As deep neural networks are deployed in safety-critical domains such as autonomous driving and medical diagnosis, stakeholders need explanations of model behavior that are not only interpretable but also trustworthy with formal guarantees. Existing XAI methods fall short of this requirement: heuristic attribution techniques (e.g., LIME, Integrated Gradients) highlight influential features for individual predictions but offer no mathematical guarantees about decision boundaries, while formal explanation methods verify robustness properties yet remain untargeted, analyzing the nearest boundary regardless of whether it represents a critical risk. In safety-critical systems, however, not all misclassifications carry equal consequences; confusing a “Stop” sign for a “60 kph” sign is far more dangerous than confusing it with a “No Passing” sign. Practitioners therefore lack a principled way to answer a fundamental safety question: how resilient is a model’s classification against a specific, high-risk alternative? We introduce ViTaX (Verified and Targeted Explanations), a formal XAI framework that addresses this gap by generating targeted semifactual explanations with mathematical guarantees. For a given input (class y) and a user-specified critical alternative (class t), ViTaX performs two key steps: (1) it identifies the minimal feature subset most sensitive to the y → t transition using class-specific sensitivity heuristics, and (2) it applies formal reachability analysis to guarantee that perturbing these features by ε is insufficient to flip the classification to t. This guarantee constitutes a verified semifactual: “even if these critical features change by ε classification y persists against t." We formalize this reasoning through Targeted ε-Robustness, a formal property that certifies whether an identified feature subset remains robust under perturbation toward a specific target class. By unifying semifactual explanations, class-specific targeting, and formal verification, ViTaX is the first method to provide formally guaranteed explanations of a model’s resilience against specific, user-identified alternatives. Our evaluations on image classification (MNIST, GTSRB, EMNIST) and regression (TaxiNet) demonstrate that ViTaX achieves significantly higher fidelity (e.g., over 30% improvement) and minimal explanation cardinality compared to existing methods. These results establish ViTaX as a scalable and trustworthy foundation for verifiable, targeted XAI.

Abstract This manuscript presents the updated version of the Neural Network Verification (NNV) tool. NNV is a formal verification software tool for deep learning models and cyber-physical systems with neural network components. NNV was first introduced as a verification framework for feedforward and convolutional neural networks, as well as for neural network control systems. Since then, numerous works have made significant improvements in the verification of new deep learning models, as well as tackling some of the scalability issues that may arise when verifying complex models. In this new version of NNV, we introduce verification support for multiple deep learning models, including neural ordinary differential equations, semantic segmentation networks and recurrent neural networks, as well as a collection of reachability methods that aim to reduce the computation cost of reachability analysis of complex neural networks. We have also added direct support for standard input verification formats in the community such as VNNLIB (verification properties), and ONNX (neural networks) formats. We present a collection of experiments in which NNV verifies safety and robustness properties of feedforward, convolutional, semantic segmentation and recurrent neural networks, as well as neural ordinary differential equations and neural network control systems. Furthermore, we demonstrate the capabilities of NNV against a commercially available product in a collection of benchmarks from control systems, semantic segmentation, image classification, and time-series data.

This interactive tutorial describes state-of-the-art methods for formally verifying neural networks and their usage within safety-critical cyber-physical systems (CPS). The inclusion of deep learning models in safety-critical applications requires to formally analyze the behavior of the system, including reasoning about the individual components (e.g., controller robustness), and their interactions and effects in the system as a whole. This tutorial begins with a lecture on this emerging research area, followed by demos of these methods implemented in software tools, specifically the Neural Network Verification (NNV) tool. Examples include systems from aerospace, automotive, and beyond.

Abstract This paper introduces robustness verification for semantic segmentation neural networks (in short, semantic segmentation networks [SSNs]), building on and extending recent approaches for robustness verification of image classification neural networks. Despite recent progress in developing verification methods for specifications such as local adversarial robustness in deep neural networks (DNNs) in terms of scalability, precision, and applicability to different network architectures, layers, and activation functions, robustness verification of semantic segmentation has not yet been considered. We address this limitation by developing and applying new robustness analysis methods for several segmentation neural network architectures, specifically by addressing reachability analysis of up-sampling layers, such as transposed convolution and dilated convolution. We consider several definitions of robustness for segmentation, such as the percentage of pixels in the output that can be proven robust under different adversarial perturbations, and a robust variant of intersection-over-union (IoU), the typical performance evaluation measure for segmentation tasks. Our approach is based on a new relaxed reachability method, allowing users to select the percentage of a number of linear programming problems (LPs) to solve when constructing the reachable set, through a relaxation factor percentage. The approach is implemented within NNV, then applied and evaluated on segmentation datasets, such as a multi-digit variant of MNIST known as M2NIST. Thorough experiments show that by using transposed convolution for up-sampling and average-pooling for down-sampling, combined with minimizing the number of ReLU layers in the SSNs, we can obtain SSNs with not only high accuracy (IoU), but also that are more robust to adversarial attacks and amenable to verification. Additionally, using our new relaxed reachability method, we can significantly reduce the verification time for neural networks whose ReLU layers dominate the total analysis time, even in classification tasks.

Abstract Verification has emerged as a means to provide formal guarantees on learning-based systems incorporating neural network before using them in safety-critical applications. This paper proposes a new verification approach for deep neural networks (DNNs) with piecewise linear activation functions using reachability analysis. The core of our approach is a collection of reachability algorithms using star sets (or shortly, stars), an effective symbolic representation of high-dimensional polytopes. The star-based reachability algorithms compute the output reachable sets of a network with a given input set before using them for verification. For a neural network with piecewise linear activation functions, our approach can construct both exact and over-approximate reachable sets of the neural network. To enhance the scalability of our approach, a star set is equipped with an outer-zonotope (a zonotope over-approximation of the star set) to quickly estimate the lower and upper bounds of an input set at a specific neuron to determine if splitting occurs at that neuron. This zonotope pre-filtering step reduces significantly the number of linear programming optimization problems that must be solved in the analysis, and leads to a reduction in computation time, which enhances the scalability of the star set approach. Our reachability algorithms are implemented in a software prototype called the neural network verification tool, and can be applied to problems analyzing the robustness of machine learning methods, such as safety and robustness verification of DNNs. Our experiments show that our approach can achieve runtimes twenty to 1400 times faster than Reluplex, a satisfiability modulo theory-based approach. Our star set approach is also less conservative than other recent zonotope and abstract domain approaches.

This paper presents the Neural Network Verification (NNV) software tool, a set-based verification framework for deep neural networks (DNNs) and learning-enabled cyber-physical systems (CPS). The crux of NNV is a collection of reachability algorithms that make use of a variety of set representations, such as polyhedra, star sets, zonotopes, and abstract-domain representations. NNV supports both exact (sound and complete) and over-approximate (sound) reachability algorithms for verifying safety and robustness properties of feed-forward neural networks (FFNNs) with various activation functions. For learning-enabled CPS, such as closed-loop control systems incorporating neural networks, NNV provides exact and over-approximate reachability analysis schemes for linear plant models and FFNN controllers with piecewise-linear activation functions, such as ReLUs. For similar neural network control systems (NNCS) that instead have nonlinear plant models, NNV supports over-approximate analysis by combining the star set analysis used for FFNN controllers with zonotope-based analysis for nonlinear plant dynamics building on CORA. We evaluate NNV using two real-world case studies: the first is safety verification of ACAS Xu networks, and the second deals with the safety verification of a deep learning-based adaptive cruise control system.

This paper proposes a new forward reachability analysis approach to verify safety of cyber-physical systems (CPS) with reinforcement learning controllers. The foundation of our approach lies on two efficient, exact and over-approximate reachability algorithms for neural network control systems using star sets, which is an efficient representation of polyhedra. Using these algorithms, we determine the initial conditions for which a safety-critical system with a neural network controller is safe by incrementally searching a critical initial condition where the safety of the system cannot be established. Our approach produces tight over-approximation error and it is computationally efficient, which allows the application to practical CPS with learning enable components (LECs). We implement our approach in NNV, a recent verification tool for neural networks and neural network control systems, and evaluate its advantages and applicability by verifying safety of a practical Advanced Emergency Braking System (AEBS) with a reinforcement learning (RL) controller trained using the deep deterministic policy gradient (DDPG) method. The experimental results show that our new reachability algorithms are much less conservative than existing polyhedra-based approaches. We successfully determine the entire region of the initial conditions of the AEBS with the RL controller such that the safety of the system is guaranteed, while a polyhedra-based approach cannot prove the safety properties of the system.