The Tractability of SHAP-scores over Deterministic and Decomposable Boolean Circuits

Scores based on Shapley values are currently widely used for providing explanations to classification results over machine learning models. A prime example of this corresponds to the influential SHAP-score, a version of the Shapley value in which the contribution of a set $S$ of features from a given entity $\mathbf{e}$ over a model $M$ is defined as the expected value in $M$ of the set of entities $\mathbf{e}'$ that coincide with $\mathbf{e}$ over all features in $S$. 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, also known as tractable probabilistic circuits. Such circuits encompass a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees and Ordered Binary Decision Diagrams (OBDDs). Moreover, we establish the computational limits of the notion of SHAP-score by showing that computing it over a class of Boolean models is always (polynomially) as hard as the model counting problem for this class (under some mild condition). This implies, for instance, that computing the SHAP-score for DNF propositional formulae is a #P-hard problem, and, thus, that determinism is essential for the circuits that we consider.