  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).
