{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T19:04:51Z","timestamp":1770750291492,"version":"3.50.0"},"publisher-location":"Cham","reference-count":29,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783032111753","type":"print"},{"value":"9783032111760","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,11,23]],"date-time":"2025-11-23T00:00:00Z","timestamp":1763856000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,11,23]],"date-time":"2025-11-23T00:00:00Z","timestamp":1763856000000},"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\/978-3-032-11176-0_26","type":"book-chapter","created":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T20:11:31Z","timestamp":1763842291000},"page":"448-458","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["PRoTECT: Parallelized ConstRuction of SafeTy BarriEr Certificates for\u00a0Nonlinear Polynomial SysTems"],"prefix":"10.1007","author":[{"given":"Ben","family":"Wooding","sequence":"first","affiliation":[]},{"given":"Viacheslav","family":"Horbanov","sequence":"additional","affiliation":[]},{"given":"Abolfazl","family":"Lavaei","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,11,23]]},"reference":[{"key":"26_CR1","doi-asserted-by":"crossref","unstructured":"Abate, A., Ahmed, D., Edwards, A., Giacobbe, M., Peruffo, A.: FOSSIL: a software tool for the formal synthesis of lyapunov functions and barrier certificates using neural networks. In: Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control, pp. 1\u201311 (2021)","DOI":"10.1145\/3447928.3456646"},{"key":"26_CR2","unstructured":"Abate, A., et al.: ARCH-COMP20 Category Report: Stochastic Models (2020)"},{"key":"26_CR3","doi-asserted-by":"crossref","unstructured":"Ames, A.D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., Tabuada, P.: Control Barrier Functions: theory and Applications. In: 2019 18th European Control Conference (ECC), pp. 3420\u20133431. IEEE (2019)","DOI":"10.23919\/ECC.2019.8796030"},{"key":"26_CR4","doi-asserted-by":"crossref","unstructured":"Anand, M., Lavaei, A., Zamani, M.: Compositional synthesis of control barrier certificates for networks of stochastic systems against $$\\omega $$-regular specifications. Nonlinear Analysis: Hybrid Systems 51 (2024)","DOI":"10.1016\/j.nahs.2023.101427"},{"key":"26_CR5","unstructured":"Andersen, M.S., Dahl, J., Vandenberghe, L., et\u00a0al.: CVXOPT: A Python package for convex optimization. Available at cvxopt.org 54 (2013)"},{"key":"26_CR6","unstructured":"ApS, M.: MOSEK Optimizer API for Python (2022)"},{"key":"26_CR7","doi-asserted-by":"crossref","unstructured":"Black, M., Fainekos, G., Hoxha, B., Okamoto, H., Prokhorov, D.: CBFKIT: A Control Barrier Function Toolbox for Robotics Applications. arXiv preprint arXiv:2404.07158 (2024)","DOI":"10.1109\/IROS58592.2024.10801431"},{"key":"26_CR8","doi-asserted-by":"crossref","unstructured":"De\u00a0Moura, L., Bj\u00f8rner, N.: Z3: An efficient SMT solver. In: Proceedings of the International conference on Tools and Algorithms for the Construction and Analysis of Systems, pp. 337\u2013340 (2008)","DOI":"10.1007\/978-3-540-78800-3_24"},{"key":"26_CR9","doi-asserted-by":"crossref","unstructured":"Edwards, A., Peruffo, A., Abate, A.: Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models. arXiv:2311.09793 (2023)","DOI":"10.1145\/3641513.3651398"},{"key":"26_CR10","doi-asserted-by":"crossref","unstructured":"Gao, S., Avigad, J., Clarke, E.M.: $$\\delta $$-complete decision procedures for satisfiability over the reals. In: Automated Reasoning, pp. 286\u2013300. Lecture Notes in Computer Science (2012)","DOI":"10.1007\/978-3-642-31365-3_23"},{"key":"26_CR11","doi-asserted-by":"crossref","unstructured":"Gardner, J., Wooding, B., Nejati, A., Lavaei, A.: TRUST: StabiliTy and Safety ContRoller Synthesis for Unknown Dynamical Models Using a Single Trajectory. In: Proceedings of the 28th ACM International Conference on Hybrid Systems: Computation and Control, pp. 1\u201316 (2025)","DOI":"10.1145\/3716863.3718036"},{"issue":"7","key":"26_CR12","doi-asserted-by":"publisher","first-page":"3097","DOI":"10.1109\/TAC.2020.3013916","volume":"66","author":"P Jagtap","year":"2020","unstructured":"Jagtap, P., Soudjani, S., Zamani, M.: Formal synthesis of stochastic systems via control barrier certificates. IEEE Trans. Autom. Control 66(7), 3097\u20133110 (2020)","journal-title":"IEEE Trans. Autom. Control"},{"key":"26_CR13","doi-asserted-by":"crossref","unstructured":"Knight, J.C.: Safety critical systems: challenges and directions. In: Proceedings of the 24th international conference on software engineering, pp. 547\u2013550 (2002)","DOI":"10.1109\/ICSE.2002.1007998"},{"issue":"1","key":"26_CR14","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1073\/pnas.53.1.8","volume":"53","author":"HJ Kushner","year":"1965","unstructured":"Kushner, H.J.: On the stability of stochastic dynamical systems. Proc. Natl. Acad. Sci. 53(1), 8\u201312 (1965)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"26_CR15","doi-asserted-by":"crossref","unstructured":"Lavaei, A., Soudjani, S., Abate, A., Zamani, M.: Automated verification and synthesis of stochastic hybrid systems: A survey. Automatica 146 (2022)","DOI":"10.1016\/j.automatica.2022.110617"},{"issue":"4","key":"26_CR16","doi-asserted-by":"publisher","first-page":"1916","DOI":"10.1109\/TCNS.2020.3014602","volume":"7","author":"L Lindemann","year":"2020","unstructured":"Lindemann, L., Dimarogonas, D.V.: Barrier function based collaborative control of multiple robots under signal temporal logic tasks. IEEE Trans. Control Netw. Syst. 7(4), 1916\u20131928 (2020)","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"26_CR17","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1109\/LCSYS.2022.3229865","volume":"7","author":"FB Mathiesen","year":"2022","unstructured":"Mathiesen, F.B., Calvert, S.C., Laurenti, L.: Safety certification for stochastic systems via neural barrier functions. IEEE Control Syst. Lett. 7, 973\u2013978 (2022)","journal-title":"IEEE Control Syst. Lett."},{"key":"26_CR18","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.ejcon.2020.04.001","volume":"57","author":"A Nejati","year":"2021","unstructured":"Nejati, A., Soudjani, S., Zamani, M.: Compositional abstraction-based synthesis for continuous-time stochastic hybrid systems. Eur. J. Control. 57, 82\u201394 (2021)","journal-title":"Eur. J. Control."},{"key":"26_CR19","doi-asserted-by":"crossref","unstructured":"Prajna, S., Jadbabaie, A., Pappas, G.J.: Stochastic safety verification using barrier certificates. In: 2004 43rd IEEE conference on decision and control (CDC). vol.\u00a01, pp. 929\u2013934. IEEE (2004)","DOI":"10.1109\/CDC.2004.1428804"},{"issue":"8","key":"26_CR20","doi-asserted-by":"publisher","first-page":"1415","DOI":"10.1109\/TAC.2007.902736","volume":"52","author":"S Prajna","year":"2007","unstructured":"Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Autom. Control 52(8), 1415\u20131428 (2007)","journal-title":"IEEE Trans. Autom. Control"},{"key":"26_CR21","doi-asserted-by":"crossref","unstructured":"Prajna, S., Papachristodoulou, A., Parrilo, P.A.: Introducing sostools: A general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002. vol.\u00a01, pp. 741\u2013746. IEEE (2002)","DOI":"10.1109\/CDC.2002.1184594"},{"key":"26_CR22","doi-asserted-by":"crossref","unstructured":"Ramos, J.A., Dos\u00a0Santos, P.L.: Mathematical modeling, system identification, and controller design of a two tank system. In: 2007 46th IEEE Conference on Decision and Control, pp. 2838\u20132843. IEEE (2007)","DOI":"10.1109\/CDC.2007.4434949"},{"key":"26_CR23","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1090\/conm\/253\/03936","volume":"253","author":"B Reznick","year":"2000","unstructured":"Reznick, B.: Some concrete aspects of Hilbert\u2019s 17th problem. Contemp. Math. 253, 251\u2013272 (2000)","journal-title":"Contemp. Math."},{"key":"26_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2023.111323","volume":"159","author":"A Salamati","year":"2024","unstructured":"Salamati, A., Lavaei, A., Soudjani, S., Zamani, M.: Data-driven verification and synthesis of stochastic systems via barrier certificates. Automatica 159, 111323 (2024)","journal-title":"Automatica"},{"key":"26_CR25","doi-asserted-by":"crossref","unstructured":"Verdier, C.F., Mazo, M.: Formal synthesis of analytic controllers for sampled-data systems via genetic programming. In: 2018 IEEE Conference on Decision and Control (CDC), pp. 4896\u20134901. IEEE (2018)","DOI":"10.1109\/CDC.2018.8619121"},{"key":"26_CR26","doi-asserted-by":"publisher","unstructured":"Wooding, B., Horbanov, V., Lavaei, A.: PRoTECT: Parallelized Construction of Safety Barrier Certificates for Nonlinear Polynomial Systems. arXiv pp. 1\u201327 (2024). https:\/\/doi.org\/10.48550\/arXiv.2404.14804","DOI":"10.48550\/arXiv.2404.14804"},{"key":"26_CR27","doi-asserted-by":"crossref","unstructured":"Xiao, W., Cassandras, C.G., Belta, C.: Safe Autonomy with Control Barrier Functions: Theory and Applications. Springer (2023)","DOI":"10.1007\/978-3-031-27576-0"},{"key":"26_CR28","unstructured":"Yuan, C.: SumOfSquares.py. https:\/\/github.com\/yuanchenyang\/SumOfSquares.py"},{"key":"26_CR29","doi-asserted-by":"crossref","unstructured":"Zhao, H., Zeng, X., Chen, T., Liu, Z.: Synthesizing barrier certificates using neural networks. In: Proceedings of the 23rd international conference on hybrid systems: Computation and control, pp. 1\u201311 (2020)","DOI":"10.1145\/3365365.3382222"}],"container-title":["Lecture Notes in Computer Science","Theoretical Aspects of Computing \u2013 ICTAC 2025"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-11176-0_26","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T11:09:22Z","timestamp":1770721762000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-11176-0_26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,23]]},"ISBN":["9783032111753","9783032111760"],"references-count":29,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-11176-0_26","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,23]]},"assertion":[{"value":"23 November 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"ICTAC","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Colloquium on Theoretical Aspects of Computing","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Marrakesh","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Morocco","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":"24 November 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"28 November 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"22","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"ictac2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/ictac2025.digital-hub.sh\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}