{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T10:30:26Z","timestamp":1787135426932,"version":"3.56.0"},"publisher-location":"Cham","reference-count":50,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783031999901","type":"print"},{"value":"9783031999918","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:00:00Z","timestamp":1761609600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:00:00Z","timestamp":1761609600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"DOI":"10.1007\/978-3-031-99991-8_10","type":"book-chapter","created":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T05:36:01Z","timestamp":1761543361000},"page":"203-220","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Abstraction-Based Proof Production in\u00a0Formal Verification of\u00a0Neural Networks (Extended Abstract)"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2309-3505","authenticated-orcid":false,"given":"Yizhak Yisrael","family":"Elboher","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-6505-6900","authenticated-orcid":false,"given":"Omri","family":"Isac","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5292-801X","authenticated-orcid":false,"given":"Guy","family":"Katz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4556-8308","authenticated-orcid":false,"given":"Tobias","family":"Ladner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5077-144X","authenticated-orcid":false,"given":"Haoze","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,10,28]]},"reference":[{"key":"10_CR1","doi-asserted-by":"crossref","unstructured":"Althoff, M.: An introduction to CORA 2015. In: Proceedings of 1st and 2nd International Workshop on Applied Verification for Continuous and Hybrid Systems (ARCH), pp. 120\u2013151 (2015)","DOI":"10.29007\/zbkv"},{"key":"10_CR2","doi-asserted-by":"crossref","unstructured":"Arnab, A., Dehghani, M., Heigold, G., Sun, C., Lu\u010di\u0107, M., Schmid, C.: ViViT: a video vision transformer. In: Proceedings of International Conference on Computer Vision (ICCV), pp. 6816\u20136826 (2021)","DOI":"10.1109\/ICCV48922.2021.00676"},{"key":"10_CR3","doi-asserted-by":"crossref","unstructured":"Ashok, P., Hashemi, V., K\u0159et\u00ednsk\u1ef3, J., Mohr, S.: DeepAbstract: neural network abstraction for accelerating verification. In: Proceedings of 18th International Symposium on Automated Technology for Verification and Analysis (ATVA), pp. 92\u2013107 (2020)","DOI":"10.1007\/978-3-030-59152-6_5"},{"key":"10_CR4","doi-asserted-by":"crossref","unstructured":"Bak, S.: nnenum: verification of ReLU neural networks with optimized abstraction refinement. In: Proceedings of 13th NASA Formal Methods Symposium (NFM), pp. 19\u201336 (2021)","DOI":"10.1007\/978-3-030-76384-8_2"},{"key":"10_CR5","doi-asserted-by":"crossref","unstructured":"Barbosa, H., et al.: Flexible proof production in an industrial-strength SMT solver. In: Proceedings of 11th International Joint Conference on Automated Reasoning (IJCAR), pp. 15\u201335 (2022)","DOI":"10.1007\/978-3-031-10769-6_3"},{"key":"10_CR6","unstructured":"Barrett, C., de\u00a0Moura, L., Fontaine, P.: Proofs in satisfiability modulo theories. In: All about Proofs, Proofs for All, pp. 23\u201344. College Publications (2015)"},{"key":"10_CR7","doi-asserted-by":"crossref","unstructured":"Brix, C., M\u00fcller, M., Bak, S., Johnson, T., Liu, C.: First three years of the international verification of neural networks competition (VNN-COMP). In: International Journal on Software Tools for Technology Transfer (STTT), pp. 1\u201311 (2023)","DOI":"10.1007\/s10009-023-00703-4"},{"key":"10_CR8","unstructured":"Chv\u00e1tal, V.: Linear Programming. Macmillan (1983)"},{"key":"10_CR9","doi-asserted-by":"crossref","unstructured":"Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Proceedings of 12th International Conference on Computer Aided Verification (CAV), pp. 154\u2013169 (2000)","DOI":"10.1007\/10722167_15"},{"key":"10_CR10","doi-asserted-by":"crossref","unstructured":"Clarke, E., Grumberg, O., Long, D.: Model checking and abstraction. ACM Trans. Programm. Lang. Syst. (TOPLAS) 1512\u20131542 (1994)","DOI":"10.1145\/186025.186051"},{"key":"10_CR11","doi-asserted-by":"crossref","unstructured":"Cohen, E., Elboher, Y.Y., Barrett, C., Katz, G.: Tighter abstract queries in neural network verification. In: Proceedings of of 24th International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR), pp. 124\u2013143 (2023)","DOI":"10.29007\/3mk7"},{"key":"10_CR12","doi-asserted-by":"crossref","unstructured":"Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of 4th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL), pp. 238\u2013252 (1977)","DOI":"10.1145\/512950.512973"},{"key":"10_CR13","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/978-3-319-68167-2_19","volume-title":"Automated Technology for Verification and Analysis","author":"R Ehlers","year":"2017","unstructured":"Ehlers, R.: Formal verification of piece-wise linear feed-forward neural networks. In: D\u2019Souza, D., Narayan Kumar, K. (eds.) ATVA 2017. LNCS, vol. 10482, pp. 269\u2013286. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-68167-2_19"},{"key":"10_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/978-3-030-53288-8_3","volume-title":"Computer Aided Verification","author":"YY Elboher","year":"2020","unstructured":"Elboher, Y.Y., Gottschlich, J., Katz, G.: An abstraction-based framework for neural network verification. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12224, pp. 43\u201365. Springer, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-53288-8_3"},{"key":"10_CR15","doi-asserted-by":"crossref","unstructured":"Elboher, Y.Y., Cohen, E., Katz, G.: Neural network verification using residual reasoning. In: Proceedings of 20th International Conference on Software Engineering and Formal Methods (SEFM), pp. 173\u2013189 (2022)","DOI":"10.1007\/978-3-031-17108-6_11"},{"key":"10_CR16","unstructured":"Elsaleh, R., Katz, G.: DelBugV: delta-debugging neural network verifiers. In: Proceedings of 23rd International Conference Formal Methods in Computer-Aided Design (FMCAD), pp. 34\u201343 (2023)"},{"key":"10_CR17","doi-asserted-by":"crossref","unstructured":"Gehr, T., Mirman, M., Drachsler-Cohen, D., Tsankov, E., Chaudhuri, S., Vechev, M.: AI2: safety and robustness certification of neural networks with abstract interpretation. In: Proceedings of 39th IEEE Symposium on Security and Privacy (S &P), pp. 3\u201318 (2018)","DOI":"10.1109\/SP.2018.00058"},{"key":"10_CR18","unstructured":"Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press Cambridge (2016)"},{"key":"10_CR19","unstructured":"Gowal, S., et al.: On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models (2019). Technical Report. https:\/\/arxiv.org\/abs\/1810.12715"},{"key":"10_CR20","doi-asserted-by":"crossref","unstructured":"Griggio, A., Roveri, M., Tonetta, S.: Certifying Proofs for SAT-Based Model Checking. Formal Methods in System Design (FMSD), pp. 178\u2013210 (2021)","DOI":"10.1007\/s10703-021-00369-1"},{"key":"10_CR21","unstructured":"Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (2024). https:\/\/www.gurobi.com"},{"key":"10_CR22","doi-asserted-by":"crossref","unstructured":"Henzinger, T., Jhala, R., Majumdar, R., McMillan, K.: Abstractions from proofs. In: Proceedings of 31st ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL), p. 232\u2013244 (2004)","DOI":"10.1145\/964001.964021"},{"key":"10_CR23","unstructured":"Isac, O., Barrett, C., Zhang, M., Katz, G.: Neural network verification with proof production. In: Proceedings of 22nd International Conference on Formal Methods in Computer-Aided Design (FMCAD), pp. 38\u201348 (2022)"},{"key":"10_CR24","unstructured":"Isac, O., Refaeli, I., Wu, H., Barrett, C., Katz, G.: Proof-Driven Clause Learning in Neural Network Verification (2025). Technical Report. http:\/\/arxiv.org\/abs\/2503.12083"},{"key":"10_CR25","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/978-3-030-88806-0_9","volume-title":"Static Analysis","author":"K Jia","year":"2021","unstructured":"Jia, K., Rinard, M.: Exploiting verified neural networks via floating point numerical error. In: Dr\u0103goi, C., Mukherjee, S., Namjoshi, K. (eds.) SAS 2021. LNCS, vol. 12913, pp. 191\u2013205. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-88806-0_9"},{"key":"10_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/978-3-319-63387-9_5","volume-title":"Computer Aided Verification","author":"G Katz","year":"2017","unstructured":"Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.: Reluplex: an efficient SMT solver for verifying deep neural networks. In: Majumdar, R., Kun\u010dak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 97\u2013117. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-63387-9_5"},{"key":"10_CR27","doi-asserted-by":"crossref","unstructured":"Katz, G., Barrett, C., Dill, D., Julian, K., Kochenderfer, M.: Reluplex: a Calculus for Reasoning about Deep Neural Networks. Formal Methods in System Design (FMSD) (2021)","DOI":"10.1007\/s10703-021-00363-7"},{"key":"10_CR28","doi-asserted-by":"crossref","unstructured":"Kochdumper, N., Schilling, C., Althoff, M., Bak, S.: Open- and closed-loop neural network verification using polynomial zonotopes. In: Proceedings of 15th NASA Formal Methods Symposium (NFM), pp. 16\u201336 (2023)","DOI":"10.1007\/978-3-031-33170-1_2"},{"key":"10_CR29","unstructured":"Krizhevsky, A., Sutskever, I., Hinton, G.: ImageNet classification with deep convolutional neural networks. In: Proceedings of Advances in Neural Information Processing Systems (NeuRIPS) (2012)"},{"key":"10_CR30","doi-asserted-by":"crossref","unstructured":"Ladner, T., Althoff, M.: Automatic abstraction refinement in neural network verification using sensitivity analysis. In: Proceedings of 26th ACM International Conference on Hybrid Systems: Computation and Control (HSCC), pp. 1\u201313 (2023)","DOI":"10.1145\/3575870.3587129"},{"key":"10_CR31","unstructured":"Ladner, T., Althoff, M.: Fully Automatic Neural Network Reduction for Formal Verification (2023). Technical Report. http:\/\/arxiv.org\/abs\/2305.01932"},{"key":"10_CR32","doi-asserted-by":"crossref","unstructured":"LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 436\u2013444 (2015)","DOI":"10.1038\/nature14539"},{"key":"10_CR33","doi-asserted-by":"crossref","unstructured":"Lipton, Z.: The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery. Queue 31\u201357 (2018)","DOI":"10.1145\/3236386.3241340"},{"key":"10_CR34","doi-asserted-by":"crossref","unstructured":"Liu, C., Arnon, T., Lazarus, C., Strong, C., Barrett, C., Kochenderfer, M.: Algorithms for verifying deep neural networks. Found. Trends Optimiz. 244\u2013404 (2021)","DOI":"10.1561\/2400000035"},{"key":"10_CR35","doi-asserted-by":"crossref","unstructured":"Liu, J., Xing, Y., Shi, X., Song, F., Xu, Z., Ming, Z.: Abstraction and refinement: towards scalable and exact verification of neural networks. ACM Trans. Softw. Eng. Methodol. (TOSEM) 1\u201335 (2024)","DOI":"10.1145\/3644387"},{"key":"10_CR36","doi-asserted-by":"crossref","unstructured":"Liu, Z., Yang, P., Zhang, L., Huang, X.: DeepCDCL: a CDCL-based neural network verification framework. In: Proceedings of 18th International Symposium on Theoretical Aspects of Software Engineering (TASE), pp. 343\u2013355 (2024)","DOI":"10.1007\/978-3-031-64626-3_20"},{"key":"10_CR37","doi-asserted-by":"crossref","unstructured":"Lopez, D., Choi, S., Tran, H.D., Johnson, T.: NNV 2.0: the neural network nerification tool. In: Proceedings of 35th International Conference on Computer Aided Verification (CAV), pp. 397\u2013412 (2023)","DOI":"10.1007\/978-3-031-37703-7_19"},{"key":"10_CR38","unstructured":"Nair, V., Hinton, G.: Rectified linear units improve restricted Boltzmann machines. In: Proceedings of 27th International Conference on Machine Learning (ICML), pp. 807\u2013814 (2010)"},{"key":"10_CR39","series-title":"Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence)","doi-asserted-by":"publisher","first-page":"366","DOI":"10.1007\/978-3-030-29436-6_22","volume-title":"Automated Deduction \u2013 CADE 27","author":"A Niemetz","year":"2019","unstructured":"Niemetz, A., Preiner, M., Reynolds, A., Zohar, Y., Barrett, C., Tinelli, C.: Towards bit-width-independent proofs in SMT solvers. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 366\u2013384. Springer, Cham (2019). https:\/\/doi.org\/10.1007\/978-3-030-29436-6_22"},{"key":"10_CR40","doi-asserted-by":"crossref","unstructured":"Ostrovsky, M., Barrett, C., Katz, G.: An abstraction-refinement approach to verifying convolutional neural networks. In: Proceedings of 20th International Symposium on Automated Technology for Verification and Analysis (ATVA), pp. 391\u2013396 (2022)","DOI":"10.1007\/978-3-031-19992-9_25"},{"key":"10_CR41","unstructured":"Radford, A., et al.: Learning transferable visual models from natural language supervision. In: Proceedings of 38th International Conference on Machine Learning (ICML) (2021)"},{"key":"10_CR42","unstructured":"Radford, A., Kim, J.W., Xu, T., Brockman, G., McLeavey, C., Sutskever, I.: Robust speech recognition via large-scale weak supervision. In: Proceedings of 40th International Conference on Machine Learning (ICML) (2023)"},{"key":"10_CR43","doi-asserted-by":"crossref","unstructured":"Rudin, C.: Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat. Mach. Intell. 206\u2013215 (2019)","DOI":"10.1038\/s42256-019-0048-x"},{"key":"10_CR44","doi-asserted-by":"crossref","unstructured":"S\u00e4lzer, M., Lange, M.: Reachability is NP-complete even for the simplest neural networks. In: Proceedings of 15th International Conference on Reachability Problems (RP), pp. 149\u2013164 (2021)","DOI":"10.1007\/978-3-030-89716-1_10"},{"key":"10_CR45","doi-asserted-by":"crossref","unstructured":"Singh, A., Sarita, Y., Mendis, C., Singh, G.: Automated verification of soundness of DNN certifiers. In: Proceedings of ACM on Programming Languages (PACMPL) (2025)","DOI":"10.1145\/3720509"},{"key":"10_CR46","doi-asserted-by":"crossref","unstructured":"Singh, G., Gehr, T., P\u00fcschel, M., Vechev, M.: An abstract domain for certifying neural networks. In: Proceedings of 46th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL), pp. 1\u201330 (2019)","DOI":"10.1145\/3290354"},{"key":"10_CR47","unstructured":"van den Oord, A., et al.: WaveNet: a generative model for raw audio. In: Proceedings of 9th ISCA Workshop on Speech Synthesis Workshop (SSW), p.\u00a0125 (2016)"},{"key":"10_CR48","unstructured":"Vaswani, A., et al.: Attention is all you need. In: Proceedings of 31st Conference on Advances in Neural Information Processing Systems (NeuRIPS) (2017)"},{"key":"10_CR49","doi-asserted-by":"crossref","unstructured":"Wu, H., et al.: Marabou 2.0: a versatile formal analyzer of neural networks. In: Proceedings of 36th International Conference on Computer Aided Verification (CAV) (2024)","DOI":"10.1007\/978-3-031-65630-9_13"},{"key":"10_CR50","unstructured":"Zombori, D., B\u00e1nhelyi, B., Csendes, T., Megyeri, I., Jelasity, M.: Fooling a complete neural network verifier. In: Proceedings of 9th International Conference on Learning Representations (ICLR) (2021)"}],"container-title":["Lecture Notes in Computer Science","AI Verification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-99991-8_10","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T10:13:46Z","timestamp":1787134426000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-99991-8_10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,28]]},"ISBN":["9783031999901","9783031999918"],"references-count":50,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-99991-8_10","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,28]]},"assertion":[{"value":"28 October 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":1,"name":"Ethics","label":"Disclosure of Interests","group":{"name":"EthicsHeading","label":"Ethics"}},{"value":"SAIV","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Symposium on AI Verification","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Zagreb","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Croatia","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"21 July 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"22 July 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"saiv2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/www.aiverification.org\/2025\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}