Abstract
It is known that the satisfiability problem for Metric Temporal Logic (MTL) is decidable over finite timed words. In this chapter we study the satisfiability problem for extensions of this logic by various process-algebraic operators. On the negative side we show that satisfiability becomes undecidable when any of hiding, renaming, or asynchronous parallel composition are added to the logic. On the positive side we show decidability with the addition of alphabetised parallel composition and fixpoint operators. We use one-clock Timed Propositional Temporal Logic (TPTL(1)) as a technical tool for the decidability results and show that { TPTL(1)} with fixpoints provides a logical characterisation of the class of languages accepted by one-clock timed alternating automata.
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Notes
- 1.
Technically speaking, [20] deals with Metric Temporal Logic rather than { TPTL(1)}. The proof techniques however carry over straightforwardly.
- 2.
This even holds for the until-free fragment of MTL, which is obtained from Definition 4 by dropping the \({\mathcal{U}}_{\mathcal{I}}\)-definition and introducing \({\square }_{\mathcal{I}}\) and \({\diamond }_{\mathcal{I}}\) as primitives.
References
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)
Alur, R., Henzinger, T.A.: Real-time logics: complexity and expressiveness. Technical report, Stanford University (1990)
Alur, R., Henzinger, T.A.: A really temporal logic. J. ACM 41(1), 181–203 (1994)
Bouyer, P., Chevalier, F., Markey, N.: On the expressiveness of TPTL and MTL. In: Proceedings of FSTTCS, volume 3821 of Lecture Notes in Computer Science, pp. 432–443. Springer (2005)
Bouyer, P., Markey, N., Ouaknine, J., Worrell, J.: On expressiveness and complexity in real-time model checking. In: Proceedings of ICALP, volume 5126 of Lecture Notes in Computer Science, pp 124–135. Springer (2008)
Brookes, S.D., Hoare, C.A.R., Roscoe, A.W.: A theory of communicating sequential processes. J. ACM 31(3), 560–599 (1984)
D’Souza, D., Prabhakar, P.: On the expressiveness of MTL in the pointwise and continuous semantics. STTT 9(1), 1–4 (2007)
Haase, C., Ouaknine, J., Worrell, J.: On extensions of metric temporal logic. Technical report, Oxford University Computing Laboratory (2009). http://www.comlab.ox.ac.uk/files/2180/how-09.pdf
Henzinger, T.A.: The temporal specification and verification of real-time systems. Ph.D, thesis, Stanford University (1992)
Henzinger, T.A.: Its about time: real-time logics reviewed. In: Proceedings of CONCUR, volume 1466 of Lecture Notes in Computer Science, pp. 439–454 (1998)
Henzinger, T.A., Raskin, J.-F., Schobbens, P.-Y.: The regular real-time languages. In: Proceedings of ICALP, volume 1443 of Lecture Notes in Computer Science, pp. 580–591. Springer (1998)
Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall (1985)
Koymans, R.: Specifying real-time properties with Metric Temporal Logic. Real-Time Syst. 2(4), 255–299 (1990)
Lange, M.: Weak automata for the linear time μ-calculus. In: Proceedings of VMCAI, volume 3385 of Lecture Notes in Computer Science, pp. 267–281. Springer (2005)
Lasota, S., Walukiewicz, I.: Alternating timed automata. ACM Trans. Comput. Logic 9(2), 1–27 (2008)
Mayer, A.J., Stockmeyer, L.J.: The complexity of PDL with interleaving. Theor. Comput. Sci. 161(1–2), 109–122 (1996)
Milner, R.: A Calculus of Communicating Systems, volume 92 of Lecture Notes in Computer Science. Springer (1980)
Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall., Upper Saddle River, NJ (1967)
Ouaknine, J., Worrell, J.: On Metric Temporal Logic and Faulty turing machines. In: Proceedings of FoSSaCS, volume 3921 of Lecture Notes in Computer Science, pp. 217–230. Springer (2006)
Ouaknine, J., Worrell, J.: On the decidability complexity of Metric Temporal Logic over finite words. Logic. Meth. Comp. Sci. 3(1) (2007)
Ouaknine, J., Worrell, J.: Some recent results in Metric Temporal Logic. In: Proceedings of FORMATS, volume 5215 of Lecture Notes in Computer Science, pp. 1–13. Springer (2008)
Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 115–125 (1959)
Reed, G.M., Roscoe, A.W.: A timed model for Communicating Sequential Processes. In: Proceedings of ICALP, volume 226 of Lecture Notes in Computer Science, pp. 314–323. Springer (1986)
Reed, G.M., Roscoe, A.W.: Metric spaces as models for real-time concurrency. In: Proceedings of MFPS, volume 298 of Lecture Notes in Computer Science, pp. 331–343. Springer (1987)
Reed, G.M., Roscoe, A.W.: The timed failures-stability model for CSP. Theor. Comput. Sci. 211(1–2), 85–127 (1999)
Sistla, A.P., Vardi, M.Y., Wolper, P.: The complementation problem for Büchi automata with applications to temporal logic (extended abstract). In: Proceedings of ICALP, volume 194 of Lecture Notes in Computer Science, pp. 465–474. Springer (1985)
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Haase, C., Ouaknine, J., Worrell, J. (2010). On Process-Algebraic Extensions of Metric Temporal Logic. In: Roscoe, A., Jones, C., Wood, K. (eds) Reflections on the Work of C.A.R. Hoare. Springer, London. https://doi.org/10.1007/978-1-84882-912-1_13
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