Core Concepts

The core message of this article is to investigate the tractability of computing expected Shapley-like scores of Boolean functions, which combine the computation of power indices and probabilistic settings, and to apply the results to probabilistic databases.

Abstract

The article focuses on the computation of expected Shapley-like scores of Boolean functions, which generalize the well-known Shapley and Banzhaf values from cooperative game theory to probabilistic settings. The authors first establish a connection between the complexity of computing expected Shapley-like scores and the computation of expected values of Boolean functions. They show that these two problems are interreducible in polynomial time, thus obtaining the same tractability landscape.
The authors then investigate a specific tractable case where Boolean functions are represented as deterministic decomposable circuits, and they design a polynomial-time algorithm for this setting. They present applications of their results to probabilistic databases through database provenance, and provide an effective implementation within the ProvSQL system, which is experimentally validated on a standard benchmark.

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Deeper Inquiries

The results on expected Shapley-like scores can be extended to other classes of Boolean functions beyond deterministic decomposable circuits by considering the tractability and closure properties of the specific class of functions. For classes that are reasonable, closed under conditioning, and closed under conjunctions or disjunctions with fresh variables, similar reductions and algorithms can be applied to compute expected Shapley-like scores efficiently. By analyzing the properties of the class of Boolean functions, such as how they handle conditioning and conjunctions/disjunctions, tailored algorithms can be developed to calculate expected Shapley-like scores within polynomial time complexity. This extension allows for a broader application of the results to a wider range of Boolean function classes, providing insights into the expected contributions of variables in various settings.

The connection between expected Shapley-like scores and expected values of Boolean functions has significant implications beyond probabilistic databases, particularly in the field of explainable machine learning. By understanding the expected contributions of variables in Boolean functions through Shapley-like scores, it becomes possible to interpret the impact of individual variables on the overall outcome of the function. This interpretability is crucial in explainable AI, where the goal is to provide transparent and understandable reasoning behind the decisions made by machine learning models. By leveraging the insights gained from expected Shapley-like scores, explanations can be generated for the predictions or outcomes of complex models, enhancing their transparency and trustworthiness. This connection opens up avenues for applying Shapley-like scores in model interpretation, feature importance analysis, and overall model explainability in the realm of machine learning.

The techniques developed in this work can be adapted to study the complexity of computing other power indices, such as the Johnston, Deegan-Packel, or Holler-Packel indices, in probabilistic settings by leveraging similar principles and computational strategies. By understanding the interplay between power indices and probabilistic settings, it is possible to explore how different power distributions or attributions change in uncertain or probabilistic environments. The computational methods and reductions used for expected Shapley-like scores can be modified and extended to accommodate the computation of other power indices in probabilistic scenarios. This adaptation allows for a deeper analysis of the impact of uncertainty on power distributions and provides insights into how different indices behave in the presence of probabilistic factors. By applying similar techniques to a broader range of power indices, a comprehensive understanding of their behavior in probabilistic settings can be achieved, leading to valuable insights in various decision-making processes and cooperative settings.

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