The supplementary material is organized as follows. Before we can prove this theorem, we will require some auxiliary lemmas. This completes the induction argument, and thus we conclude the proof of the lemma. We are now ready to prove Proposition 2. We will use notation We start by explaining the high-level idea of the proof. We now define the problem Minimal-Expected-Clauses .

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models.

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models. Inside formal XAI, one of the most studied cases is that of explaining the choices taken by decision trees, as they are traditionally deemed as one of the most interpretable classes of models. Recent work has focused on studying the computation of sufficient reasons, a kind of explanation in which given a decision tree and an instance, one explains the decision () by providing a subset of the features of such that for any other instance compatible with, it holds that () = (), intuitively meaning that the features in are already enough to fully justify the classification of by. It has been argued, however, that sufficient reasons constitute a restrictive notion of explanation. For such a reason, the community has started to study their probabilistic counterpart, in which one requires that the probability of () = () must be at least some value (0, 1], where is a random instance that is compatible with. Our paper settles the computational complexity of -sufficient-reasons over decision trees, showing that both (1) finding -sufficient-reasons that are minimal in size, and (2) finding -sufficient-reasons that are minimal inclusion-wise, do not admit polynomial-time algorithms (unless PTIME = NP). This is in stark contrast with the deterministic case (= 1) where inclusion-wise minimal sufficient-reasons are easy to compute. By doing this, we answer two open problems originally raised by Izza et al., and extend the hardness of explanations for Boolean circuits presented by Wäldchen et al. to the more restricted case of decision trees. On the positive side, we identify structural restrictions of decision trees that make the problem tractable, and show how SAT solvers might be able to tackle these problems in practical settings.