[{"page":"1-13","quality_controlled":"1","ec_funded":1,"publisher":"Institute of Electrical and Electronics Engineers","author":[{"last_name":"Chatterjee","first_name":"Krishnendu","full_name":"Chatterjee, Krishnendu","orcid":"0000-0002-4561-241X","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Wolfgang","last_name":"Dvorak","full_name":"Dvorak, Wolfgang"},{"orcid":"0000-0002-5008-6530","full_name":"Henzinger, Monika H","first_name":"Monika H","last_name":"Henzinger","id":"540c9bbd-f2de-11ec-812d-d04a5be85630"},{"last_name":"Svozil","first_name":"Alexander","full_name":"Svozil, Alexander"}],"_id":"10002","scopus_import":"1","title":"Symbolic time and space tradeoffs for probabilistic verification","publication_status":"published","date_created":"2021-09-12T22:01:24Z","department":[{"_id":"KrCh"}],"article_processing_charge":"No","acknowledgement":"The authors are grateful to the anonymous referees for their valuable comments. A. S. is fully supported by the Vienna Science and Technology Fund (WWTF) through project ICT15–003. K. C. is supported by the Austrian Science Fund (FWF) NFN Grant No S11407-N23 (RiSE/SHiNE) and by the ERC CoG 863818 (ForM-SMArt). For M. H. the research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement no. 340506.","isi":1,"external_id":{"isi":["000947350400089"],"arxiv":["2104.07466"]},"date_updated":"2025-07-14T09:10:07Z","year":"2021","citation":{"apa":"Chatterjee, K., Dvorak, W., Henzinger, M. H., &#38; Svozil, A. (2021). Symbolic time and space tradeoffs for probabilistic verification. In <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i> (pp. 1–13). Rome, Italy: Institute of Electrical and Electronics Engineers. <a href=\"https://doi.org/10.1109/LICS52264.2021.9470739\">https://doi.org/10.1109/LICS52264.2021.9470739</a>","ama":"Chatterjee K, Dvorak W, Henzinger MH, Svozil A. Symbolic time and space tradeoffs for probabilistic verification. In: <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>. Institute of Electrical and Electronics Engineers; 2021:1-13. doi:<a href=\"https://doi.org/10.1109/LICS52264.2021.9470739\">10.1109/LICS52264.2021.9470739</a>","ieee":"K. Chatterjee, W. Dvorak, M. H. Henzinger, and A. Svozil, “Symbolic time and space tradeoffs for probabilistic verification,” in <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>, Rome, Italy, 2021, pp. 1–13.","chicago":"Chatterjee, Krishnendu, Wolfgang Dvorak, Monika H Henzinger, and Alexander Svozil. “Symbolic Time and Space Tradeoffs for Probabilistic Verification.” In <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>, 1–13. Institute of Electrical and Electronics Engineers, 2021. <a href=\"https://doi.org/10.1109/LICS52264.2021.9470739\">https://doi.org/10.1109/LICS52264.2021.9470739</a>.","short":"K. Chatterjee, W. Dvorak, M.H. Henzinger, A. Svozil, in:, Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, Institute of Electrical and Electronics Engineers, 2021, pp. 1–13.","mla":"Chatterjee, Krishnendu, et al. “Symbolic Time and Space Tradeoffs for Probabilistic Verification.” <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>, Institute of Electrical and Electronics Engineers, 2021, pp. 1–13, doi:<a href=\"https://doi.org/10.1109/LICS52264.2021.9470739\">10.1109/LICS52264.2021.9470739</a>.","ista":"Chatterjee K, Dvorak W, Henzinger MH, Svozil A. 2021. Symbolic time and space tradeoffs for probabilistic verification. Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS: Symposium on Logic in Computer Science, 1–13."},"abstract":[{"lang":"eng","text":"We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal end-component (MEC) decomposition of Markov decision processes (MDPs). This problem generalizes the SCC decomposition problem of graphs and closed recurrent sets of Markov chains. The model of symbolic algorithms is widely used in formal verification and model-checking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of one-step neighborhood). For an input MDP with  n  vertices and  m  edges, the classical symbolic algorithm from the 1990s for the MEC decomposition requires  O(n2)  symbolic operations and  O(1)  symbolic space. The only other symbolic algorithm for the MEC decomposition requires  O(nm−−√)  symbolic operations and  O(m−−√)  symbolic space. A main open question is whether the worst-case  O(n2)  bound for symbolic operations can be beaten. We present a symbolic algorithm that requires  O˜(n1.5)  symbolic operations and  O˜(n−−√)  symbolic space. Moreover, the parametrization of our algorithm provides a trade-off between symbolic operations and symbolic space: for all  0<ϵ≤1/2  the symbolic algorithm requires  O˜(n2−ϵ)  symbolic operations and  O˜(nϵ)  symbolic space ( O˜  hides poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure winning regions of  ω -regular objectives for MDPs. We consider the canonical parity objectives for  ω -regular objectives, and for parity objectives with  d -priorities we present an algorithm that computes the almost-sure winning region with  O˜(n2−ϵ)  symbolic operations and  O˜(nϵ)  symbolic space, for all  0<ϵ≤1/2 ."}],"arxiv":1,"doi":"10.1109/LICS52264.2021.9470739","day":"07","language":[{"iso":"eng"}],"keyword":["Computer science","Computational modeling","Markov processes","Probabilistic logic","Formal verification","Game Theory"],"conference":{"start_date":"2021-06-29","name":"LICS: Symposium on Logic in Computer Science","end_date":"2021-07-02","location":"Rome, Italy"},"publication":"Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science","month":"07","oa_version":"Preprint","project":[{"name":"Game Theory","grant_number":"S11407","call_identifier":"FWF","_id":"25863FF4-B435-11E9-9278-68D0E5697425"},{"grant_number":"863818","name":"Formal Methods for Stochastic Models: Algorithms and Applications","call_identifier":"H2020","_id":"0599E47C-7A3F-11EA-A408-12923DDC885E"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2104.07466"}],"date_published":"2021-07-07T00:00:00Z","type":"conference","oa":1,"publication_identifier":{"eisbn":["978-1-6654-4895-6"],"issn":["1043-6871"],"isbn":["978-1-6654-4896-3"]}},{"keyword":["Computer science","Heuristic algorithms","Memory management","Automata","Markov processes","Probability distribution","Complexity theory"],"language":[{"iso":"eng"}],"conference":{"end_date":"2021-07-02","location":"Rome, Italy","start_date":"2021-06-29","name":"LICS: Symposium on Logic in Computer Science"},"publication":"Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science","month":"07","project":[{"_id":"0599E47C-7A3F-11EA-A408-12923DDC885E","call_identifier":"H2020","grant_number":"863818","name":"Formal Methods for Stochastic Models: Algorithms and Applications"}],"oa_version":"Preprint","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2104.07278"}],"type":"conference","date_published":"2021-07-07T00:00:00Z","oa":1,"publication_identifier":{"isbn":["978-1-6654-4896-3"],"eisbn":["978-1-6654-4895-6"],"issn":["1043-6871"]},"quality_controlled":"1","ec_funded":1,"page":"1-13","publisher":"Institute of Electrical and Electronics Engineers","author":[{"orcid":"0000-0002-4561-241X","full_name":"Chatterjee, Krishnendu","first_name":"Krishnendu","last_name":"Chatterjee","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Doyen","first_name":"Laurent","full_name":"Doyen, Laurent"}],"scopus_import":"1","_id":"10004","title":"Stochastic processes with expected stopping time","date_created":"2021-09-12T22:01:25Z","department":[{"_id":"KrCh"}],"article_processing_charge":"No","publication_status":"published","acknowledgement":"We are grateful to the anonymous reviewers of LICS 2021 and of a previous version of this paper for insightful comments that helped improving the presentation. This research was partially supported by the grant ERC CoG 863818 (ForM-SMArt).","external_id":{"isi":["000947350400036"],"arxiv":["2104.07278"]},"isi":1,"citation":{"chicago":"Chatterjee, Krishnendu, and Laurent Doyen. “Stochastic Processes with Expected Stopping Time.” In <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>, 1–13. Institute of Electrical and Electronics Engineers, 2021. <a href=\"https://doi.org/10.1109/LICS52264.2021.9470595\">https://doi.org/10.1109/LICS52264.2021.9470595</a>.","ieee":"K. Chatterjee and L. Doyen, “Stochastic processes with expected stopping time,” in <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>, Rome, Italy, 2021, pp. 1–13.","ama":"Chatterjee K, Doyen L. Stochastic processes with expected stopping time. In: <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>. Institute of Electrical and Electronics Engineers; 2021:1-13. doi:<a href=\"https://doi.org/10.1109/LICS52264.2021.9470595\">10.1109/LICS52264.2021.9470595</a>","apa":"Chatterjee, K., &#38; Doyen, L. (2021). Stochastic processes with expected stopping time. In <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i> (pp. 1–13). Rome, Italy: Institute of Electrical and Electronics Engineers. <a href=\"https://doi.org/10.1109/LICS52264.2021.9470595\">https://doi.org/10.1109/LICS52264.2021.9470595</a>","ista":"Chatterjee K, Doyen L. 2021. Stochastic processes with expected stopping time. Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS: Symposium on Logic in Computer Science, 1–13.","short":"K. Chatterjee, L. Doyen, in:, Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, Institute of Electrical and Electronics Engineers, 2021, pp. 1–13.","mla":"Chatterjee, Krishnendu, and Laurent Doyen. “Stochastic Processes with Expected Stopping Time.” <i>Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science</i>, Institute of Electrical and Electronics Engineers, 2021, pp. 1–13, doi:<a href=\"https://doi.org/10.1109/LICS52264.2021.9470595\">10.1109/LICS52264.2021.9470595</a>."},"year":"2021","date_updated":"2025-07-14T09:10:08Z","abstract":[{"text":"Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.","lang":"eng"}],"day":"07","doi":"10.1109/LICS52264.2021.9470595","arxiv":1}]