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Widespread use of artificial intelligence (AI) algorithms and machinelearning (ML) models on the one hand and a number of crucial issues pertainingto them warrant the need for explainable artificial intelligence (XAI). A keyexplainability question is: given this decision was made, what are the inputfeatures which contributed to the decision? Although a range of XAI approachesexist to tackle this problem, most of them have significant limitations.Heuristic XAI approaches suffer from the lack of quality guarantees, and oftentry to approximate Shapley values, which is not the same as explaining whichfeatures contribute to a decision. A recent alternative is so-called formalfeature attribution (FFA), which defines feature importance as the fraction offormal abductive explanations (AXp's) containing the given feature. Thismeasures feature importance from the view of formally reasoning about themodel's behavior. It is challenging to compute FFA using its definition becausethat involves counting AXp's, although one can approximate it. Based on theseresults, this paper makes several contributions. First, it gives compellingevidence that computing FFA is intractable, even if the set of contrastiveformal explanations (CXp's) is provided, by proving that the problem is#P-hard. Second, by using the duality between AXp's and CXp's, it proposes anefficient heuristic to switch from CXp enumeration to AXp enumerationon-the-fly resulting in an adaptive explanation enumeration algorithmeffectively approximating FFA in an anytime fashion. Finally, experimentalresults obtained on a range of widely used datasets demonstrate theeffectiveness of the proposed FFA approximation approach in terms of the errorof FFA approximation as well as the number of explanations computed and theirdiversity given a fixed time limit.

Anytime Approximate Formal Feature Attribution