Abstract Neural networks (NNs) are the core of AI-based technologies. However, the degree of reliability in performing the task is an open problem. The explainability of a central task of NNs, classification, is of immense importance. While at the rise of AI-based reasoning, explainability of the NN classification has mostly been done using statistical methods, nowadays, a more reliable trend of formal logic-based methods is gaining popularity. The advantage of the formal approach is that it gives strict and provable guarantees of the classification. Formal methods is a mature field that has delivered a number of efficient computational solutions already applied in the analysis of software and hardware systems. Formal explainability methods naturally have the ability to reuse existing techniques and tools for a newly emerging field of formal explainability of NN classification. This paper surveys existing efforts to compute explanations of neural network classification based on logical abductive reasoning. The abduction approach is crucial for generalizing the results, capturing the underlying behavior of the classifier. We present the existing techniques as instances of a general formalization that allows contrasting them against each other. In addition, we discuss the issue of the quality of explanations, focusing on their key metrics and factors. As an illustrative example, the paper also presents a practical framework, SpEXplAIn , which automatically computes Space Explanations, the most general abduction-based explanations for classifying NNs with provable guarantees of the behavior of the network in continuous areas of the input feature space. The tool leverages an SMT solver compatible with a range of flexible Craig interpolation algorithms and unsatisfiable core generation, and is applicable to a wide range of applications.

We present a novel algorithm for parallel solving of SMT problems based on a partitioning process that divides the original problem into a tree structure in an iterative way. By enabling node revisiting, the new method addresses the problem of partitioning divergence found in prior approaches that frequently leads to longer runtimes compared to sequential results. The resulting algorithm is highly flexible, offers a combination of partitioning, portfolio solving, and clause sharing, allows the use of various partitioning functions, and scales gracefully with the available resources. We implemented the new approach in the tool SMTS on top of the efficient sequential SMT solver OpenSMT . Our experimental results demonstrate a substantial improvement over OpenSMT in logics QF_LRA and QF_LIA even when the partitioning approach utilizes just a single solver. Notably, SMTS has consistently dominated several divisions of the annual competition of parallel SMT solvers.

Abstract We present a novel logic-based concept called Space Explanations for classifying neural networks that gives provable guarantees of the behavior of the network in continuous areas of the input feature space. To automatically generate space explanations, we leverage a range of flexible Craig interpolation algorithms and unsatisfiable core generation. Based on real-life case studies, ranging from small to medium to large size, we demonstrate that the generated explanations are more meaningful than those computed by state-of-the-art.