NONE 20260716095818829 wseas/wseas content-registration-form@crossref.org content-registration-form International Journal of Applied Mathematics Computational Science and Systems Engineering 2766-9823 Moses Adeolu Agoi Lagos State University of Education Lagos NIGERIA https://orcid.org/0000-0002-8910-2876 Emmanuel Taiwo Agoi Loughborough University UNITED KINGDOM Oluwanifemi Opeyemi Agoi Obafemi Awolowo University Osun NIGERIA Samuel Olayiwola Ajaga Lagos State University of Education Lagos NIGERIA The use of Artificial Intelligence (AI) is becoming established as a component of safety-critical engineering systems such as the autonomous transportation systems, energy infrastructure, industrial automation, and structural monitoring. As much as AI models and especially deep neural networks prove to be more accurate in prediction, their black box nature create serious issues when it comes to interpretability and safety guarantees. In safety critical areas explainability is not only a good idea, but a precursor to regulatory compliance, accountability and mitigation of hazards. This paper creates a comprehensive mathematical conceptualization of using interpretability metrics and formal safety verification protocols of AI-based engineering systems. Based on the recent developments in explainable artificial intelligence (XAI), mechanistic interpretability, and formal verification theory, we operationalize interpretability as a measurable functional characterization of model structures and safety verification as meet-in-the-middle of reachable state spaces. Our abstraction of causal circuits and verification of time logics constraints using Shapley-based attribution functions make our integrated optimisation model offer explanation fidelity and safety invariants. The framework illustrates how the measures of interpretability can be integrated in the form of the side-constraints in formal verification pipelines, such that the model decision-making can be transparent and safely proven. Findings suggest that the metrics of coupling explanation coupled with reachability analysis can minimize unsafe decision regions and enhance calibration of the trust. The paper adds a mathematically based architecture that can be able to fill in the gap between heuristic XAI methods and serious engineering safety requirements. It is discussed in implications to aerospace, autonomous systems and industrial control environment as well as computational trade-offs and scalability issues. The model suggested creates a direction towards the certifiable AI systems in which interpretability and safety are not set at cross purposes. 07 16 2026 07 16 2026 87 9 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/232026.2026.8.9 https://wseas.com/journals/amcse/2026/a18amcse-009(2026).pdf Albahri, O. S., Albahri, A. S., Mohammed, K. I., Zaidan, A. A., & Zaidan, B. B. (2025). Intrinsic interpretability: A comprehensive survey of models, evaluation metrics, and engineering applications. Journal of AI Systems, 12(1), 45-82. Alsaleh, M., Madumal, P., & Pinto dos Santos, B. (2025). Explainable artificial intelligence: A systematic review of progress and challenges. Journal of Artificial Intelligence Research, 1-32. https://doi.org/10.1016/j.jair.2025.01.001 Athavale, A., Bartocci, E., Christakis, M., Maffei, M., & Nickovic, D. (2025). Verifying global two-safety properties in neural networks with confidence. In International Conference on Formal Techniques for Distributed Objects, Components, and Systems (pp. 329-351). Springer. https://doi.org/10.1007/978-3-031-65630- 9_17 Bereska, L., & Gavves, E. (2024). Mechanistic interpretability for AI safety: A review. arXiv preprint arXiv:2404.14082, pp. 1-38. https://doi.org/10.48550/arXiv.2404.14082 De Bock, K. W., Coussement, K., & Lessmann, S. (2024). Shapley values for explainable AI: Axiomatic foundations and engineering stability. International Journal of Information Management, 74, 102712. Fan, F. L., Xiong, J., Li, M., & Wang, G. (2021). On interpretability of artificial neural networks: A survey. IEEE Transactions on Radiation and Plasma Medical Sciences, 5(6), 741-760. https://doi.org/10.1109/TRPMS.2021.30664 28 Geiger, A., Ibeling, D., Zur, A., Chaudhary, M., Chauhan, S., Huang, J., Arora, A., Wu, Z., Goodman, N., Potts, C., & Icard, T. (2023). Causal abstraction: A theoretical foundation for mechanistic interpretability. arXiv preprint arXiv:2301.04709. https://doi.org/10.48550/arXiv.2301.04709 International Journal of Approximate Reasoning. (2024). On the failings of Shapley values for explainability. International Journal of Approximate Reasoning, 171, Article 109112. https://doi.org/10.1016/j.ijar.2023.109112 Islam, M., et al. (2022). Explainability of deep neural networks: A review of methods and applications. Neurocomputing. https://doi.org/10.1016/j.neucom.2024.12820 4 ISO/IEC TR 29119-11:2020. (2020). Software and systems engineering: Software testing_ Part 11: Guidelines on the testing of AI-based systems. ISO. Kästner, L., & Crook, B. (2024). Explaining AI through mechanistic interpretability. European Journal for Philosophy of Science, 14(52). https://doi.org/10.1007/s13194-024- 00614-4 Klüver, C., Greisbach, A., Kindermann, M., & Püttmann, B. (2024). A requirements model for AI algorithms in functional safetycritical systems. Security and Safety, 3, 2024020, pp. 1-18. https://doi.org/10.1051/sands/2024020 Kuznietsov, A., Gyevnar, B., Wang, C., & Peters, S. (2024). Explainable AI for safe and trustworthy autonomous driving: A systematic review. arXiv preprint arXiv:2402.10086. https://doi.org/10.48550/arXiv.2402.10086 Linardatos, P., Papastefanopoulos, V., & Kotsiantis, S. (2021). Explainable AI: A review of machine learning interpretability methods. Entropy, 23(1), 18. https://doi.org/10.3390/e23010018 McDermid, J. A., Jia, Y., Porter, Z., & Habli, I. (2021). Artificial intelligence explainability: The technical and ethical dimensions. Philosophical Transactions of the Royal Society A, 379(2207), 20200363, pp. 1-15. https://doi.org/10.1098/rsta.2020.0363 Mersha, M., Lam, K., Wood, J., AlShami, A., & Kalita, J. (2024). Explainable artificial intelligence: A survey of needs, techniques, applications, and future direction. arXiv preprint arXiv:2409.00265, pp. 1-45. https://doi.org/10.48550/arXiv.2409.00265 Molnar, C. (2022). Interpretable machine learning (2nd ed.). Lulu Press. https://christophm.github.io/interpretable-mlbook/ Newcomb, A., & Ochoa, O. (2026). Formal methods for safety-critical machine learning: A systematic literature review. Frontiers in Artificial Intelligence, 9, Article 1749956. https://doi.org/10.3389/frai.2026.1749956 Olah, C., et al. (2018). Feature visualization and mechanistic interpretability. Distill. https://distill.pub/2018/feature-visualization Parameshwaran, A., & Wang, Y. (2025). Scalable and explainable verification of image-based neural network controllers for autonomous vehicles. arXiv preprint arXiv:2501.14009. https://arxiv.org/abs/2501.14009 PMC. (2023). Explainable AI evaluation comparisons: Comparative evaluation of six XAI techniques. National Institutes of Health. https://www.ncbi.nlm.nih.gov/pmc/articles/P MC12647901/ Ribeiro, M. T., Singh, S., & Guestrin, C. (2016). "Why should I trust you?” Explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1135-1144). Vonderhaar, L., Couder, J., Procko, T. T., Lueddeke, E., Cisneros, D., & Ochoa, O. (2026). Verifying machine learning interpretability and explainability requirements through provenance. Software, 5(1), 9. https://doi.org/10.3390/software5010009 Wang, X., & Kee, T. (2025). Integrating Shapley value and least core attribution for robust explainable AI. Buildings, 15(17), 3133. https://doi.org/10.3390/buildings15173133