--- _id: '10002' 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 .' 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. article_processing_charge: No author: - first_name: Krishnendu full_name: Chatterjee, Krishnendu id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87 last_name: Chatterjee orcid: 0000-0002-4561-241X - first_name: Wolfgang full_name: Dvorak, Wolfgang last_name: Dvorak - first_name: Monika H full_name: Henzinger, Monika H id: 540c9bbd-f2de-11ec-812d-d04a5be85630 last_name: Henzinger orcid: 0000-0002-5008-6530 - first_name: Alexander full_name: Svozil, Alexander last_name: Svozil citation: ama: 'Chatterjee K, Dvorak W, Henzinger MH, Svozil A. Symbolic time and space tradeoffs for probabilistic verification. In: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science. Institute of Electrical and Electronics Engineers; 2021:1-13. doi:10.1109/LICS52264.2021.9470739' apa: 'Chatterjee, K., Dvorak, W., Henzinger, M. H., & Svozil, A. (2021). Symbolic time and space tradeoffs for probabilistic verification. In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (pp. 1–13). Rome, Italy: Institute of Electrical and Electronics Engineers. https://doi.org/10.1109/LICS52264.2021.9470739' chicago: Chatterjee, Krishnendu, Wolfgang Dvorak, Monika H Henzinger, and Alexander Svozil. “Symbolic Time and Space Tradeoffs for Probabilistic Verification.” In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, 1–13. Institute of Electrical and Electronics Engineers, 2021. https://doi.org/10.1109/LICS52264.2021.9470739. ieee: K. Chatterjee, W. Dvorak, M. H. Henzinger, and A. Svozil, “Symbolic time and space tradeoffs for probabilistic verification,” in Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, Rome, Italy, 2021, pp. 1–13. 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.' mla: Chatterjee, Krishnendu, et al. “Symbolic Time and Space Tradeoffs for Probabilistic Verification.” Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, Institute of Electrical and Electronics Engineers, 2021, pp. 1–13, doi:10.1109/LICS52264.2021.9470739. 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. conference: end_date: 2021-07-02 location: Rome, Italy name: 'LICS: Symposium on Logic in Computer Science' start_date: 2021-06-29 date_created: 2021-09-12T22:01:24Z date_published: 2021-07-07T00:00:00Z date_updated: 2023-08-14T06:51:33Z day: '07' department: - _id: KrCh doi: 10.1109/LICS52264.2021.9470739 ec_funded: 1 external_id: arxiv: - '2104.07466' isi: - '000947350400089' isi: 1 keyword: - Computer science - Computational modeling - Markov processes - Probabilistic logic - Formal verification - Game Theory language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2104.07466 month: '07' oa: 1 oa_version: Preprint page: 1-13 project: - _id: 25863FF4-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: S11407 name: Game Theory - _id: 0599E47C-7A3F-11EA-A408-12923DDC885E call_identifier: H2020 grant_number: '863818' name: 'Formal Methods for Stochastic Models: Algorithms and Applications' publication: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science publication_identifier: eisbn: - 978-1-6654-4895-6 isbn: - 978-1-6654-4896-3 issn: - 1043-6871 publication_status: published publisher: Institute of Electrical and Electronics Engineers quality_controlled: '1' scopus_import: '1' status: public title: Symbolic time and space tradeoffs for probabilistic verification type: conference user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 year: '2021' ...