilistic databases through database provenance, and an effective implementation of this algorithm within the ProvSQL system, which experimentally validates its feasibility over a standard benchmark.