## Abstract

We present a logic for stating properties such as, “after a request for service there is at least a 98% probability that the service will be carried out within 2 seconds”. The logic extends the temporal logic CTL by Emerson, Clarke and Sistla with time and probabilities. Formulas are interpreted over discrete time Markov chains. We give algorithms for checking that a given Markov chain satisfies a formula in the logic. The algorithms require a polynomial number of arithmetic operations, in size of both the formula and the Markov chain. A simple example is included to illustrate the algorithms.

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## References

Ajmone Marsan, M., Balbo, G. and Conte, G.:

*Performance Models of Multiprocessor Systems*. MIT Press, 1986.Abrahamson, K.:

*Decidability and Expressiveness of Logics of Processes*. PhD thesis, Univ. of Washington, 1980.Alur, R., Courcoubetis, C. and Dill, D.: Model-checking for real-time systems. In

*Proc. 5*^{th}*IEEE Int. Symp. on Logic in Computer Science*, pages 414–425, 1990.Alur, R., Courcoubetis, C. and Dill, D.: Model-checking for probabilistic real-time systems. In

*Proc. 18*^{th}*Int. Coll. on Automata Languages and Programming (ICALP)*, volume 510 of*Lecture Notes in Computer Science*, pages 115–126. Springer Verlag, 1991.Alur, R., Courcoubetis, C. and Dill, D.: Verifying Automata Specifications of Probabilistic Real-Time Systems. In J. de Bakker, C. Huizing, W.-P. de Roever, and G. Rozenberg, editors,

*Real-Time: Theory in Practice*, volume 600 of*Lecture Notes in Computer Science*, pages 28–44. Springer Verlag, 1992.Alur, R. and Dill, D.: Automata for modeling real-time systems. In

*Proc. 17*^{th}*Int. Coll. on Automata Languages and Programming (ICALP)*, volume 443 of*Lecture Notes in Computer Science*, Springer Verlag, 1990.Alur, R. and Henzinger, T.: A really temporal logic. In

*Proc. 30*^{th}*IEEE Annual Symp. Foundations of Computer Science*, pages 164–169, 1989.Alur, R. and Henzinger, T.: Logics and Models of Real Time: A Survey. In J. de Bakker, C. Huizing, W.-P. de Roever, and G. Rozenberg, editors,

*Real-Time: Theory in Practice*, volume 600 of*Lecture Notes in Computer Science*, pages 28–44. Springer Verlag, 1992.Aho, A.V., Hopcroft, J.E. and Ullman, J.D.:

*The Design and Analysis of Computer Algorithms.*Addison-Wesley Publishing Company, 1974.Bernstein, A. and Harter, P.K.: Proving real-time properties of programs with temporal logic. In

*Proc. 8*^{th}*ACM Symp. on Operating System Principles*, pages 1–11, Pacific Grove, California, 1981.Bartlett, K., Scantlebury, R. and Wilkinson, P.: A note on reliable full-duplex transmissions over half duplex lines.

*Communications of the ACM*, 2(5):260–261, 1969.Christoff, L. and Christoff, I: Reasoning about safety and liveness properties for probabilistic processes. In R. Shyamasundar, editor,

*Proc. 12*^{th}*Conf. on Foundations of Software Technology and Theoretical Computer Science*, volume 652 of*Lecture Notes in Computer Science*, pages 342–355. Springer-Verlag, 1992.Clarke, E.M., Emerson, E.A. and Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specification.

*ACM Trans. on Programming Languages and Systems*, 8(2):244–263, April 1986.Chiola, G.: A software package for the analysis of generalized stochastic Petri net models. In

*Proc. Int. Workshop on Time Petri Nets*, pages 136–143, July 1985.Ciardo, G., Muppala, J. and Trivedi, K.S.: Spnp: Stochastic petri net package. In

*Proc. of the third International Workshop on Petri Nets and Performance Models*. IEEE Computer Society Press, Kyoto, Japan, December 1989.Courcoubetis, C., Vardi, M. and Wolper, P.: Reasoning about fair concurrent programs. In

*Proc. 18*^{th}*ACM Symp. on Theory of Computing*, pages 283–294, 1986.Courcoubetis, C. and Yannakakis, C.: The complexity of probabilistic verification. In

*Proc. 29*^{th}*IEEE Annual Symp. Foundations of Computer Science*, pages 338–345, 1988.Courcoubetis, C. and Yannakakis, C.: The complexity of probabilistic verification. Bell labs Murry Hill, 1989.

de Bakker, J., Huizing, C., de Roever, W-.P. and Rozenberg, G.: editors.

*Real-Time: Theory in Practice*, volume 600 of*Lecture Notes in Computer Science*. Springer Verlag, 1992.Emerson, E.A. and Clarke, E.M.: Using branching time Temporal Logic to synthesize synchronization skeletons.

*Science of Computer Programming*, 2(3):241–266, 1982.Emerson, A.: Real-Time and the Mu-Calculus. In J. de Bakker, C. Huizing, W-.P. de Roever, and G. Rozenberg, editors,

*Real-Time: Theory in Practice*, volume 600 of*Lecture Notes in Computer Science*, pages 176–194. Springer Verlag, 1992.Emerson, A., Mok, A., Sistla, A. and Srinivasan, J.: Quantitative temporal reasoning.

*Real-Time Systems — The International Journal of Time-Critical Computing Systems*, 4:331–352, 1992.Feldman, Y.A.: A decidable propositional probabilistic dynamic logic. In

*Proc. 15*^{th}*ACM Symp. on Theory of Computing*, pages 298–309, Boston, 1983.Gibbons, A.:

*Algorithmic Graph Theory*. Cambridge University Press, 1985.Hansson, H.:

*Time and Probabilities in Formal Design of Distributed Systems*. PhD thesis, Department of Computer Systems, Uppsala University, 1991. Available as report DoCS 91/27, Department of Computer Systems, Uppsala University, Sweden, and as report 05 in SICS dissertation series, SICS, Kista, Sweden. A revised version of the thesis will appear in the Elsevier book series Real-Time Safety Critical Systems.Hansson, H. and Jonsson, B.: A calculus for communicating systems with time and probabilities. In

*Proc. 11*^{th}*IEEE Real -Time Systems Symp.*, pages 278–287, Orlando, Fl., December 1990. IEEE Computer Society Press.Hooman, J.:

*Specification and Compositional Verification of Real-Time Systems*, volume 558 of*Lecture Notes in Computer Science*. North-Holland, 1991.Hart, S. and Sharir, M.: Probabilistic temporal logics for finite and bounded models. In

*Proc. 16*^{th}*ACM Symp. on Theory of Computing*, pages 1–13, 1984.Hart, S., Sharir, M. and Pnueli, A.: Termination of probabilistic concurrent programs.

*ACM Trans. on Programming Languages and Systems*, 5:356–380, 1983.Holliday, M.A. and Vernon, M.K.: The GTPN Analyzer: numerical methods and user interface. Technical Report 639, Dept. of Computer Science, Univ. of Wisconsin — Madison, Apr. 1986.

Holliday, M.A. and Vernon, M.K.: Exact performance estimates for multiprocessor memory and bus interface.

*IEEE Trans. on Computers*, C-36:76–85, Jan. 1987.Holliday, M.A. and Vernon, M.K.: A generalized timed Petri net model for performance analysis.

*IEEE Trans. on Software Engineering*, SE-13(12), 1987.Jahanian, F. and Mok, K.-L.: Safety analysis of timing properties in real-time systems.

*IEEE Trans. on Software Engineering*, SE-12(9):890–904, Sept. 1986.Jahanian, F. and Mok, A.K.: A graph-theoretic approach for timing analysis and its implementation.

*IEEE Trans, on Computers*, 36(8):961–975, August 1987.Joseph, M.: editor.

*Formal Techniques in Real-Time and Fault-Tolerant Systems*, volume 331 of*Lecture Notes in Computer Science*. Springer Verlag, 1988.Koymans, R., Vytopil, J. and de Roever, W.P.: Real-time programming and asynchronous message passing. In

*Proc. 2*^{nd}*ACM Symp. on Principles of Distributed Computing*, pages 187–197, Montréal, Canada, 1983.Lehmann, D. and Shelah, S.: Reasoning with time and chance.

*Information and Control*, 53:165–198, 1982.Larsen, K.G. and Skou, A.: Bisimulation through probabilistic testing. In

*Proc. 16*^{th}*ACM Symp. on Principles of Programming Languages*, pages 344–352, 1989.Milner, R.:

*Communication and Concurrency*. Prentice-Hall, 1989.Molloy, M.K.: Performance analysis using stochastic petri nets.

*IEEE Trans. on Computers*, C-31(9):913–917, Sept. 1982.Owicki, S. and Lamport, L.: Proving liveness properties of concurrent programs.

*ACM Trans. on Programming Languages and Systems*, 4(3):455–495, 1982.Ostroff, J.: Automatic verification of timed transition models. In Sifakis, editor,

*Workshop on automatic verification methods for finite state systems*, volume 407 of*Lecture Notes in Computer Science*, pages 247–256. Springer Verlag, 1989.Ostroff, J. and Wonham, W.: Modelling, specifying and verifying real-time embedded computer systems. In

*Proc. IEEE Real-time Systems Symp.*, pages 124–132, Dec. 1987.Parrow, J.:

*Fairness Properties in Process Algebra*. PhD thesis, Uppsala University, Uppsala, Sweden, 1985. Available as report DoCS 85/03, Department of Computer Systems, Uppsala University, Sweden.Pnueli, A. and Harel, E.: Applications of temporal logic to the specification of real-time systems. In M. Joseph, editor,

*Proc. Symp. on Formal Techniques in Real-Time and Fault-Tolerant Systems*, volume 331 of*Lecture Notes in Computer Science*, pages 84–98. Springer Verlag, 1988.Pnueli, A.: The temporal semantics of concurrent programs.

*Theoretical Computer Science*, 13:45–60, 1982.Pnueli, A. and Zuck, L.: Verification of multiprocess probabilistic protocols.

*Distributed Computing*, 1(1):53–72, 1986.Razouk, R.R.: The derivation of performance expressions for communication protocols from timed Petri net models. In

*Proc. ACM SIGCOMM '84*, pages 210–217, Montréal, Québec, 1984.Razouk, R.R. and Phelps, C.V.: Performance analysis of timed Petri net models. In

*Proc. IFIP WG 6.2 Symp. on Protocol Specification, Testing, and Verification IV*, pages 126–129. North-Holland, June 1984.Shankar, A.U. and Lam, S.S.: Time dependent distributed systems: Proving safety, liveness and real-time properties.

*Distributed Computing*, 2:61–79, 1987.Sanders, W.H. and Meyer, J.F.: Metasan: a performability evaluation tool based on stochastic activity networks. In

*Proc of the ACM-IEEE Comp. Soc. Fall Joint Conf.*IEEE Computer Society Press, November 1986.Vardi, M: Automatic verification of probabilistic concurrent finite-state programs. In

*Proc. 26*^{th}*IEEE Annual Symp. Foundations of Computer Science*, pages 327–337, 1985.Vernon, M.K. and Holliday, M.A.: Performance analysis of multiprocessor cache consistency protocols using generalized timed Petri nets. In

*Proc. of Performance 86 and ACM SIGMETRICS 1986 Joint conf. on Computer Performance Modelling, Measurement, and Evaluation*, pages 9–17. ACM press, May 1986.Vardi, M.Y. and Wolper, P.: An automata-theoretic approach to automatic program verification. In

*Proc. IEEE Symp. on Logic in Computer Science*, pages 332–344, June 1986.Vytopil, P.: editor.

*Formal Techniques in Real-Time and Fault-Tolerant Systems*, volume 571 of*Lecture Notes in Computer Science.*Springer Verlag, 1991.Zuberek, W.: Performance evaluation using extended timed Petri nets. In

*Proc. International Workshop on Timed Petri Nets*, pages 272–278, Torino Italy, 1985. IEEE Computer Society Press.

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Hansson, H., Jonsson, B. A logic for reasoning about time and reliability.
*Formal Aspects of Computing* **6**, 512–535 (1994). https://doi.org/10.1007/BF01211866

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DOI: https://doi.org/10.1007/BF01211866