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Duration Calculus

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Logics of Specification Languages

Abstract

Duration Calculus (DC) is an interval logic which was introduced to express and reason about models of real-time systems. DC was introduced by Zhou Chaochen, Tony Hoare and A.P. Ravn during the ProCoS I Project (ESPRIT BRA 3104, 1989–1991) [6]. Formal techniques for the construction of safetycritical systems were investigated in this project, and in an early case study of gas burner systems, conducted by E.V. Sørensen, A.P. Ravn and H. Rischel, it turned out that certain requirements for such systems were not expressible in the real-time formalisms which were available at that time.

This work is partially funded by The Danish Council for Strategic Research under project MoDES.

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References

  1. J.F. Allen (1983) Maintaining Knowledge about Temporal Intervals. Communications of the ACM 26(11):832–843, 1983

    Article  MATH  Google Scholar 

  2. J.F. Allen (1984) Towards a general theory of action and time. Artificial Intelligence 23:123–154

    Article  MATH  Google Scholar 

  3. R. Alur, C. Courcoubetis, T. Henzinger, P.H. Ho (1993) Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. In Hybrid Systems, LNCS 736, Springer, Berlin Heidelberg New York: 209–229

    Chapter  Google Scholar 

  4. R. Alur, D. Dill (1992) The theory of timed automata. In Real-Time: Theory in Practice, LNCS 600, Springer, Berlin Heidelberg New York: 45–73

    Chapter  Google Scholar 

  5. R. Barua, S. Roy, and C. Zhou (2000) Completeness of neighbourhood logic. Journal of Logic and Computation 10(2): 271–295

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Bjørner (1992) Trusted computing systems: the ProCoS experience. In Proceedings of the 14th international conference on Software engineering, Melbourne, Australia, ACM Press: 15–34

    Google Scholar 

  7. P. Blackburn, M. de Rijke, Y. Venema (2001) Modal Logic. Cambridge University Press

    Google Scholar 

  8. T. Bolander, J.U. Hansen, M.R. Hansen (2006) Decidability of hybrid duration calculus. In InternationalWorkshop on Hybrid Logics, Seattle 2006. To appear in Electronic Notes in Theoretical Computer Science.

    Google Scholar 

  9. V.A. Braberman, D.V. Hung (1998) On checking timed automata for linear duration invariants. In Proceedings of the 19th IEEE Real-Time Systems Symposium, IEEE Computer Society Press: 264–273

    Google Scholar 

  10. D. Basin, S. Matthews, L. Vigano (1997) Labelled propositional modal logics: theory and practice. Journal of logic and computation 7(6):685–717

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Basin, S. Matthews, L. Vigano(1998) Labelled modal logics: quantifiers. Journal of logic, language and information 7(3):237–263

    Article  MathSciNet  MATH  Google Scholar 

  12. J.R. Buchi (1960) Weak second-order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 6:66–92

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Chan, D.V. Hung (1995) Duration calculus specification of scheduling for tasks with shared resources. In Asian Computing Science Conference 1995, LNCS 1023, Springer, Berlin Heidelberg New York: 365–380

    Google Scholar 

  14. Z. Chen, J. Wang, C. Zhou (1995) An abstraction of hybrid control systems. In IEEE Singapore Intl. Conf. on Intelligent Control and Instrumentation. IEEE Press:1–6

    Google Scholar 

  15. S. Colin, G. Mariano, V. Poirriez (2004) Duration Calculus: A Real-Time Semantics for B. In Theoretical Aspects of Computing — ICTAC 2004, LNCS 3407, Springer, Berlin Heidelberg New York: 431–446

    Chapter  Google Scholar 

  16. S. Dong, Q. Xu, N. Zhan (1999) A formal proof of the rate monotonic scheduler. In Proceedings of the Sixth International Conference on Real-Time Computing Systems and Applications. IEEE Computer Society Press:500–507

    Google Scholar 

  17. B. Dutertre (1995) Complete proof systems for first order interval temporal logic. In Tenth Annual IEEE Symposium on Logic in Computer Science, IEEE Press:36–43

    Google Scholar 

  18. B. Dutertre (1995) On first order interval temporal logic. Technical report, Report No. CSD-TR-94-3, Department of Computer Science, Royal Holloway, University of London

    Google Scholar 

  19. C.C. Elgot (1961) Decision problems of finite automata design and related arithmetics. Transactions of the American Mathematical Society 98:21–52

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Engel, M. Kubica, J. Madey, D.L. Parnas, A.P. Ravn, A.J. van Schouwen (1993) A formal approach to computer systems requirements documentation. In Hybrid Systems, LNCS 736, Springer, Berlin Heidelberg New York: 452–474

    Chapter  Google Scholar 

  21. M. Engel, H. Rischel (1994) Dagstuhl-seminar specification problem-a duration calculus solution. Technical report, Department of Computer Science, Technical University of Denmark — Personal Communication

    Google Scholar 

  22. M. Fränzle, M.R. Hansen (2005) A Robust Interpretation of Duration Calculus. In Theoretical Aspects of Computing — ICTAC 2005, LNCS 3722, Springer, Berlin Heidelberg New York: 257–271

    Chapter  Google Scholar 

  23. M. Fränzle, M.R. Hansen (2007) Deciding an Interval Logic with Accumulated Durations. In Tools and algorithms for the construction and analysis of systems — TACAS 2007, LNCS 4424, Springer, Berlin Heidelberg New York: 201–215

    Google Scholar 

  24. M. Fränzle (1996) Synthesizing controllers from duration calculus. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 1135, Springer, Berlin Heidelberg New York:168–187

    Chapter  Google Scholar 

  25. M. Fränzle (1997) Controller Design from Temporal Logic: Undecidability Need Not Matter. PhD Thesis Institut für Informatik und Praktische Mathematik der Christian-Albrechts-Universität Kiel

    Google Scholar 

  26. M. Fränzle (2002) Take it NP-easy: Bounded model construction for duration calculus. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 2469, Springer, Berlin Heidelberg New York:245–264

    Chapter  Google Scholar 

  27. V. Goranko, A. Montanari, G. Sciavicco (2004) A Road Map of Propositional Interval Temporal Logics and Duration Calculi. Journal of Applied Nonclassical Logics, Special issue on Interval Temporal Logics and Duration Calculi, 14(1–2):11–56

    MATH  Google Scholar 

  28. D.P. Guelev (1998) A calculus of durations on abstract domains: Completeness and extensions. Technical report 139, UNU/IIST (UN University, Inst. of Software Technologz)

    Google Scholar 

  29. D.P. Guelev (2000) Complete fragment of higher-order duration μ-calculus. In Foundations of Software Technology and Theoretical Computer Science (FST&TCS’2000), LNCS 1974, Springer, Berlin Heidelberg New York:264–276

    Google Scholar 

  30. D.V. Guelev, D.V. Hung (1999) On the completeness and decidability of duration calculus with iteration. In Advances in Computing Science, LNCS 1742, Springer, Berlin Heidelberg New York:139–150

    Google Scholar 

  31. D.V. Guelev, D.V. Hung (2002) Prefix and projection onto state in duration calculus. In Proceedings of Workshop on Theory and Practice of Timed Systems, Electronic Notes in Theoretical Computer Science

    Google Scholar 

  32. J. Halpern, B. Moskowski, Z. Manna (1983) A hardware semantics based on temporal intervals. In ICALP’83, LNCS 154, Springer, Berlin Heidelberg New York:278–291

    Google Scholar 

  33. J. Halpern, Y. Shoham (1986) A propositional modal logic of time intervals. In Proceedings of the First IEEE Symposium on Logic in Computer Science, IEEE Press:279–292

    Google Scholar 

  34. J. Halpern, Y. Shoham (1991) A propositional modal logic of time intervals. Journal of the ACM 33(4):935–962

    Article  MathSciNet  MATH  Google Scholar 

  35. K.M. Hansen, A.P. Ravn, V. Stavridou (1996) From safety analysis to formal specification. Technical report, Department of Information Technology, Technical University of Denmark

    Google Scholar 

  36. M.R. Hansen (1994) Model-checking discrete duration calculus. Formal Aspects of Computing 6A:826–845

    Article  MATH  Google Scholar 

  37. M.R. Hansen, C. Zhou (1992) Semantics and completeness of duration calculus. In Real-Time: Theory in Practice, LNCS 600, Springer, Berlin Heidelberg New York:209–225

    Chapter  Google Scholar 

  38. M.R. Hansen, C. Zhou (1997) Duration calculus: Logical foundations. Formal Aspects of Computing 9:283–330

    Article  MATH  Google Scholar 

  39. M.R. Hansen, C. Zhou, J. Staunstrup(1992) A real-time duration semantics for circuits. In TAU’92: 1992 Workshop on Timing Issues in the Specification and Synthesis of Digital Systems, Princeton Univ., NJ. ACM/SIGDA

    Google Scholar 

  40. M.R. Hansen, P.K. Pandya, C. Zhou (1995) Finite divergence. Theoretical Computer Science 138:113–139

    Article  MathSciNet  MATH  Google Scholar 

  41. M.R. Hansen, R. Sharp (2003) Using interval logics for temporal analysis of security protocols. In First ACM Workshop on Formal Methods in Security Engineering (FMSE’03), ACM Press:24–31

    Google Scholar 

  42. D. Harel, O. Lichtenstein, A. Pnueli (1990) Explicit clock temporal logic. In Symposium on Logic in Computer Science, IEEE Press:402–413

    Google Scholar 

  43. J. Harrison (1998) Theorem Proving with the Real Numbers. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  44. J. Hoenicke, E.R. Olderog (2002) CSP-OZ-DC: A combination of specification techniques for processes, data and time. Nordic Journal of Computing 9(4):301–334

    MathSciNet  MATH  Google Scholar 

  45. G.E. Hughes, M.J. Crestwell (1968) An Introduction to Modal Logic. Routledge

    Google Scholar 

  46. G.E. Hughes, M.J Crestwell (1984) A Companion to Modal Logic. Methuen

    Google Scholar 

  47. Z. Huibiao, J.F. He (2000) A DC-based semantics for Verilog. In Proceedings of International Conference on Software: Theory and Practice. Publishing House of Electronics Industry:421–432

    Google Scholar 

  48. D.V. Hung (1998) Modelling and verification of biphase mark protocols in duration calculus using PVS/DC. In Application of Concurrency to System Design (CSD’98), IEEE Computer Society Press:88–98

    Google Scholar 

  49. D.V. Hung, C. Zhou (1999) Probabilistic duration calculus for continuous time. Formal Aspects of Computing, 11(1):21–44

    Article  MATH  Google Scholar 

  50. D.V. Hung, P.H. Giang (1996) A sampling semantics of duration calculus. In Formal Techniques for Real-Time and Fault Tolerant Systems, LNCS 1135. Springer, Berlin Heidelberg New York: 188–207

    Chapter  Google Scholar 

  51. D.V. Hung, D.P. Guelev (1999) Completeness and decidability of a fragment of duration calculus with iteration. In Advances in Computing Science, LNCS 1742. Springer, Berlin Heidelberg New York:139–150

    Google Scholar 

  52. D.V. Hung, J. Wand (1996) On design of hybrid control systems using i/o automata models. In Foundations of Software Technology and Theoretical Computer Science, LNCS 1180. Springer, Berlin Heidelberg New York:156–167

    Chapter  Google Scholar 

  53. R. Inal (1994) Modular specification of real-time systems. In 1994 Euromicro Workshop on Real-Time Systems. IEEE Computer Society Press

    Google Scholar 

  54. J.F. He (1994) From CSP to hybrid systems. In A Classical Mind: Essays in Honour of C.A.R. Hoare. Prentice Hall International:171–190

    Google Scholar 

  55. J.F. He, J. Bowen (1992) Time interval semantics and implementation of a real-time programming language. In 1992 Euromicro Workshop on Real-Time Systems. IEEE Computer Society Press

    Google Scholar 

  56. J.F. He, C.A.R. Hoare, M. Fränzle, M. Müller-Olm, E.R. Olderog, M. Schenke, M.R. Hansen, A.P. Ravn, H. Rischel (1994) Provably correct systems. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 863. Springer, Berlin Heidelberg New York:288–335

    Chapter  Google Scholar 

  57. J.F. He, Q. Xu (2000) Advanced features of the duration calculus. In Millennial Perspectives in Computer Science. Palgrave:133–146

    Google Scholar 

  58. W. He, C. Zhou (1995) A case study of optimization. The Computer Journal 38(9):734–746

    Article  Google Scholar 

  59. K.T. Hong, D.V. Hung (2001) Formal design of hybrid control systems: Duration calculus approach. In Proceedings of the Twenty-Fifth Annual International Computer Software and Applications Conference. IEEE Computer Society Press:423–428

    Google Scholar 

  60. F. Jahanian, A.K.L Mok (1986) Safety analysis of timing properties in real-time systems. IEEE Transaction on Software Engineering 12(9)

    Google Scholar 

  61. J. Gao, Q. Xu (1997) Rigorous design of a fault diagnosis and isolation algorithm. In Proceedings of the Fifth International Workshop on Hybrid Systems

    Google Scholar 

  62. Y. Kesten, A. Pnueli, J. Sifakis, S. Yovine (1993) Integration graphs: A class of decidable hybrid systems. In Hybrid Systems, LNCS 736. Springer, Berlin Heidelberg New York:179–208

    Chapter  Google Scholar 

  63. C. Kleuker (2000) Constraint Diagrams. PhD Thesis, Oldenburg University, Germany

    MATH  Google Scholar 

  64. N. Klarlund, A. Moller (2001) MONA Version 1.4: User Manual. BRICS, Department of Computer Science, University of Aarhus, Denmark

    Google Scholar 

  65. M. Kääramees (1995) Transformation of duration calculus specifications to DISCO language. Master’s Thesis, Automation and Systems Engineering, Tallinn Technical University, Estonia

    Google Scholar 

  66. M. Kääramees(1995) Transforming designs towards implementations. In 1995 Euromicro Workshop on Real-Time Systems, IEEE Computer Society Press:197–204

    Google Scholar 

  67. R. Koymans (1990) Specifying real-time properties with metric temporal logic. Real-Time Systems 2(4):255–299

    Article  Google Scholar 

  68. L. Lamport (1993) Hybrid systems in tla+. In Hybrid Systems, LNCS 736. Springer p. 77–102

    Google Scholar 

  69. X. Li (1993) A Mean Value Calculus. PhD Thesis, Software Institute, Academia Sinica

    Google Scholar 

  70. X. Li, J. Wang (1996) Specifying optimal design of a steam-boiler system. In Formal Methods for Industrial Applications, LNCS 1165. Springer Berlin Heidelberg New York:359–378

    Chapter  Google Scholar 

  71. X. Li, D.V. Hung (1996) Checking linear duration invariants by linear programming. In Concurrency and Parallelism, Programming, Networking, and Security, LNCS 1179. Springer Berlin Heidelberg New York:321–332

    Chapter  Google Scholar 

  72. X. Li, D.V. Hung, T. Zheng T (1997) Checking hybrid automata for linear duration invariants. In Advances in Computing Science, LNCS 1997. Springer Berlin Heidelberg New York:166–180

    Google Scholar 

  73. L. Li, J.F. He (1999) A denotational semantics of timed rsl using duration calculus. In Proceedings of the Sixth International Conference on Real-Time Computing Systems and Applications. IEEE Computer Society Press:492–503

    Google Scholar 

  74. C.L. Liu, J.W. Layland (1973) Scheduling algorithm for multiprogramming in a hard real-time environment. Journal of the ACM 20(1):46–61

    Article  MathSciNet  MATH  Google Scholar 

  75. Z. Liu, A.P. Ravn, E.V. Sørensen, C. Zhou (1993) A probabilistic duration calculus. In Dependable Computing and Fault-Tolerant Systems Vol. 7: Responsive Computer Systems. Springer Berlin Heidelberg New York:30–52

    Google Scholar 

  76. Z. Liu, A.P. Ravn, E.V. Sørensen, C. Zhou (1994) Towards a calculus of systems dependability. High Integrity Systems 1(1):49–75

    Google Scholar 

  77. Z. Liu, J. Nordahl, E.V. Sørensen (1995) Composition and refinement of probabilistic real-time systems. In Mathematics of Dependable Systems, Oxford University Press:149–163

    Google Scholar 

  78. Z. Liu (1996) Specification and verification in DC. In Mathematics of Dependable Systems, International Series in Computer Science, Prentice Hall p.182–228

    Google Scholar 

  79. Z. Manna, A. Pnueli (1993) Verifying hybrid systems. In Hybrid Systems, LNCS 736. Springer, Berlin Heidelberg New York:4–35

    Chapter  Google Scholar 

  80. X. Mao, Q. Xu, D.V. Hung, J. Wang (1996) Towards a proof assistant for interval logics. Technical report, UNU/IIST Report No. 77, UN University, International Institute for Software Technology, Macau

    Google Scholar 

  81. P.C. Masiero, A.P. Ravn, H. Rischel (1993) Refinement of real-time specifications. Technical Report ProCoS, Technical Report ID/DTH PCM 1/1, Department of Computer Science, Technical University of Denmark

    Google Scholar 

  82. C. Meadows (2001) A cost-based framework for analysis of denial-of-service in networks. Journal of Computer Security 9(1/2):143–164

    Article  Google Scholar 

  83. C.A. Middelburg (1998) Truth of duration calculus formulae in timed frames. Fundamenta Informaticae Journal 36(2/3):235–263

    MathSciNet  MATH  Google Scholar 

  84. S. Mørk, J.C. Godskesen, M.R. Hansen, R. Sharp (1996) A timed semantics for sdl. In Formal Description Techniques IX: Theory, application and tools. Chapman & Hall:295–309

    Google Scholar 

  85. B. Moszkowski (1985) A temporal logic for multilevel reasoning about hardware. IEEE Computer 18(2):10–19

    Article  Google Scholar 

  86. B. Moszkowski (1995) Compositional reasoning about projected and infinite time. In Engineering of Complex Computer Systems, IEEE Computer Society Press:238–245

    Google Scholar 

  87. X. Nicollin, A. Olivero, J. Sifakis, S. Yovine (1993) An approach to the description and analysis of hybrid systems. In Hybrid Systems, LNCS 736. Springer, Berlin Heidelberg New York:149–178

    Chapter  MATH  Google Scholar 

  88. E.R. Olderog, A.P. Ravn, J.U. Skakkebæk (1996) Refining system requirements to program specifications. In Formal Methods in Real-Time Systems, Trends in Software-Engineering, chapter 5. Wiley

    Google Scholar 

  89. S. Owre, N. Shankar, J. Rushby (1993) Users guide for the pvs specification and verification system, language, and proof checker (beta release) (three volumes). Technical report, Computer Science Laboratory, SRI International, Menlo Park, CA

    Google Scholar 

  90. P. Øhrstrøm, P.F. Hasle (1995) Temporal Logic: From Ancient Ideas to Artificial Intelligence. Kluwer Academic

    Google Scholar 

  91. P.K. Pandya (1996) Some extensions to propositional mean value calculus: Expressiveness and decidability. In Computer Science Logic, CSL’95, LNCS 1092. Springer, Berlin Heidelberg New York:434–451

    Chapter  Google Scholar 

  92. P.K. Pandya (1996) Weak chop inverses and liveness in duration calculus. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 1135. Springer, Berlin Heidelberg New York:148–167

    Chapter  Google Scholar 

  93. P.K. Pandya (2000) Specifying and deciding quantified discrete-time duration calculus formulae using DCVALID. Technical report TCS00-PKP-1, Tata Institute of Fundamental Research, India.

    Google Scholar 

  94. P.K. Pandya (1999) DCVALID 1.3: The user manual. Technical report TCS-99/1, Computer Science Group, TIFR, Bombay

    Google Scholar 

  95. P.K. Pandya, D.V. Hung (1998) Duration calculus with weakly monotonic time. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 1486. Springer, Berlin Heidelberg New York:55–64

    Chapter  Google Scholar 

  96. P.K. Pandya, Y.S. Ramakrishna, R.K. Shyamasundar (1995) A compositional semantics of Esterel in duration calculus. Technical report, Computer Science Group, TIFR, Bombay

    Google Scholar 

  97. L.C. Paulson (1994) Isabelle, A Generic Theorem Prover, LNCS 828. Springer Berlin Heidelberg New York

    MATH  Google Scholar 

  98. L.C. Paulson (1997) Proving properties of security protocols by induction. In Proceedings of the 10th IEEE Computer Security Foundations Workshop, Rockport, Mass. IEEE Press:70–83

    Google Scholar 

  99. J.L. Petersen, H. Rischel (1994) Formalizing requirements and design for a production cell system. In Symposium ADPM’ 94: Automatisation des Processus Mixtes: Les Systemes Dynamiques Hybrides. Belgian Institute of Automatic Control, IBRA:37–46

    Google Scholar 

  100. H. Pilegaard (2002) Modelling properties of security protocols. MA Thesis, Informatics and Mathematical Modelling, Technical University of Denmark

    Google Scholar 

  101. H. Pilegaard, M.R. Hansen, R. Sharp (2003) An approach to analyzing availability properties of security protocols. Nordic Journal of Computing 10:337–373

    MathSciNet  MATH  Google Scholar 

  102. Z. Qiu, C. Zhou (1998) A combination of interval logic and linear temporal logic. In Programming Concepts and Methods. Chapman & Hall:444–461

    Google Scholar 

  103. A. Rabinovich (1998) Non-elementary lower bound for propositional duration calculus. Information Processing Letters: 7–11

    Google Scholar 

  104. A. Rabinovich (1998) On the decidability of continuous time specification formalism. Journal of Logic and Computation 8(5):669–678.

    Article  MathSciNet  MATH  Google Scholar 

  105. T.M. Rasmussen (1999) Signed interval logic. In Computer Science Logic, CSL’99, LNCS 1683. Springer Berlin Heidelberg New York:157–171

    Chapter  Google Scholar 

  106. T.M. Rasmussen (2001) Automated proof support for interval logics. In LPAR 2001, LNAI 2250. Springer Berlin Heidelberg New York:317–326

    Google Scholar 

  107. T.M. Rasmussen (2001) Labelled natural deduction for interval logics. In Computer Science Logic, CSL’01, LNCS 2142. Springer Berlin Heidelberg New York:308–323

    Chapter  Google Scholar 

  108. T.M. Rasmussen (2002) Interval Logic: Proof Theory and Theorem Proving. PhD Thesis, Informatics and Mathematical Modelling, Technical University of Denmark

    Google Scholar 

  109. A.P. Ravn, T.J. Eriksen, M. Holdgaard, H. Rischel (1998) Engineering of real-time systems with an experiment in hybrid control. In Embedded Systems, LNCS 1494. Springer Berlin Heidelberg New York:316–352

    Google Scholar 

  110. A.P. Ravn (1995) Design of Embedded Real-Time Computing Systems. Doctoral Dissertation, Department of Computer Science, Technical University of Denmark

    Google Scholar 

  111. A.P. Ravn, H. Rischel (1991) Requirements capture for embedded real-time systems. In Proceedings of IMACS-MCTS’91 Symposium on Modelling and Control of Technological Systems, Villeneuve d’Ascq, France, volume 2. IMACS:147–152

    Google Scholar 

  112. A.P. Ravn, H. Rischel, K.M. Hansen (1993) Specifying and verifying requirements of real-time systems. IEEE Transactions on Software Engineering 19(1):41–55

    Article  Google Scholar 

  113. H. Rischel (1992) A duration calculus proof of fischer’s mutual exclusion protocol. ProCoS II, ESPRIT BRA 7071, Report No. DTH HR 4/1, Department of Computer Science, Technical University of Denmark

    Google Scholar 

  114. S. Roy, C. Zhou (1997) Notes on neighbourhood logic. UNU/IIST Report No. 97, International Institute for Software Technology, Macau

    Google Scholar 

  115. S. Satpathy, D.V. Hung, P.K. Pandya (1998) Some results on the decidability of duration calculus under synchronous interpretation. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 1486. Springer, Berlin Heidelberg New York:186–197

    Chapter  Google Scholar 

  116. M. Schenke (1994) Specification and transformation of reactive systems with time restrictions and concurrency. In Techniques in Real-Time and Fault-Tolerant Systems, LNCS 863. Springer, Berlin Heidelberg New York:605–620

    Chapter  Google Scholar 

  117. M. Schenke (1995) Requirements to programs: A development methodology for real time systems, part 2. Technical report, Fachbereich Informatik, Universität Oldenburg, Germany

    Google Scholar 

  118. M. Schenke, E.R. Olderog (1995) Requirements to programs: A development methodology for real time systems, part 1. Technical report, Fachbereich Informatik, Universität Oldenburg, Germany

    Google Scholar 

  119. M. Schenke, A.P. Ravn (1996) Refinement from a control problem to programs. In Formal Methods for Industrial Applications, LNCS 1165. Springer Berlin Heidelberg New York:403–427

    Chapter  Google Scholar 

  120. G. Schneider, Q. Xu (1998) Towards a formal semantics of verilog using duration calculus. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 1486. Springer, Berlin Heidelberg New York:282–293

    Chapter  Google Scholar 

  121. A. Schäfer (2004). A Calculus for Shapes in Time and Space. In Theoretical Aspects of Computing, ICTAC 2004, LNCS 3407. Springer Berlin Heidelberg New York:463–477

    Chapter  Google Scholar 

  122. J.U. Skakkebæk (1994) Liveness and fairness in duration calculus. In CONCUR’94: Concurrency Theory, LNCS 836. Springer, Berlin Heidelberg New York: 283–298

    Chapter  Google Scholar 

  123. J.U. Skakkebæk (1994) A Verification Assistant for a Real-Time Logic. PhD Thesis, Department of Computer Science, Technical University of Denmark

    Google Scholar 

  124. J.U. Skakkebæk, A.P. Ravn, H. Rischel, C. Zhou (1992) Specification of embedded, real-time systems. In Proceedings of 1992 Euromicro Workshop on Real-Time Systems. IEEE Computer Society Press

    Google Scholar 

  125. J.U. Skakkebæk, P. Sestoft (1994) Checking validity of duration calculus formulas. Technical report, ProCoS II, ESPRIT BRA 7071, report no. ID/DTH JUS 3/1, Department of Computer Science, Technical University of Denmark

    Google Scholar 

  126. J.U. Skakkebæk, N. Shankar (1994) Towards a duration calculus proof assistant in pvs. In Techniques in Real-Time and Fault-Tolerant Systems, LNCS 863. Springer, Berlin Heidelberg New York:660–679

    Chapter  Google Scholar 

  127. E.V. Sørensen, A.P. Ravn, H. Rischel (1990) Control program for a gas burner: Part 1: Informal requirements, ProCoS case study 1. Technical Report ID/DTH EVS2

    Google Scholar 

  128. A. Tarski (1948) A decision method for elementary algebra and geometry. RAND Corporation, Santa Monica, California

    MATH  Google Scholar 

  129. Y. Venema (1990) Expressiveness and completeness of an interval tense logic. Notre Dame Journal of Formal Logic, 31(4):529–547

    Article  MathSciNet  MATH  Google Scholar 

  130. Y. Venema (1991) A modal logic for chopping intervals. Journal of Logic and Computation 1(4):453–476

    Article  MathSciNet  MATH  Google Scholar 

  131. L. Vigano (2000) Labelled non-classical logics. Kluwer Academic Punlishers

    Google Scholar 

  132. J. Wang, H. Weidong (1996) Formal specification of stability in hybrid control systems. In Hybrid Systems III, LNCS 1066. Springer, Berlin Heidelberg New York:294–303

    Chapter  Google Scholar 

  133. B.H. Widjaja, W. He, Z. Chen, C. Zhou (1996) A cooperative design for hybrid control systems. In Proceedings of Logic and Software Engineering International Workshop in Honor of Chih-Sung Tang. World Scientific:127–150

    Google Scholar 

  134. Q. Xu, M. Swarup (1998) Compositional reasoning using assumption-commitment paradigm. In Compositionality-The Significant Difference, LNCS 1536. Springer Berlin Heidelberg New York:565–583

    Chapter  Google Scholar 

  135. Q. Xu, H. Weidong (1996) Hierarchical design of a chemical concentration control system. In Hybrid Systems III: Verification and Control, LNCS 1066. Springer Berlin Heidelberg New York:270–281

    Chapter  Google Scholar 

  136. Q. Xu, Z. Yang (1996) Derivation of control programs: a heating system. UNU/IIST Report No. 73, International Institute for Software Technology, Macau

    Google Scholar 

  137. X. Yu, J. Wang, C. Zhou, and P.K. Pandya (1994) Formal design of hybrid systems. In Techniques in Real-Time and Fault-Tolerant Systems, LNCS 863. Springer, Berlin Heidelberg New York:738–755

    Chapter  Google Scholar 

  138. J. Zhao, D.V. Hung (1998) On checking real-time parallel systems for linear duration properties. In Formal Techniques in Real-Time and Fault-Tolerant Systems, LNCS 1486. Springer, Berlin Heidelberg New York:241–250

    Chapter  Google Scholar 

  139. Y. Zheng, C. Zhou (1994) A formal proof of the deadline driven scheduler. In Techniques in Real-Time and Fault-Tolerant Systems, LNCS 863. Springer, Berlin Heidelberg New York:756–775

    Chapter  Google Scholar 

  140. C. Zhou (1993) Duration Calculi: An overview. In Proceedings of Formal Methods in Programming and Their Applications, LNCS 735. Springer Berlin Heidelberg New York:256–266

    Chapter  Google Scholar 

  141. C. Zhou, D.P. Guelev, N. Zhan (2000) A higher-order duration calculus. In Millennial Perspectives in Computer Science. Palgrave:407–416.

    Google Scholar 

  142. C. Zhou, M.R. Hansen (1996) Chopping a point. In BCS-FACS 7th Refinement Workshop. Electronic Workshops in Computing, Springer Berlin Heidelberg New York

    Google Scholar 

  143. C. Zhou, M.R. Hansen (1998) An adequate first order logic of intervals. In Compositionality: The Significant Difference, LNCS 1536. Springer Berlin Heidelberg New York:584–608

    Google Scholar 

  144. C. Zhou, M.R. Hansen, A.P. Ravn, H. Rischel (1991) Duration specifications for shared processors. In Symposium on Formal Techniques in Real-Time and Fault Tolerant Systems, LNCS 571. Springer Berlin Heidelberg New York:21–32

    Google Scholar 

  145. C. Zhou, M.R. Hansen (2004) Duration Calculus: A formal approach to realtime systems. Springer

    Google Scholar 

  146. C. Zhou, M.R. Hansen, P. Sestoft (1993) Decidability and undecidability results for duration calculus. In STACS’93, LNCS 665. Springer Berlin Heidelberg New York:58–68

    Google Scholar 

  147. C. Zhou, C.A.R. Hoare, A.P. Ravn (1991) A calculus of durations. Information Processing Letters 40(5):269–276

    Article  MathSciNet  MATH  Google Scholar 

  148. C. Zhou, D.V. Hung, X. Li (1995) A duration calculus with infinite intervals. In Fundamentals of Computation Theory, LNCS, Springer p. 16–41

    Google Scholar 

  149. X. Zhou, J. Wang, A.P. Ravn (1996) A formal description of hybrid systems. In Hybrid Systems III, LNCS 1066. Springer Berlin Heidelberg New York:511–530

    Chapter  Google Scholar 

  150. C. Zhou, J. Zhang, L. Yang, X. Li (1994) Linear duration invariants. In Techniques in Real-Time and Fault-Tolerant Systems, LNCS 863. Springer, Berlin Heidelberg New York:86–109

    Chapter  Google Scholar 

  151. C. Zhou, A.P. Ravn, M.R. Hansen (1993) An extended duration calculus for hybrid systems. In Hybrid Systems, LNCS 736. Springer, Berlin Heidelberg New York:36–59

    Chapter  Google Scholar 

  152. C. Zhou, X. Li (1994) A mean value calculus of durations. In A Classical Mind: Essays in Honour of C.A.R. Hoare. Prentice Hall International:431–451

    Google Scholar 

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Hansen, M.R. (2008). Duration Calculus. In: Bjørner, D., Henson, M.C. (eds) Logics of Specification Languages. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74107-7_6

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