ABSTRACT
Reachability analysis plays a central role in system design and verification. The reachability problem, denoted ◊jΦ, asks whether the system will meet the property Φ after some time in a given time interval j. Recently, it has been considered on a novel kind of real-time systems — quantum continuous-time Markov chains (QCTMCs), and embedded into the model-checking algorithm. In this paper, we further study the repeated reachability problem in QCTMCs, denoted □Ι◊jΦ, which concerns whether the system starting from each absolute time in Ι meet the property Φ after some coming relative time in j. First of all, we reduce it to the real root isolation of a class of real-valued functions (exponential polynomials), whose solvability is conditional to Schanuel’s conjecture being true. To speed up the procedure, we employ the strategy of sampling. The original problem is shown to be equivalent to the existence of a finite collection of satisfying samples. We then present a sample-driven procedure, which can effectively refine the sample space after each time of sampling, no matter whether the sample itself is satisfying or conflicting. The improvement on efficiency is validated by randomly generated instances. Hence the proposed method would be promising to attack the repeated reachability problems together with checking other ω -regular properties in a wide scope of real-time systems.
- Melanie Achatz, Scott McCallum, and Volker Weispfenning. 2008. Deciding polynomial–exponential problems. In Proc. 33rd International Symposium on Symbolic and Algebraic Computation, ISSAC 2008. ACM Press, New York, 215–222.Google Scholar
- Dorit Aharonov, Amnon Ta-Shma, Umesh V. Vazirani, and Andrew C.-C. Yao. 2000. Quantum bit escrow. In Proc. 32nd Annual ACM Symposium on Theory of Computing, STOC 2000. ACM Press, New York, 705–714.Google ScholarDigital Library
- Marco Antoniotti and Bud Mishra. 1995. Descrete events models + temporal logic = supervisory controller: Automatic synthesis of locomotion controllers. In Proc. 1995 International Conference on Robotics and Automation. IEEE Computer Society, Washington, 1441–1446.Google Scholar
- James Ax. 1971. On Schanuel’s conjectures. Annals of Mathematics 93, 2 (1971), 252–268.Google ScholarCross Ref
- Adnan Aziz, Kumud Sanwal, Vigyan Singhal, and Robert Brayton. 1996. Verifying continuous time Markov chains. In Computer Aided Verification: 8th International Conference, CAV’96(LNCS, Vol. 1102), Rajeev Alur and Thomas A. Henzinger (Eds.). Springer, Berlin, 269–276.Google ScholarCross Ref
- Christel Baier, Boudewijn R. Haverkort, Holger Hermanns, and Joost-Pieter Katoen. 2003. Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering 29, 6 (2003), 524–541.Google ScholarDigital Library
- Christel Baier, Joost-Pieter Katoen, and Holger Hermanns. 1999. Approximate symbolic model checking of continuous-time Markov chains. In CONCUR’99 Concurrency Theory(LNCS, Vol. 1664), Jos C. M. Baeten and Sjouke Mauw (Eds.). Springer, Berlin, 146–161.Google Scholar
- Alan Baker. 1975. Transcendental Number Theory. Cambridge University Press, London.Google Scholar
- Charles H. Bennett and Gilles Brassard. 1984. An update on quantum cryptography. In Advances in Cryptology, Proceedings of CRYPTO’84(LNCS, Vol. 196), George R. Blakley and David Chaum (Eds.). Springer, Berlin, 475–480.Google Scholar
- Julian C. Bradfield and Igor Walukiewicz. 2018. The mu-calculus and model checking. In Handbook of Model Checking, Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem (Eds.). Springer, Berlin, 871–919.Google Scholar
- Doron Bustan, Sasha Rubin, and Moshe Y. Vardi. 2004. Verifying ω -regular properties of Markov chains. In Computer Aided Verification, 16th International Conference, CAV 2004(LNCS, Vol. 3114), Rajeev Alur and Doron A. Peled (Eds.). Springer, Berlin, 189–201.Google Scholar
- Francesca Cairoli, Nicola Paoletti, and Luca Bortolussi. 2023. Conformal quantitative predictive monitoring of STL requirements for stochastic processes. In Proc. 26th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2023. ACM Press, New York, 1:1–1:11.Google ScholarDigital Library
- Edmund M. Clarke, Ernst Allen Emerson, and Aravinda Prasad Sistla. 1986. Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Transactions on Programming Languages and Systems 8, 2 (1986), 244–263.Google ScholarDigital Library
- Henri Cohen. 1996. A Course in Computational Algebraic Number Theory. Springer, Berlin.Google ScholarDigital Library
- Mohan Dantam and Amaury Pouly. 2021. On the decidability of reachability in continuous time linear time-invariant systems. In HSCC ’21: 24th ACM International Conference on Hybrid Systems: Computation and Control. ACM Press, New York, 15:1–15:12.Google ScholarDigital Library
- Yuan Feng, Ernst M. Hahn, Andrea Turrini, and Shenggang Ying. 2017. Model checking ω -regular properties for quantum Markov chains. In 28th International Conference on Concurrency Theory, CONCUR 2017(LIPIcs, Vol. 85). Schloss Dagstuhl, Heidelberg, 35:1–35:16.Google Scholar
- Yuan Feng, Nengkun Yu, and Mingsheng Ying. 2013. Model checking quantum Markov chains. J. Comput. System Sci. 79, 7 (2013), 1181–1198.Google ScholarCross Ref
- Yuan Feng and Lijun Zhang. 2017. Precisely deciding CSL formulas through approximate model checking for CTMCs. J. Comput. System Sci. 89 (2017), 361–371.Google ScholarCross Ref
- Ting Gan, Mingshuai Chen, Yangjia Li, Bican Xia, and Naijun Zhan. 2018. Reachability analysis for solvable dynamical systems. IEEE Trans. Automat. Control 63, 7 (2018), 2003–2018.Google ScholarCross Ref
- Vittorio Gorini, Andrzej Kossakowski, and E. C. George Sudarshan. 1976. Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 5 (1976), 821–825.Google ScholarCross Ref
- Ji Guan and Nengkun Yu. 2022. A probabilistic logic for verifying continuous-time Markov chains. In Tools and Algorithms for the Construction and Analysis of Systems - 28th International Conference, TACAS 2022, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022, Part II(LNCS, Vol. 13244), Dana Fisman and Grigore Rosu (Eds.). Springer, Berlin, 3–21.Google Scholar
- Hans Hansson and Bengt Jonsson. 1994. A logic for reasoning about time and reliability. Formal Aspects of Computing 6, 5 (1994), 512–535.Google ScholarDigital Library
- Cheng-Chao Huang, Jing-Cao Li, Ming Xu, and Zhi-Bin Li. 2018. Positive root isolation for poly-powers by exclusion and differentiation. Journal of Symbol Computation 85 (2018), 148–169.Google ScholarCross Ref
- Thomas Kailath. 1980. Linear Systems. Prentice Hall, New Jersey.Google Scholar
- Attila Kondacs and John Watrous. 1997. On the power of quantum finite state automata. In Proc. 38th Annual Symposium on Foundations of Computer Science, FOCS’97. IEEE Computer Society, Washington, 66–75.Google ScholarCross Ref
- Arjen K. Lenstra, Hendrik W. Lenstra, and László Lovász. 1982. Factoring polynomials with rational coefficients. Math. Ann. 261, 4 (1982), 515–534.Google ScholarCross Ref
- Goran Lindblad. 1976. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics 48, 2 (1976), 119–130.Google ScholarCross Ref
- Savvas G. Loizou and Kostas J. Kyriakopoulos. 2004. Automatic synthesis of multi-agent motion tasks based on LTL specifications. In Proc. 43rd IEEE Conference on Decision and Control, CDC 2004. IEEE Computer Society, Washington, 153–158.Google Scholar
- Oded Maler and Dejan Nickovic. 2004. Monitoring temporal properties of continuous signals. In Formal Techniques, Modelling and Analysis of Timed and Fault-Tolerant Systems(LNCS, Vol. 3253), Yassine Lakhnech and Sergio Yovine (Eds.). Springer, Berlin, 152–166.Google Scholar
- Jingyi Mei, Ming Xu, Ji Guan, Yuxin Deng, and Nengkun Yu. 2022. Checking continuous stochastic logic against quantum continuous-time Markov chains. CoRR abs/2202.05412 (2022), 1–21. https://arxiv.org/abs/2202.05412.Google Scholar
- Michael A. Nielsen and Isaac L. Chuang. 2000. Quantum Computation and Quantum Information. Cambridge University Press, London.Google ScholarDigital Library
- Joël Ouaknine and James Worrell. 2005. On the decidability of metric temporal logic. In Proc. 20th IEEE Symposium on Logic in Computer Science, LICS 2005. IEEE Computer Society, Washington, 188–197.Google ScholarDigital Library
- Mahmoud Salamati, Sadegh Soudjani, and Rupak Majumdar. 2020. A Lyapunov approach for time-bounded reachability of CTMCs and CTMDPs. ACM Transactions on Modeling and Performance Evaluation of Computing Systems 5, 1 (2020), 2:1–2:29.Google ScholarDigital Library
- William J. Stewart. 1994. Introduction to the Numerical Solution of Markov Chains. Princeton University Press, New Jersey.Google Scholar
- Alan M. Turing. 1937. On computable numbers, with an application to the entscheidungsproblem. Proceedings of the London Mathematical Society s2-42, 1 (1937), 230–265.Google ScholarCross Ref
- Moshe Y. Vardi. 1985. Automatic verification of probabilistic concurrent finite-state programs. In Proc. 26th Annual Symposium on Foundations of Computer Science, FOCS’85. IEEE Computer Society, Washington, 327–338.Google ScholarDigital Library
- Alex J. Wilkie. 1996. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. Journal of the American Mathematical Society 9, 4 (1996), 1051–1094.Google ScholarCross Ref
- Ming Xu, Jianling Fu, Jingyi Mei, and Yuxin Deng. 2022. Model checking QCTL plus on quantum Markov chains. Theoretical Computer Science 913 (2022), 43–72.Google ScholarDigital Library
- Ming Xu, Cheng-Chao Huang, and Yuan Feng. 2021. Measuring the constrained reachability in quantum Markov chains. Acta Informatica 58, 6 (2021), 653–674.Google ScholarDigital Library
- Ming Xu, Jingyi Mei, Ji Guan, and Nengkun Yu. 2021. Model checking quantum continuous-time Markov chains. In 32nd International Conference on Concurrency Theory, CONCUR 2021(LIPIcs, Vol. 203). Schloss Dagstuhl, Heidelberg, 13:1–13:17.Google Scholar
- Ming Xu, Lijun Zhang, David N. Jansen, Huibiao Zhu, and Zongyuan Yang. 2016. Multiphase until formulas over Markov reward models: An algebraic approach. Theoretical Computer Science 611 (2016), 116–135.Google ScholarDigital Library
- Shenggang Ying, Yuan Feng, Nengkun Yu, and Mingsheng Ying. 2013. Reachability probabilities of quantum Markov chains. In CONCUR 2013: Concurrency Theory - 24th International Conference(LNCS, Vol. 8052), Pedro R. D’Argenio and Hernán C. Melgratti (Eds.). Springer, Berlin, 334–348.Google ScholarDigital Library
- Shenggang Ying and Mingsheng Ying. 2018. Reachability analysis of quantum Markov decision processes. Information and Computation 263 (2018), 31–51.Google ScholarCross Ref
- Lijun Zhang, David N. Jansen, Flemming Nielson, and Holger Hermanns. 2011. Automata-based CSL model checking. In Automata, Languages and Programming: 38th International Colloquium, ICALP 2011, Part II(LNCS, Vol. 6756), Luca Aceto, Monika Henzinger, and Jirí Sgall (Eds.). Springer, Berlin, 271–282.Google Scholar
Index Terms
- A Sample-Driven Solving Procedure for the Repeated Reachability of Quantum Continuous-time Markov Chains
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