Complexity of the Abductive Shapley Value

Determine the computational complexity of computing the abductive Shapley value defined for Boolean classifiers, specifically whether computing this value is #P-hard.

Background

The paper introduces an abductive Shapley value whose coalition value is determined by whether the selected literals form an abductive explanation. It proves that the value can be computed in polynomial time when a d-DNNF representation of the dual-rail encoding is available. However, the paper explicitly leaves unresolved whether computing this specific Shapley value is #P-hard in general; the subsequent #P-hardness result concerns counting abductive explanations for OBDDs, not a formal hardness proof for the Shapley value itself.

References

Although we do not know of a formal proof of #P-hardness on this specific Shapley value, the following observation regarding the complexity of counting abductive explanations ensures that the method used in the proof of Theorem \ref{th:drshap} is out of reach even for an $\mathtt{OBDD}$ representation of $f$.

— Solving Hard XAI Queries Based on a Compiled Dual-Rail Encoding  (2609.04931 - Ledaguenel et al., 4 Sep 2026) in Section 4, paragraph “Shapley values”