{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T14:09:59Z","timestamp":1756994999299,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"19","license":[{"start":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T00:00:00Z","timestamp":1653868800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T00:00:00Z","timestamp":1653868800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Programme d\u2019Ivestissement d\u2019Avenir"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput & Applic"],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s00521-022-07363-6","type":"journal-article","created":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T10:04:24Z","timestamp":1653905064000},"page":"17129-17144","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Coupling algebraic topology theory, formal methods and safety requirements toward a new coverage metric for artificial intelligence models"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0100-9352","authenticated-orcid":false,"given":"Faouzi","family":"Adjed","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8324-0891","authenticated-orcid":false,"given":"Mallek","family":"Mziou-Sallami","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3110-8359","authenticated-orcid":false,"given":"Fr\u00e9d\u00e9ric","family":"Pelliccia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mehdi","family":"Rezzoug","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7946-9685","authenticated-orcid":false,"given":"Lucas","family":"Schott","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Bohn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2831-877X","authenticated-orcid":false,"given":"Yesmina","family":"Jaafra","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,5,30]]},"reference":[{"key":"7363_CR1","unstructured":"Goodfellow IJ, Shlens J, Szegedy C (2014) Explaining and harnessing adversarial examples, arXiv preprint arXiv:1412.6572"},{"key":"7363_CR2","doi-asserted-by":"publisher","first-page":"4566","DOI":"10.1109\/ACCESS.2020.3045078","volume":"9","author":"X Liu","year":"2020","unstructured":"Liu X, Xie L, Wang Y, Zou J, Xiong J, Ying Z, Vasilakos AV (2020) Privacy and security issues in deep learning: a survey. IEEE Access 9:4566","journal-title":"IEEE Access"},{"key":"7363_CR3","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.inffus.2019.12.012","volume":"58","author":"AB Arrieta","year":"2020","unstructured":"Arrieta AB, D\u00edaz-Rodr\u00edguez N, Del Ser J, Bennetot A, Tabik S, Barbado A, Garc\u00eda S, Gil-L\u00f3pez S, Molina D, Benjamins R, Chatila R, Franscisco H (2020) Explainable artificial intelligence (XAI): concepts, taxonomies, opportunities and challenges toward responsible AI. Inf Fusion 58:82","journal-title":"Inf Fusion"},{"key":"7363_CR4","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.inffus.2021.01.008","volume":"71","author":"A Holzinger","year":"2021","unstructured":"Holzinger A, Malle B, Saranti A, Pfeifer B (2021) Towards multi-modal causability with graph neural networks enabling information fusion for explainable AI. Inf Fusion 71:28","journal-title":"Inf Fusion"},{"issue":"3","key":"7363_CR5","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1145\/3236386.3241340","volume":"16","author":"ZC Lipton","year":"2018","unstructured":"Lipton ZC (2018) The mythos of model interpretability: in machine learning, the concept of interpretability is both important and slippery. Queue 16(3):31","journal-title":"Queue"},{"key":"7363_CR6","unstructured":"Urban C, Min\u00e9 A (2021) A review of formal methods applied to machine learning, arXiv preprint arXiv:2104.02466"},{"key":"7363_CR7","doi-asserted-by":"crossref","unstructured":"Katz G, Barrett C, Dill DL, Julian K, Kochenderfer MJ (2017) Reluplex: an efficient SMT solver for verifying deep neural networks. In: International conference on computer aided verification, pp 97\u2013117. Springer","DOI":"10.1007\/978-3-319-63387-9_5"},{"key":"7363_CR8","unstructured":"Sotoudeh M, Thakur A (2019) Correcting deep neural networks with small, generalizing patches. In: Workshop on safety and robustness in decision making"},{"key":"7363_CR9","doi-asserted-by":"crossref","unstructured":"Naseer M, Minhas MF, Khalid F, Hanif MA, Hasan O, Shafique M (2020) Fannet: formal analysis of noise tolerance, training bias and input sensitivity in neural networks. In: 2020 Design, automation & test in europe conference & exhibition (DATE), pp 666\u2013669. IEEE","DOI":"10.23919\/DATE48585.2020.9116247"},{"key":"7363_CR10","doi-asserted-by":"crossref","unstructured":"Li J, Liu J, Yang P, Chen L, Huang X, Zhang L (2019) Analyzing deep neural networks with symbolic propagation: towards higher precision and faster verification. In: International static analysis symposium, pp 296\u2013319. Springer","DOI":"10.1007\/978-3-030-32304-2_15"},{"key":"7363_CR11","doi-asserted-by":"crossref","unstructured":"Huang X, Kwiatkowska M, Wang S, Wu M (2017) Safety verification of deep neural networks. In: International conference on computer aided verification, pp 3\u201329. Springer","DOI":"10.1007\/978-3-319-63387-9_1"},{"key":"7363_CR12","doi-asserted-by":"crossref","unstructured":"Pei K, Cao Y, Yang J, Jana S (2017) Deepxplore: automated whitebox testing of deep learning systems. In: Proceedings of the 26th symposium on operating systems principles, pp 1\u201318. ACM","DOI":"10.1145\/3132747.3132785"},{"key":"7363_CR13","doi-asserted-by":"crossref","unstructured":"Tian Y, Pei K, Jana S, Ray B (2018) Deeptest: automated testing of deep-neural-network-driven autonomous cars. In: Proceedings of the 40th international conference on software engineering, pp 303\u2013314","DOI":"10.1145\/3180155.3180220"},{"key":"7363_CR14","doi-asserted-by":"crossref","unstructured":"Yu J, Fu Y, Zheng Y, Wang Z, Ye X (2019) Test4deep: an effective white-box testing for deep neural networks. In: 2019 IEEE international conference on computational science and engineering (CSE) and IEEE international conference on embedded and ubiquitous computing (EUC), pp 16\u201323. IEEE","DOI":"10.1109\/CSE\/EUC.2019.00013"},{"key":"7363_CR15","doi-asserted-by":"crossref","unstructured":"Cousot P, Cousot R (1977) Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on principles of programming languages, pp 238\u2013252. ACM","DOI":"10.1145\/512950.512973"},{"issue":"2\u20133","key":"7363_CR16","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/0743-1066(92)90030-7","volume":"13","author":"P Cousot","year":"1992","unstructured":"Cousot P, Cousot R (1992) Abstract interpretation and application to logic programs. J Logic Program 13(2\u20133):103","journal-title":"J Logic Program"},{"key":"7363_CR17","unstructured":"Singh G, Gehr T, Mirman M, P\u00fcschel M, Vechev M (2018) Fast and effective robustness certification. In: Advances in neural information processing systems, pp 10825\u201310836"},{"key":"7363_CR18","doi-asserted-by":"crossref","unstructured":"Gehr T, Mirman M, Drachsler-Cohen D, Tsankov P, Chaudhuri S, Vechev M (2018) Ai2: safety and robustness certification of neural networks with abstract interpretation. In: 2018 IEEE symposium on security and privacy (SP), pp 3\u201318. IEEE","DOI":"10.1109\/SP.2018.00058"},{"key":"7363_CR19","unstructured":"Singh G, Gehr T, P\u00fcschel M, Vechev MT (2019) Boosting robustness certification of neural networks. In: ICLR (Poster)"},{"issue":"POPL","key":"7363_CR20","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1145\/3290354","volume":"3","author":"G Singh","year":"2019","unstructured":"Singh G, Gehr T, P\u00fcschel M, Vechev M (2019) An abstract domain for certifying neural networks. Proc ACM Program Lang 3(POPL):41","journal-title":"Proc ACM Program Lang"},{"key":"7363_CR21","unstructured":"Blanchet B (2002) Introduction to abstract interpretation, lecture script"},{"key":"7363_CR22","doi-asserted-by":"crossref","unstructured":"Ghorbal K, Goubault E, Putot S (2009) The zonotope abstract domain Taylor1+. In: International conference on computer aided verification, pp 627\u2013633. Springer","DOI":"10.1007\/978-3-642-02658-4_47"},{"key":"7363_CR23","doi-asserted-by":"crossref","unstructured":"Khalsi R, Mziou-Sallami M, Smati I, Ghorbel F (2022) Contourverifier: a novel system for the robustness evaluation of deepcontour classifiers. In: Proceedings of the 14th international conference on agents and artificial intelligence, vol\u00a02","DOI":"10.5220\/0010994500003116"},{"key":"7363_CR24","doi-asserted-by":"crossref","unstructured":"Mziou-Sallami M, Khedher MI, Trabelsi A, Kerboua-Benlarbi S, Bettebghor D (2019) Safety and robustness of deep neural networks object recognition under generic attacks. In: International conference on neural information processing, pp 274\u2013286. Springer","DOI":"10.1007\/978-3-030-36808-1_30"},{"key":"7363_CR25","doi-asserted-by":"publisher","unstructured":"Mziou-Sallami M, Adjed F (2022) Towards a certification of deep image classifiers against convolutional attacks. In: Proceedings of the 14th international conference on agents and artificial intelligence\u2014volume 2: ICAART, pp 419\u2013428. INSTICC (SciTePress). https:\/\/doi.org\/10.5220\/0010870400003116","DOI":"10.5220\/0010870400003116"},{"key":"7363_CR26","doi-asserted-by":"crossref","unstructured":"Holzinger A (2014) In: Interactive knowledge discovery and data mining in biomedical informatics, pp 331\u2013356. Springer","DOI":"10.1007\/978-3-662-43968-5_19"},{"key":"7363_CR27","doi-asserted-by":"crossref","unstructured":"Edelsbrunner H, Harer J (2010) Computational topology: an introduction. American Mathematical Society","DOI":"10.1090\/mbk\/069"},{"key":"7363_CR28","unstructured":"Khrulkov V, Oseledets IV (2018) Geometry score: a method for comparing generative adversarial networks. CoRR abs\/1802.02664"},{"key":"7363_CR29","unstructured":"Maria C (2014) Algorithms and data structures in computational topology. Ph.D. thesis, Universit\u00e9 Nice Sophia Antipolis"},{"key":"7363_CR30","unstructured":"Sutton RS, Barto AG et al (1998) Introduction to reinforcement learning, vol 135. MIT Press Cambridge"},{"issue":"5","key":"7363_CR31","first-page":"679","volume":"6","author":"R Bellman","year":"1957","unstructured":"Bellman R (1957) A Markovian decision process. J Math Mech 6(5):679","journal-title":"J Math Mech"},{"key":"7363_CR32","doi-asserted-by":"crossref","unstructured":"Adjed F, Pelliccia F, Rezzoug M, Schott L (2021) Certification of deep reinforcement learning with multiple outputs using abstract interpretation and safety critical systems. In: Proceedings of the 31st European safety and reliability conference, pp 3185\u20133191","DOI":"10.3850\/978-981-18-2016-8_572-cd"},{"key":"7363_CR33","unstructured":"Leurent E (2018) An environment for autonomous driving decision-making. https:\/\/github.com\/eleurent\/highway-env"},{"key":"7363_CR34","unstructured":"Schulman J, Wolski F, Dhariwal P, Radford A, Klimov O (2017) Proximal policy optimization algorithms, arXiv preprint arXiv:1707.06347"},{"key":"7363_CR35","doi-asserted-by":"crossref","unstructured":"Zhao C, Tang Y, Sun Q, Vasilakos A.V (2021) Deep direct visual odometry. In: IEEE transactions on intelligent transportation systems","DOI":"10.1109\/TITS.2021.3071886"},{"issue":"4","key":"7363_CR36","doi-asserted-by":"publisher","first-page":"2663","DOI":"10.1109\/JIOT.2019.2942165","volume":"7","author":"J Chen","year":"2019","unstructured":"Chen J, Zhou J, Cao Z, Vasilakos AV, Dong X, Choo KKR (2019) Lightweight privacy-preserving training and evaluation for discretized neural networks. IEEE Internet Things J 7(4):2663","journal-title":"IEEE Internet Things J"},{"key":"7363_CR37","doi-asserted-by":"crossref","unstructured":"Karras T, Laine S, Aila T (2019) A style-based generator architecture for generative adversarial networks. In: Proceedings of the IEEE\/CVF conference on computer vision and pattern recognition, pp 4401\u20134410","DOI":"10.1109\/CVPR.2019.00453"},{"key":"7363_CR38","unstructured":"Wu M, Xiong N, Vasilakos A.V, Leung V.C, Chen C.P (2020) RNN-K: a reinforced newton method for consensus-based distributed optimization and control over multiagent systems. IEEE Trans Cybern"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-022-07363-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-022-07363-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-022-07363-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T13:54:19Z","timestamp":1663854859000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-022-07363-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,30]]},"references-count":38,"journal-issue":{"issue":"19","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["7363"],"URL":"https:\/\/doi.org\/10.1007\/s00521-022-07363-6","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"type":"print","value":"0941-0643"},{"type":"electronic","value":"1433-3058"}],"subject":[],"published":{"date-parts":[[2022,5,30]]},"assertion":[{"value":"30 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing relevant interest to declare about the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}