{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","intvolume":"        51","publication_status":"epub_ahead","status":"public","isi":1,"article_number":"101430","date_updated":"2023-12-13T12:58:56Z","quality_controlled":"1","title":"Symbolic control for stochastic systems via finite parity games","date_published":"2023-09-27T00:00:00Z","article_processing_charge":"No","acknowledgement":"We thank Daniel Hausmann and Nir Piterman for their valuable comments on an earlier version of the manuscript of our other paper [22] where we present, among other things, the parity fixpoint for 2 1/2-player games (for a slightly more general class of games) with a different and indirect proof of correctness. Based on their comments we observed that, unlike the other fixpoints that we present in [22], the parity fixpoint does not follow the exact same structure as its counterpart for 2-player games, which we also use int his paper.\r\nWe also thank Thejaswini Raghavan for observing that our symbolic parity fixpoint algorithm can be solved in quasi-polynomial time using recent improved algorithms for solving \r\n-calculus expressions. This significantly improved the complexity bounds of our algorithm in this paper.\r\nThe work of R. Majumdar and A.-K. Schmuck are partially supported by DFG, Germany project 389792660 TRR 248–CPEC. A.-K. Schmuck is additionally funded through DFG, Germany project (SCHM 3541/1-1). K. Mallik is supported by the ERC project ERC-2020-AdG 101020093. S. Soudjani is supported by the following projects: EPSRC EP/V043676/1, EIC 101070802, and ERC 101089047.","type":"journal_article","volume":51,"_id":"14400","doi":"10.1016/j.nahs.2023.101430","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1016/j.nahs.2023.101430"}],"date_created":"2023-10-08T22:01:15Z","oa":1,"publisher":"Elsevier","external_id":{"isi":["001093188100001"],"arxiv":["2101.00834"]},"day":"27","article_type":"original","language":[{"iso":"eng"}],"year":"2023","author":[{"full_name":"Majumdar, Rupak","last_name":"Majumdar","first_name":"Rupak"},{"orcid":"0000-0001-9864-7475","first_name":"Kaushik","last_name":"Mallik","full_name":"Mallik, Kaushik","id":"0834ff3c-6d72-11ec-94e0-b5b0a4fb8598"},{"last_name":"Schmuck","full_name":"Schmuck, Anne Kathrin","first_name":"Anne Kathrin"},{"full_name":"Soudjani, Sadegh","last_name":"Soudjani","first_name":"Sadegh"}],"department":[{"_id":"ToHe"}],"project":[{"grant_number":"101020093","call_identifier":"H2020","name":"Vigilant Algorithmic Monitoring of Software","_id":"62781420-2b32-11ec-9570-8d9b63373d4d"}],"month":"09","publication_identifier":{"issn":["1751-570X"]},"ec_funded":1,"citation":{"ieee":"R. Majumdar, K. Mallik, A. K. Schmuck, and S. Soudjani, “Symbolic control for stochastic systems via finite parity games,” <i>Nonlinear Analysis: Hybrid Systems</i>, vol. 51. Elsevier, 2023.","mla":"Majumdar, Rupak, et al. “Symbolic Control for Stochastic Systems via Finite Parity Games.” <i>Nonlinear Analysis: Hybrid Systems</i>, vol. 51, 101430, Elsevier, 2023, doi:<a href=\"https://doi.org/10.1016/j.nahs.2023.101430\">10.1016/j.nahs.2023.101430</a>.","ama":"Majumdar R, Mallik K, Schmuck AK, Soudjani S. Symbolic control for stochastic systems via finite parity games. <i>Nonlinear Analysis: Hybrid Systems</i>. 2023;51. doi:<a href=\"https://doi.org/10.1016/j.nahs.2023.101430\">10.1016/j.nahs.2023.101430</a>","ista":"Majumdar R, Mallik K, Schmuck AK, Soudjani S. 2023. Symbolic control for stochastic systems via finite parity games. Nonlinear Analysis: Hybrid Systems. 51, 101430.","chicago":"Majumdar, Rupak, Kaushik Mallik, Anne Kathrin Schmuck, and Sadegh Soudjani. “Symbolic Control for Stochastic Systems via Finite Parity Games.” <i>Nonlinear Analysis: Hybrid Systems</i>. Elsevier, 2023. <a href=\"https://doi.org/10.1016/j.nahs.2023.101430\">https://doi.org/10.1016/j.nahs.2023.101430</a>.","short":"R. Majumdar, K. Mallik, A.K. Schmuck, S. Soudjani, Nonlinear Analysis: Hybrid Systems 51 (2023).","apa":"Majumdar, R., Mallik, K., Schmuck, A. K., &#38; Soudjani, S. (2023). Symbolic control for stochastic systems via finite parity games. <i>Nonlinear Analysis: Hybrid Systems</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.nahs.2023.101430\">https://doi.org/10.1016/j.nahs.2023.101430</a>"},"publication":"Nonlinear Analysis: Hybrid Systems","abstract":[{"text":"We consider the problem of computing the maximal probability of satisfying an \r\n-regular specification for stochastic, continuous-state, nonlinear systems evolving in discrete time. The problem reduces, after automata-theoretic constructions, to finding the maximal probability of satisfying a parity condition on a (possibly hybrid) state space. While characterizing the exact satisfaction probability is open, we show that a lower bound on this probability can be obtained by (I) computing an under-approximation of the qualitative winning region, i.e., states from which the parity condition can be enforced almost surely, and (II) computing the maximal probability of reaching this qualitative winning region.\r\nThe heart of our approach is a technique to symbolically compute the under-approximation of the qualitative winning region in step (I) via a finite-state abstraction of the original system as a \r\n-player parity game. Our abstraction procedure uses only the support of the probabilistic evolution; it does not use precise numerical transition probabilities. We prove that the winning set in the abstract -player game induces an under-approximation of the qualitative winning region in the original synthesis problem, along with a policy to solve it. By combining these contributions with (a) a symbolic fixpoint algorithm to solve \r\n-player games and (b) existing techniques for reachability policy synthesis in stochastic nonlinear systems, we get an abstraction-based algorithm for finding a lower bound on the maximal satisfaction probability.\r\nWe have implemented the abstraction-based algorithm in Mascot-SDS, where we combined the outlined abstraction step with our tool Genie (Majumdar et al., 2023) that solves \r\n-player parity games (through a reduction to Rabin games) more efficiently than existing algorithms. We evaluated our implementation on the nonlinear model of a perturbed bistable switch from the literature. We show empirically that the lower bound on the winning region computed by our approach is precise, by comparing against an over-approximation of the qualitative winning region. Moreover, our implementation outperforms a recently proposed tool for solving this problem by a large margin.","lang":"eng"}],"oa_version":"Published Version"}