Concisely Explaining the Doubt: Minimum-Size Abductive Explanations for Linear Models with a Reject Option

Trustworthiness in artificial intelligence depends not only on what a model decides, but also on how it handles and explains cases in which a reliable decision cannot be made. In critical domains such as healthcare and finance, a reject option allows the model to abstain when evidence is insufficient, making it essential to explain why an instance is rejected in order to support informed human intervention. In these settings, explanations must not only be interpretable, but also faithful to the underlying model and computationally efficient enough to support real-time decision making. Abductive explanations guarantee fidelity, but their exact computation is known to be NP-hard for many classes of models, limiting their practical applicability. Computing \textbf{minimum-size} abductive explanations is an even more challenging problem, as it requires reasoning not only about fidelity but also about optimality. Prior work has addressed this challenge in restricted settings, including log-linear-time algorithms for computing minimum-size abductive explanations in linear models without rejection, as well as a polynomial-time method based on linear programming for computing abductive explanations, without guarantees of minimum size, for linear models with a reject option. In this work, we bridge these lines of research by computing minimum-size abductive explanations for linear models with a reject option. For accepted instances, we adapt the log-linear algorithm to efficiently compute optimal explanations. For rejected instances, we formulate a 0-1 integer linear programming problem that characterizes minimum-size abductive explanations of rejection. Although this formulation is NP-hard in theory, our experimental results show that it is consistently more efficient in practice than the linear-programming-based approach that does not guarantee minimum-size explanations.

Generalizing Logic-based Explanations for Machine Learning Classifiers via Optimization

Machine learning models support decision-making, yet the reasons behind their predictions are opaque. Clear and reliable explanations help users make informed decisions and avoid blindly trusting model outputs. However, many existing explanation methods fail to guarantee correctness. Logic-based approaches ensure correctness but often offer overly constrained explanations, limiting coverage. Recent work addresses this by incrementally expanding explanations while maintaining correctness. This process is performed separately for each feature, adjusting both its upper and lower bounds. However, this approach faces a trade-off: smaller increments incur high computational costs, whereas larger ones may lead to explanations covering fewer instances. To overcome this, we propose two novel methods. Onestep builds upon this prior work, generating explanations in a single step for each feature and each bound, eliminating the overhead of an iterative process. \textit{Twostep} takes a gradual approach, improving coverage. Experimental results show that Twostep significantly increases explanation coverage (by up to 72.60\% on average across datasets) compared to Onestep and, consequently, to prior work.

Bound Propagation meets Constraint Simplification: Improving Logic-based XAI for Neural Networks

Logic-based methods for explaining neural network decisions offer formal guarantees of correctness and non-redundancy, but they often suffer from high computational costs, especially for large networks. In this work, we improve the efficiency of such methods by combining bound propagation with constraint simplification. These simplifications, derived from the propagation, tighten neuron bounds and eliminate unnecessary binary variables, making the explanation process more efficient. Our experiments suggest that combining these techniques reduces explanation time by up to 89.26\%, particularly for larger neural networks.

Enhancing Framingham Cardiovascular Risk Score Transparency through Logic-Based XAI

Cardiovascular disease (CVD) remains one of the leading global health challenges, accounting for more than 19 million deaths worldwide. To address this, several tools that aim to predict CVD risk and support clinical decision making have been developed. In particular, the Framingham Risk Score (FRS) is one of the most widely used and recommended worldwide. However, it does not explain why a patient was assigned to a particular risk category nor how it can be reduced. Due to this lack of transparency, we present a logical explainer for the FRS. Based on first-order logic and explainable artificial intelligence (XAI) fundaments, the explainer is capable of identifying a minimal set of patient attributes that are sufficient to explain a given risk classification. Our explainer also produces actionable scenarios that illustrate which modifiable variables would reduce a patient's risk category. We evaluated all possible input combinations of the FRS (over 22,000 samples) and tested them with our explainer, successfully identifying important risk factors and suggesting focused interventions for each case. The results may improve clinician trust and facilitate a wider implementation of CVD risk assessment by converting opaque scores into transparent and prescriptive insights, particularly in areas with restricted access to specialists.

Reliable XAI Explanations in Sudden Cardiac Death Prediction for Chagas Cardiomyopathy

Sudden cardiac death (SCD) is unpredictable, and its prediction in Chagas cardiomyopathy (CC) remains a significant challenge, especially in patients not classified as high risk. While AI and machine learning models improve risk stratification, their adoption is hindered by a lack of transparency, as they are often perceived as \textit{black boxes} with unclear decision-making processes. Some approaches apply heuristic explanations without correctness guarantees, leading to mistakes in the decision-making process. To address this, we apply a logic-based explainability method with correctness guarantees to the problem of SCD prediction in CC. This explainability method, applied to an AI classifier with over 95\% accuracy and recall, demonstrated strong predictive performance and 100\% explanation fidelity. When compared to state-of-the-art heuristic methods, it showed superior consistency and robustness. This approach enhances clinical trust, facilitates the integration of AI-driven tools into practice, and promotes large-scale deployment, particularly in endemic regions where it is most needed.

Slice and Explain: Logic-Based Explanations for Neural Networks through Domain Slicing

Neural networks (NNs) are pervasive across various domains but often lack interpretability. To address the growing need for explanations, logic-based approaches have been proposed to explain predictions made by NNs, offering correctness guarantees. However, scalability remains a concern in these methods. This paper proposes an approach leveraging domain slicing to facilitate explanation generation for NNs. By reducing the complexity of logical constraints through slicing, we decrease explanation time by up to 40\% less time, as indicated through comparative experiments. Our findings highlight the efficacy of domain slicing in enhancing explanation efficiency for NNs.

Comparing Neural Network Encodings for Logic-based Explainability

Providing explanations for the outputs of artificial neural networks (ANNs) is crucial in many contexts, such as critical systems, data protection laws and handling adversarial examples. Logic-based methods can offer explanations with correctness guarantees, but face scalability challenges. Due to these issues, it is necessary to compare different encodings of ANNs into logical constraints, which are used in logic-based explainability. This work compares two encodings of ANNs: one has been used in the literature to provide explanations, while the other will be adapted for our context of explainability. Additionally, the second encoding uses fewer variables and constraints, thus, potentially enhancing efficiency. Experiments showed similar running times for computing explanations, but the adapted encoding performed up to 18\% better in building logical constraints and up to 16\% better in overall time.

An incremental MaxSAT-based model to learn balanced rules

The increasing advancements in the field of machine learning have led to the development of numerous applications that effectively address a wide range of problems with accurate predictions. However, in certain cases, accuracy alone may not be sufficient. Many real-world problems also demand explanations and interpretability behind the predictions. One of the most popular interpretable models that are classification rules. This work aims to propose an incremental model for learning interpretable and balanced rules based on MaxSAT, called IMLIB. This new model was based on two other approaches, one based on SAT and the other on MaxSAT. The one based on SAT limits the size of each generated rule, making it possible to balance them. We suggest that such a set of rules seem more natural to be understood compared to a mixture of large and small rules. The approach based on MaxSAT, called IMLI, presents a technique to increase performance that involves learning a set of rules by incrementally applying the model in a dataset. Finally, IMLIB and IMLI are compared using diverse databases. IMLIB obtained results comparable to IMLI in terms of accuracy, generating more balanced rules with smaller sizes.

Logic-based Explanations for Linear Support Vector Classifiers with Reject Option

Support Vector Classifier (SVC) is a well-known Machine Learning (ML) model for linear classification problems. It can be used in conjunction with a reject option strategy to reject instances that are hard to correctly classify and delegate them to a specialist. This further increases the confidence of the model. Given this, obtaining an explanation of the cause of rejection is important to not blindly trust the obtained results. While most of the related work has developed means to give such explanations for machine learning models, to the best of our knowledge none have done so for when reject option is present. We propose a logic-based approach with formal guarantees on the correctness and minimality of explanations for linear SVCs with reject option. We evaluate our approach by comparing it to Anchors, which is a heuristic algorithm for generating explanations. Obtained results show that our proposed method gives shorter explanations with reduced time cost.