On the Computation of Necessary and Sufficient Explanations

TL;DR

This paper explores the computation of necessary and sufficient explanations for decisions, introduces efficient methods for certain classes of models, and discusses their applications in explainability and bias detection.

Contribution

It introduces a semantic distinction between necessary and sufficient reasons, derives closed-form solutions for decision trees and graphs, and provides algorithms for enumerating shortest reasons.

Findings

Efficient closed-form complete reasons for decision trees and graphs.

Polynomial-time enumeration of shortest necessary reasons.

Empirical evidence of algorithm efficiency for shortest sufficient reasons.

Abstract

The complete reason behind a decision is a Boolean formula that characterizes why the decision was made. This recently introduced notion has a number of applications, which include generating explanations, detecting decision bias and evaluating counterfactual queries. Prime implicants of the complete reason are known as sufficient reasons for the decision and they correspond to what is known as PI explanations and abductive explanations. In this paper, we refer to the prime implicates of a complete reason as necessary reasons for the decision. We justify this terminology semantically and show that necessary reasons correspond to what is known as contrastive explanations. We also study the computation of complete reasons for multi-class decision trees and graphs with nominal and numeric features for which we derive efficient, closed-form complete reasons. We further investigate the computation of shortest necessary and sufficient reasons for a broad class of complete reasons, which include the derived closed forms and the complete reasons for Sentential Decision Diagrams (SDDs). We provide an algorithm which can enumerate their shortest necessary reasons in output polynomial time. Enumerating shortest sufficient reasons for this class of complete reasons is hard even for a single reason. For this problem, we provide an algorithm that appears to be quite efficient as we show empirically.