{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T11:02:55Z","timestamp":1761562975429,"version":"3.37.3"},"reference-count":63,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T00:00:00Z","timestamp":1685404800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T00:00:00Z","timestamp":1685404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100016378","name":"Technische Universit\u00e4t Dortmund","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100016378","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Softw Tools Technol Transfer"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we present an algebraic approach to the precise and global verification and explanation of <jats:italic>Rectifier Neural Networks<\/jats:italic>, a\u00a0subclass of <jats:italic>Piece-wise Linear Neural Networks<\/jats:italic> (PLNNs), i.e., networks that semantically represent piece-wise affine functions. Key to our approach is the symbolic execution of these networks that allows the construction of semantically equivalent <jats:italic>Typed Affine Decision Structures<\/jats:italic> (TADS). Due to their deterministic and sequential nature, TADS can, similarly to decision trees, be considered as white-box models and therefore as precise solutions to the model and outcome explanation problem. TADS are linear algebras, which allows one to elegantly compare Rectifier Networks for equivalence or similarity, both with precise diagnostic information in case of failure, and to characterize their classification potential by precisely characterizing the set of inputs that are specifically classified, or the set of inputs where two network-based classifiers differ. All phenomena are illustrated along a detailed discussion of a minimal, illustrative example: the continuous XOR function.<\/jats:p>","DOI":"10.1007\/s10009-023-00700-7","type":"journal-article","created":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T14:02:36Z","timestamp":1685455356000},"page":"301-327","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Towards rigorous understanding of neural networks via semantics-preserving transformations"],"prefix":"10.1007","volume":"25","author":[{"given":"Maximilian","family":"Schl\u00fcter","sequence":"first","affiliation":[]},{"given":"Gerrit","family":"Nolte","sequence":"additional","affiliation":[]},{"given":"Alnis","family":"Murtovi","sequence":"additional","affiliation":[]},{"given":"Bernhard","family":"Steffen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,30]]},"reference":[{"key":"700_CR1","unstructured":"Arora, R., Basu, A., Mianjy, P., Mukherjee, A.: Understanding deep neural networks with rectified linear units. arXiv preprint (2016). arXiv:1611.01491"},{"key":"700_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/b97662","volume-title":"Linear Algebra Done Right","author":"S. Axler","year":"1997","unstructured":"Axler, S.: Linear Algebra Done Right. Springer, Berlin (1997)"},{"issue":"7","key":"700_CR3","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0130140","volume":"10","author":"S. Bach","year":"2015","unstructured":"Bach, S., Binder, A., Montavon, G., Klauschen, F., M\u00fcller, K.R., Samek, W.: On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation. PLoS ONE 10(7), e0130140 (2015)","journal-title":"PLoS ONE"},{"key":"700_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2020.113816","volume":"165","author":"C. Badue","year":"2021","unstructured":"Badue, C., Guidolini, R., Carneiro, R.V., Azevedo, P., Cardoso, V.B., Forechi, A., Jesus, L., Berriel, R., Paixao, T.M., Mutz, F., et al.: Self-driving cars: a survey. Expert Syst. Appl. 165, 113816 (2021)","journal-title":"Expert Syst. Appl."},{"issue":"2","key":"700_CR5","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1023\/A:1008699807402","volume":"10","author":"R.I. Bahar","year":"1997","unstructured":"Bahar, R.I., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., Somenzi, F.: Algebric decision diagrams and their applications. Form. Methods Syst. Des. 10(2), 171\u2013206 (1997)","journal-title":"Form. Methods Syst. Des."},{"key":"700_CR6","unstructured":"Bak, S., Liu, C., Johnson, T.: The second international verification of neural networks competition (VNN-COMP 2021): summary and results. arXiv preprint (2021). arXiv:2109.00498"},{"key":"700_CR7","unstructured":"Berner, C., Brockman, G., Chan, B., Cheung, V., D\u0119biak, P., Dennison, C., Farhi, D., Fischer, Q., Hashme, S., Hesse, C., et\u00a0al.: Dota 2 with large scale deep reinforcement learning. arXiv preprint (2019). arXiv:1912.06680"},{"key":"700_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1148-8","volume-title":"An Introduction to Convex Polytopes","author":"A. Brondsted","year":"1983","unstructured":"Brondsted, A.: An Introduction to Convex Polytopes, first edn. Springer, New York (1983). https:\/\/doi.org\/10.1007\/978-1-4612-1148-8","edition":"1"},{"key":"700_CR9","first-page":"1877","volume":"33","author":"T. Brown","year":"2020","unstructured":"Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J.D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al.: Language models are few-shot learners. Adv. Neural Inf. Process. Syst. 33, 1877\u20131901 (2020)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"700_CR10","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1109\/SP.2017.49","volume-title":"2017 IEEE Symposium on Security and Privacy (SP)","author":"N. Carlini","year":"2017","unstructured":"Carlini, N., Wagner, D.: Towards evaluating the robustness of neural networks. In: 2017 IEEE Symposium on Security and Privacy (SP), pp.\u00a039\u201357. IEEE (2017)"},{"key":"700_CR11","doi-asserted-by":"publisher","first-page":"4774","DOI":"10.1109\/ICASSP.2018.8462105","volume-title":"2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","author":"C.C. Chiu","year":"2018","unstructured":"Chiu, C.C., Sainath, T.N., Wu, Y., Prabhavalkar, R., Nguyen, P., Chen, Z., Kannan, A., Weiss, R.J., Rao, K., Gonina, E., et al.: State-of-the-art speech recognition with sequence-to-sequence models. In: 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.\u00a04774\u20134778. IEEE (2018)"},{"key":"700_CR12","doi-asserted-by":"publisher","first-page":"1244","DOI":"10.1145\/3219819.3220063","volume-title":"Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining","author":"L. Chu","year":"2018","unstructured":"Chu, L., Hu, X., Hu, J., Wang, L., Pei, J.: Exact and consistent interpretation for piecewise linear neural networks: a closed form solution. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp.\u00a01244\u20131253 (2018)"},{"key":"700_CR13","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1109\/TSE.1976.233817","volume":"3","author":"L.A. Clarke","year":"1976","unstructured":"Clarke, L.A.: A system to generate test data and symbolically execute programs. IEEE Trans. Softw. Eng. 3, 215\u2013222 (1976)","journal-title":"IEEE Trans. Softw. Eng."},{"key":"700_CR14","unstructured":"Doran, D., Schulz, S., Besold, T.R.: What does explainable AI really mean? A\u00a0new conceptualization of perspectives. arXiv preprint (2017). arXiv:1710.00794"},{"key":"700_CR15","series-title":"JMLR Workshop and Conference Proceedings","first-page":"315","volume-title":"Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics","author":"X. Glorot","year":"2011","unstructured":"Glorot, X., Bordes, A., Bengio, Y.: Deep sparse rectifier neural networks. In: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics. JMLR Workshop and Conference Proceedings, pp.\u00a0315\u2013323 (2011)"},{"key":"700_CR16","unstructured":"Goodfellow, I.J., Shlens, J., Szegedy, C.: Explaining and harnessing adversarial examples. arXiv preprint (2014). arXiv:1412.6572"},{"key":"700_CR17","volume-title":"Deep Learning","author":"I. Goodfellow","year":"2016","unstructured":"Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016). http:\/\/www.deeplearningbook.org"},{"key":"700_CR18","unstructured":"Gopinath, D., Wang, K., Zhang, M., Pasareanu, C.S., Khurshid, S.: Symbolic execution for deep neural networks. arXiv preprint (2018). arXiv:1807.10439"},{"key":"700_CR19","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1109\/ICSE-Companion.2019.00115","volume-title":"2019 IEEE\/ACM 41st International Conference on Software Engineering: Companion Proceedings (ICSE-Companion)","author":"D. Gopinath","year":"2019","unstructured":"Gopinath, D., Pasareanu, C.S., Wang, K., Zhang, M., Khurshid, S.: Symbolic execution for attribution and attack synthesis in neural networks. In: 2019 IEEE\/ACM 41st International Conference on Software Engineering: Companion Proceedings (ICSE-Companion), pp.\u00a0282\u2013283. IEEE (2019)"},{"issue":"3","key":"700_CR20","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1080\/02331939408844018","volume":"31","author":"V.V. Gorokhovik","year":"1994","unstructured":"Gorokhovik, V.V., Zorko, O.I., Birkhoff, G.: Piecewise affine functions and polyhedral sets. Optimization 31(3), 209\u2013221 (1994)","journal-title":"Optimization"},{"key":"700_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/s10009-021-00635-x","author":"F. Gossen","year":"2021","unstructured":"Gossen, F., Steffen, B.: Algebraic aggregation of random forests: towards explainability and rapid evaluation. Int. J. Softw. Tools Technol. Transf. (2021). https:\/\/doi.org\/10.1007\/s10009-021-00635-x","journal-title":"Int. J. Softw. Tools Technol. Transf."},{"issue":"6","key":"700_CR22","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1109\/MITP.2021.3123495","volume":"23","author":"F. Gossen","year":"2021","unstructured":"Gossen, F., Margaria, T., Steffen, B.: Formal methods boost experimental performance for explainable AI. IT Prof. 23(6), 8\u201312 (2021)","journal-title":"IT Prof."},{"issue":"5","key":"700_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3236009","volume":"51","author":"R. Guidotti","year":"2018","unstructured":"Guidotti, R., Monreale, A., Ruggieri, S., Turini, F., Giannotti, F., Pedreschi, D.: A survey of methods for explaining black box models. ACM Comput. Surv. 51(5), 1\u201342 (2018). https:\/\/doi.org\/10.1145\/3236009","journal-title":"ACM Comput. Surv."},{"key":"700_CR24","series-title":"Proceedings of Machine Learning Research","first-page":"2596","volume-title":"Proceedings of the 36th International Conference on Machine Learning","author":"B. Hanin","year":"2019","unstructured":"Hanin, B., Rolnick, D.: Complexity of linear regions in deep networks. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol.\u00a097, pp.\u00a02596\u20132604 (2019), PMLR. https:\/\/proceedings.mlr.press\/v97\/hanin19a.html"},{"key":"700_CR25","series-title":"Advances in Neural Information Processing Systems","volume-title":"Deep ReLU networks have surprisingly few activation patterns","author":"B. Hanin","year":"2019","unstructured":"Hanin, B., Rolnick, D.: Deep ReLU networks have surprisingly few activation patterns. Advances in Neural Information Processing Systems, vol.\u00a032 (2019)"},{"key":"700_CR26","unstructured":"He, J., Li, L., Xu, J., Zheng, C.: ReLU deep neural networks and linear finite elements. arXiv preprint (2018). arXiv:1807.03973"},{"key":"700_CR27","unstructured":"Hinz, P.: Using activation histograms to bound the number of affine regions in ReLU feed-forward neural networks (2021). arXiv:2103.17174 [abs]"},{"key":"700_CR28","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/978-3-319-63387-9_5","volume-title":"International Conference on 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: International Conference on Computer Aided Verification, pp.\u00a097\u2013117. Springer, Berlin (2017)"},{"issue":"7","key":"700_CR29","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1145\/360248.360252","volume":"19","author":"J.C. King","year":"1976","unstructured":"King, J.C.: Symbolic execution and program testing. Commun. ACM 19(7), 385\u2013394 (1976)","journal-title":"Commun. ACM"},{"key":"700_CR30","unstructured":"Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. arXiv preprint (2014). arXiv:1412.6980"},{"issue":"1","key":"700_CR31","doi-asserted-by":"publisher","first-page":"18","DOI":"10.3390\/e23010018","volume":"23","author":"P. Linardatos","year":"2021","unstructured":"Linardatos, P., Papastefanopoulos, V., Kotsiantis, S.: Explainable AI: a review of machine learning interpretability methods. Entropy 23(1), 18 (2021)","journal-title":"Entropy"},{"key":"700_CR32","series-title":"Advances in Neural Information Processing Systems","volume-title":"A unified approach to interpreting model predictions","author":"S.M. Lundberg","year":"2017","unstructured":"Lundberg, S.M., Lee, S.I.: A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, vol.\u00a030 (2017)"},{"key":"700_CR33","unstructured":"Madry, A., Makelov, A., Schmidt, L., Tsipras, D., Vladu, A.: Towards deep learning models resistant to adversarial attacks. arXiv preprint (2017). arXiv:1706.06083"},{"issue":"3","key":"700_CR34","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s11263-016-0911-8","volume":"120","author":"A. Mahendran","year":"2016","unstructured":"Mahendran, A., Vedaldi, A.: Visualizing deep convolutional neural networks using natural pre-images. Int. J. Comput. Vis. 120(3), 233\u2013255 (2016)","journal-title":"Int. J. Comput. Vis."},{"key":"700_CR35","volume-title":"Perceptrons","author":"M. Minsky","year":"1969","unstructured":"Minsky, M., Papert, S.: Perceptrons (1969)"},{"key":"700_CR36","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/j.patcog.2016.11.008","volume":"65","author":"G. Montavon","year":"2017","unstructured":"Montavon, G., Lapuschkin, S., Binder, A., Samek, W., M\u00fcller, K.R.: Explaining nonlinear classification decisions with deep Taylor decomposition. Pattern Recognit. 65, 211\u2013222 (2017)","journal-title":"Pattern Recognit."},{"key":"700_CR37","series-title":"Advances in Neural Information Processing Systems","volume-title":"On the number of linear regions of deep neural networks","author":"G.F. Montufar","year":"2014","unstructured":"Montufar, G.F., Pascanu, R., Cho, K., Bengio, Y.: On the number of linear regions of deep neural networks. Advances in Neural Information Processing Systems, vol.\u00a027 (2014)"},{"key":"700_CR38","unstructured":"Mundhenk, T.N., Chen, B.Y., Friedland, G.: Efficient saliency maps for explainable AI. arXiv preprint (2019). arXiv:1911.11293"},{"key":"700_CR39","doi-asserted-by":"publisher","unstructured":"Murtovi, A., Nolte, G., Schl\u00fcter, M., Bernhard, S.: Forest Gump: a\u00a0tool for verification and explanation. Int. J. Softw. Tools. Technol. Transf. (2023, in this issue). https:\/\/doi.org\/10.1007\/s10009-023-00702-5","DOI":"10.1007\/s10009-023-00702-5"},{"key":"700_CR40","doi-asserted-by":"publisher","unstructured":"Nolte, G., Schl\u00fcter, M., Murtovi, A., Bernhard, S.: The power of Typed Affine Decision Structures: a case study. Int. J. Softw. Tools. Technol. Transf. (2023, in this issue). https:\/\/doi.org\/10.1007\/s10009-023-00701-6","DOI":"10.1007\/s10009-023-00701-6"},{"issue":"6","key":"700_CR41","doi-asserted-by":"publisher","first-page":"1311","DOI":"10.1016\/j.patcog.2004.01.013","volume":"37","author":"K.S. Oh","year":"2004","unstructured":"Oh, K.S., Jung, K.: GPU implementation of neural networks. Pattern Recognit. 37(6), 1311\u20131314 (2004)","journal-title":"Pattern Recognit."},{"issue":"5","key":"700_CR42","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1016\/j.ejc.2009.11.005","volume":"31","author":"S. Ovchinnikov","year":"2010","unstructured":"Ovchinnikov, S.: Discrete piecewise linear functions. Eur. J. Comb. 31(5), 1283\u20131294 (2010). https:\/\/doi.org\/10.1016\/j.ejc.2009.11.005","journal-title":"Eur. J. Comb."},{"key":"700_CR43","unstructured":"Pascanu, R., Montufar, G., Bengio, Y.: On the number of response regions of deep feed forward networks with piece-wise linear activations. arXiv preprint (2013). arXiv:1312.6098"},{"key":"700_CR44","first-page":"2847","volume-title":"International Conference on Machine Learning","author":"M. Raghu","year":"2017","unstructured":"Raghu, M., Poole, B., Kleinberg, J., Ganguli, S., Sohl-Dickstein, J.: On the expressive power of deep neural networks. In: International Conference on Machine Learning, pp.\u00a02847\u20132854. PMLR (2017)"},{"key":"700_CR45","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1145\/2939672.2939778","volume-title":"Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining","author":"M.T. Ribeiro","year":"2016","unstructured":"Ribeiro, M.T., Singh, S., Guestrin, C.: \u201cWhy should i trust you?\u201d Explaining the predictions of any classifier. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp.\u00a01135\u20131144 (2016)"},{"key":"700_CR46","unstructured":"Ruder, S.: An overview of gradient descent optimization algorithms. arXiv preprint (2016). arXiv:1609.04747"},{"key":"700_CR47","first-page":"618","volume-title":"Proceedings of the IEEE International Conference on Computer Vision","author":"R.R. Selvaraju","year":"2017","unstructured":"Selvaraju, R.R., Cogswell, M., Das, A., Vedantam, R., Parikh, D., Batra, D.: Grad-CAM: visual explanations from deep networks via gradient-based localization. In: Proceedings of the IEEE International Conference on Computer Vision, pp.\u00a0618\u2013626 (2017)"},{"key":"700_CR48","first-page":"4558","volume-title":"International Conference on Machine Learning","author":"T. Serra","year":"2018","unstructured":"Serra, T., Tjandraatmadja, C., Ramalingam, S.: Bounding and counting linear regions of deep neural networks. In: International Conference on Machine Learning, pp.\u00a04558\u20134566. PMLR (2018)"},{"issue":"7676","key":"700_CR49","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1038\/nature24270","volume":"550","author":"D. Silver","year":"2017","unstructured":"Silver, D., Schrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., Hubert, T., Baker, L., Lai, M., Bolton, A., et al.: Mastering the game of go without human knowledge. Nature 550(7676), 354\u2013359 (2017)","journal-title":"Nature"},{"key":"700_CR50","unstructured":"Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. arXiv preprint (2014). arXiv:1409.1556"},{"key":"700_CR51","unstructured":"Simonyan, K., Vedaldi, A., Zisserman, A.: Deep inside convolutional networks: visualising image classification models and saliency maps. arXiv preprint (2013). arXiv:1312.6034"},{"key":"700_CR52","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107705937","volume-title":"Ockham\u2019s Razors","author":"E. Sober","year":"2015","unstructured":"Sober, E.: Ockham\u2019s Razors. Cambridge University Press, Cambridge (2015)"},{"key":"700_CR53","unstructured":"Sudjianto, A., Knauth, W., Singh, R., Yang, Z., Zhang, A.: Unwrapping the black box of deep ReLU networks: interpretability, diagnostics, and simplification (2020). arXiv:2011.04041 [abs]"},{"key":"700_CR54","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1145\/3238147.3238172","volume-title":"Proceedings of the 33rd ACM\/IEEE International Conference on Automated Software Engineering","author":"Y. Sun","year":"2018","unstructured":"Sun, Y., Wu, M., Ruan, W., Huang, X., Kwiatkowska, M., Kroening, D.: Concolic testing for deep neural networks. In: Proceedings of the 33rd ACM\/IEEE International Conference on Automated Software Engineering, pp.\u00a0109\u2013119 (2018)"},{"key":"700_CR55","unstructured":"Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., Fergus, R.: Intriguing properties of neural networks. arXiv preprint (2013). arXiv:1312.6199"},{"key":"700_CR56","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1145\/37401.37421","volume-title":"Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques","author":"W.C. Thibault","year":"1987","unstructured":"Thibault, W.C., Naylor, B.F.: Set operations on polyhedra using binary space partitioning trees. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, pp.\u00a0153\u2013162 (1987)"},{"issue":"11","key":"700_CR57","doi-asserted-by":"publisher","first-page":"4793","DOI":"10.1109\/TNNLS.2020.3027314","volume":"32","author":"E. Tjoa","year":"2020","unstructured":"Tjoa, E., Guan, C.: A survey on explainable artificial intelligence (XAI): toward medical XAI. IEEE Trans. Neural Netw. Learn. Syst. 32(11), 4793\u20134813 (2020)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"5","key":"700_CR58","doi-asserted-by":"publisher","first-page":"945","DOI":"10.1016\/S0005-1098(02)00308-4","volume":"39","author":"P. T\u00f8ndel","year":"2003","unstructured":"T\u00f8ndel, P., Johansen, T.A., Bemporad, A.: Evaluation of piecewise affine control via binary search tree. Automatica 39(5), 945\u2013950 (2003). https:\/\/doi.org\/10.1016\/S0005-1098(02)00308-4","journal-title":"Automatica"},{"key":"700_CR59","first-page":"670","volume-title":"International Symposium on Formal Methods","author":"H.D. Tran","year":"2019","unstructured":"Tran, H.D., Manzanas Lopez, D., Musau, P., Yang, X., Nguyen, L.V., Xiang, W., Johnson, T.T.: Star-based reachability analysis of deep neural networks. In: International Symposium on Formal Methods, pp.\u00a0670\u2013686. Springer, Berlin (2019)"},{"issue":"7782","key":"700_CR60","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1038\/s41586-019-1724-z","volume":"575","author":"O. Vinyals","year":"2019","unstructured":"Vinyals, O., Babuschkin, I., Czarnecki, W.M., Mathieu, M., Dudzik, A., Chung, J., Choi, D.H., Powell, R., Ewalds, T., Georgiev, P., et al.: Grandmaster level in StarCraft II using multi-agent reinforcement learning. Nature 575(7782), 350\u2013354 (2019)","journal-title":"Nature"},{"key":"700_CR61","unstructured":"Wang, S., Zhang, H., Xu, K., Lin, X., Jana, S., Hsieh, C.J., Kolter, J.Z.: Beta-crown: efficient bound propagation with per-neuron split constraints for complete and incomplete neural network verification. arXiv preprint (2021). arXiv:2103.06624"},{"key":"700_CR62","doi-asserted-by":"crossref","unstructured":"Woo, S., Lee, C.L.: Decision boundary formation of deep convolution networks with ReLU. In: 2018 IEEE 16th Intl. Conf. on Dependable, Autonomic and Secure Computing, 16th Intl. Conf. on Pervasive Intelligence and Computing, 4th Intl. Conf. on Big Data Intelligence and Computing and Cyber Science and Technology Congress (DASC\/PiCom\/DataCom\/CyberSciTech), pp.\u00a0885\u2013888. IEEE (2018)","DOI":"10.1109\/DASC\/PiCom\/DataCom\/CyberSciTec.2018.00-13"},{"key":"700_CR63","unstructured":"Zhang, X., Wu, D.: Empirical studies on the properties of linear regions in deep neural networks. arXiv preprint (2020). arXiv:2001.01072"}],"container-title":["International Journal on Software Tools for Technology Transfer"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10009-023-00700-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10009-023-00700-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10009-023-00700-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,19]],"date-time":"2023-09-19T09:11:10Z","timestamp":1695114670000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10009-023-00700-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,30]]},"references-count":63,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["700"],"URL":"https:\/\/doi.org\/10.1007\/s10009-023-00700-7","relation":{},"ISSN":["1433-2779","1433-2787"],"issn-type":[{"type":"print","value":"1433-2779"},{"type":"electronic","value":"1433-2787"}],"subject":[],"published":{"date-parts":[[2023,5,30]]},"assertion":[{"value":"20 May 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 May 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}