The Tractability of SHAP-Score-Based Explanations over Deterministic and Decomposable Boolean Circuits

TL;DR

This paper proves that SHAP-score explanations can be computed efficiently over deterministic and decomposable Boolean circuits, extending previous results and establishing computational limits for these explanations.

Contribution

It demonstrates polynomial-time computability of SHAP-scores over a broad class of Boolean circuits and clarifies the importance of circuit properties for tractability.

Findings

SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits.

Computing SHAP-score is #P-hard if either determinism or decomposability is absent.

The results extend to various Boolean circuit classes, including decision trees and decision diagrams.

Abstract

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard).