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An Abstraction Framework for Mixed Non-deterministic and Probabilistic Systems

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Validation of Stochastic Systems

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2925))

Abstract

We study abstraction techniques for model checking systems that combine non-deterministic with probabilistic behavior, emphasizing the discrete case. Existing work on abstraction offers a host of isolated techniques which we discuss uniformly through the formulation of abstracted model-checking problems (MCPs). Although this conceptualization is primarily meant to be a useful focal point for surveying the literature on abstraction-based model checking even beyond such combined systems, it also opens up new research opportunities and challenges for abstract model checking of mixed systems. In particular, we sketch how quantitative domain theory may be used to specify the precision of answers obtained from abstract model checks.

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References

  1. Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford Univ. Press, Oxford (1994)

    Google Scholar 

  2. Baeten, J.C.M., Weijland, W.P.: Process Algebra. Cambridge Tracts in Theoretical Computer Science, vol. 18. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  3. Baier, C., Engelen, B., Majster-Cederbaum, M.: Deciding Bisimilarity and Similarity for Probabilistic Processes. Journal of Computer and System Sciences 60, 187–231 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baier, C., Stoelinga, M.I.A.: Norm functions for probabilistic bisimulations with delays. In: Tiuryn, J. (ed.) FOSSACS 2000. LNCS, vol. 1784, pp. 1–16. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  5. Baier, C., Kwiatkowska, M., Norman, G.: Computing probability bounds for linear time formulas over concurrent probabilistic systems. In: Kwiatkowska, M., Baier, C., Huth, M., Ryan, M. (eds.) Electronic Notes in Theoretical Computer Science, vol. 22. Elsevier Science Publishers, Amsterdam (2000)

    Google Scholar 

  6. Bradfield, J.C.: Verifying Temporal Properties Of Systems. Birkhäuser, Boston (1991)

    Google Scholar 

  7. Bruns, G., Godefroid, P.: Model Checking Partial State Spaces with 3-Valued Temporal Logics. In: Halbwachs, N., Peled, D. (eds.) CAV 1999. LNCS, vol. 1633, pp. 274–287. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  8. Bruns, G., Godefroid, P.: Generalized Model Checking: Reasoning about Partial State Spaces. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 168–182. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  9. Bryant, R.R.: Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams. ACM Computing Surveys 24(3), 293–318 (1992)

    Article  Google Scholar 

  10. Burch, J.R., Clarke, E.M., Dill, D.L., McMillan, K.L., Hwang, J.: Symbolic model checking: 1020 states and beyond. In: Proceedings of the Fifth Annual Symposium on Logic in Computer Science (June 1990)

    Google Scholar 

  11. Cattani, S., Segala, R.: Decision algorithms for probabilistic bisimulation. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 371–385. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  12. Cerans, K., Godskesen, J.C., Larsen, K.G.: Timed modal specification - theory and tools. In: Computer Aided Verification, pp. 253–267 (1993)

    Google Scholar 

  13. Chen, Y.-F., Gansner, E.R., Koutsofios, E.: A C++ data model supporting reachability analysis and dead code detection. In: Jazayeri, M., Schauer, H. (eds.) Proceedings of the Sixth European Software Engineering Conference (ESEC/FSE 1997), pp. 414–431. Springer, Heidelberg (1997)

    Google Scholar 

  14. Clark, D., Hankin, C., Hunt, S., Nagarajan, R.: Possibilistic Information Flow is safe for Probabilistic Non-Interference. In: Workshop on Issues in the Theory of Security WITS 2000, Geneva, Switzerland, July 7-8 (2000)

    Google Scholar 

  15. Clarke, E., Biere, A., Raimi, R., Zhu, Y.: Bounded Model Checking Using Satisfiability Solving. Formal Methods in System Design 19(1), 7–34 (2001)

    Article  MATH  Google Scholar 

  16. Clarke, E.M., Fujita, M., Zhao, X.: Representations of discrete functions, chapter Multi-terminal binary decision diagrams and hybrid decision diagrams, pp. 93–108. Kluwer academic publishers, Dordrecht (1996)

    Google Scholar 

  17. Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2000)

    Google Scholar 

  18. Clarke, E.M., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Transactions on Programming Languages and Systems 16(5), 1512–1542 (1994)

    Article  Google Scholar 

  19. Cleaveland, R., Smolka, S.A., Zwarico, A.E.: Testing preorders for probabilistic processes. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 708–719. Springer, Heidelberg (1992)

    Google Scholar 

  20. Courcoubetis, C., Yannakakis, M.: The Complexity of Probabilistic Verification. Journal of the Association of Computing Machinery 42(4), 857–907 (1995)

    MATH  MathSciNet  Google Scholar 

  21. Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs. In: Proc. 4th ACM Symp. on Principles of Programming Languages, pp. 238–252. ACM Press, New York (1977)

    Google Scholar 

  22. Cousot, P., Cousot, R.: Temporal abstract interpretation. In: Conference Record of the 27th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Boston, Mass, pp. 12–25. ACM Press, New York (2000)

    Google Scholar 

  23. Dams, D.: Abstract Interpretation and Partition Refinement for Model Checking. PhD thesis, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (July 1996)

    Google Scholar 

  24. D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability Analysis of Probabilistic Systems by Successive Refinements. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  25. D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reduction and Refinement Strategies for Probabilistic Analysis. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, pp. 57–76. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  26. Denning, D.: A Lattice Model of Secure Information Flow. Communications of the ACM 19(5), 236–243 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  27. Denning, D.: Certification of Programs for Secure Information Flow. Communications of the ACM 20(7), 504–513 (1977)

    Article  MATH  Google Scholar 

  28. Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for Labelled Markov Processes. Journal of Information and Computation 179(2), 163–193 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating Labeled Markov Processes. In: 15th Annual IEEE Symposium on Logic in Computer Science LICS 2000, Santa Barbara, California. IEEE Computer Society Press, Los Alamitos (2000)

    Google Scholar 

  30. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Weak bisimulation is sound and complete for PCTL*. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 355–370. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  31. Dwyer, M.B., Schmidt, D.A.: Limiting State Explosion with Filter-Based Refinement. In: Proceedings of the ILPS 1997 Workshop on Verification, Model Checking, and Abstraction (1997)

    Google Scholar 

  32. http://www.fsel.com/fdr2_download.html

  33. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Heidelberg (1980)

    MATH  Google Scholar 

  34. Gilmore, S., Ryan, M. (eds.): Language Constructs for Describing Features. In: Proc. of the FIREworks workshop. Springer, Heidelberg (2001)

    Google Scholar 

  35. Godefroid, P.: Model Checking for Programming Languages using VeriSoft. In: Proceedings of the 24th ACM Symposium on Principles of Programming Languages, Paris, January 1997, pp. 174–186 (1997)

    Google Scholar 

  36. Graf, S., Sifakis, J.: Readiness Semantics for Regular Processes with Silent Actions. In: Proc. of ICALP 1987, pp. 115–125 (1987)

    Google Scholar 

  37. Halmos, P.R.: Measure Theory. In: Graduate Texts in Mathematics 18. Springer, Heidelberg (1950)

    Google Scholar 

  38. Hansson, H.: Time and Probability in Formal Design of Distributed Systems. PhD thesis, Department of Computer Science, Uppsala University, Uppsala, Sweden (1991)

    Google Scholar 

  39. Hansson, H.A., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)

    Article  MATH  Google Scholar 

  40. Harding, A., Ryan, M., Schobbens, P.-Y.: Approximating ATL* in ATL. In: Cortesi, A. (ed.) VMCAI 2002. LNCS, vol. 2294, pp. 289–301. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  41. Hatcliff, J., Corbett, J.C., Dwyer, M.B., Sokolowski, S., Zheng, H.: A formal study of slicing for multi-threaded programs with JVM concurrency primitives. In: Static Analysis Symposium, pp. 1–18 (1999)

    Google Scholar 

  42. Hermanns, H., Herzog, U., Katoen, J.-P.: Process algebra for performance evaluation. Theoretical Computer Science 274(1–2), 43–87 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  43. Hermanns, H., Katoen, J.-P.: Performance Evaluation : = (Process Algebra + Model Checking) × Markov Chains. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 59–81. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  44. Hermanns, H.: Interactive Markov Chains. LNCS, vol. 2428, p. 57. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  45. Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  46. Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall, Englewood Cliffs (1985)

    MATH  Google Scholar 

  47. Huth, M.: A Unifying Framework for Model Checking Labeled Kripke Structures, Modal Transition Systems, and Interval Transition Systems. In: Proceedings of the 19th International Conference on the Foundations of Software Technology & Theoretical Computer Science, IIT Chennai, India. LNCS, pp. 369–380. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  48. Huth, M.: Domains of view: a foundation for specification and analysis. In: Domains and Processes, pp. 183–218. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  49. Huth, M.: Probabilistic Abstraction-Based Model Checking. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, pp. 115–134. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  50. Huth, M., Kwiatkowska, M.: Quantitative analysis and model checking. In: Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science, Warsaw, Poland, pp. 111–122. IEEE Computer Society Press, Los Alamitos (1997)

    Google Scholar 

  51. Huth, M.: The interval domain: A matchmaker for aCTL and aPCTL. In: Mislove, M., Cleaveland, R., Mulry, P. (eds.) Electronic Notes in Theoretical Computer Science, vol. 14. Elsevier Science Publishers, Amsterdam (2000)

    Google Scholar 

  52. Huynh, T., Tian, L.: On some Equivalence Relations for Probabilistic Processes. Fundamenta Informaticae 17, 211–234 (1992)

    MATH  MathSciNet  Google Scholar 

  53. Jifeng, H., Seidel, K., McIver, A.: Probabilistic models for the guarded command language. Science of Computer Programming 28(2–3), 171–192 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  54. Jones, C.: Probabilistic Nondeterminism. PhD thesis, Laboratory for the Foundations of Computer Science, University of Edinburgh, Edinburgh, Scotland, Monograph ECS-LFCS-90-105 (1990)

    Google Scholar 

  55. Jones, C., Plotkin, G.: A probabilistic powerdomain of evaluations. In: Proceedings of the IEEE 4th Annual Symposium on Logic in Computer Science, pp. 186–195. IEEE Computer Society Press, Los Alamitos (1989)

    Chapter  Google Scholar 

  56. Jonsson, B., Larsen, K.G.: Specification and Refinement of Probabilistic Processes. In: 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, The Netherlands, July 15-18, pp. 266–277. IEEE Computer Society Press, Los Alamitos (1991)

    Chapter  Google Scholar 

  57. Kannelakis, P., Smolka, S.: CCS Expressions, Finite State Processes and Three Problems of Equivalence. Journal of Information and Computation 86, 43–68 (1990)

    Article  Google Scholar 

  58. McMillan, K.L.: Symbolic Model Checking. Kluwer Academic Publishers, Dordrecht (1993)

    MATH  Google Scholar 

  59. Kozen, D.: Semantics of Probabilistic Programs. Computer and System Sciences 22, 328–350 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  60. Kozen, D.: A Probabilistic PDL. Computer and System Sciences 22(2), 162–178 (1985)

    Article  MathSciNet  Google Scholar 

  61. Kwiatkowska, M., Norman, G., Sproston, J.: Probabilistic Model Checking of the IEEE 802.11 Wireless Local Area Network Protocol. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, pp. 169–187. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  62. Larsen, K.G.: Modal Specifications. In: Sifakis, J. (ed.) Automatic Verification Methods for Finite State Systems. International Workshop, Grenoble, France. LNCS, vol. 407, pp. 232–246. Springer, Heidelberg (1989)

    Google Scholar 

  63. Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94(1), 1–28 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  64. Larsen, K.G., Thomsen, B.: A Modal Process Logic. In: Third Annual Symposium on Logic in Computer Science, pp. 203–210. IEEE Computer Society Press, Los Alamitos (1988)

    Chapter  Google Scholar 

  65. Lassaigne, R., Peyronnet, S.: Approximate Verification of Probabilistic Systems. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, pp. 213–214. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  66. Martin, K.: The measurement process in domain theory. In: Proc. of Automata, Languages and Programming ICALP 2000, pp. 116–126 (2000)

    Google Scholar 

  67. McIver, A., Morgan, C.: Almost-certain eventualities and abstract probabilities in quantitative temporal logic. In: Fidge, C. (ed.) Electronic Notes in Theoretical Computer Science, vol. 42. Elsevier Science Publishers, Amsterdam (2001)

    Google Scholar 

  68. McIver, A., Morgan, C.: Games, probability and the quantitative μ-calculus qMμ. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, pp. 292–310. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  69. Milner, R.: An algebraic definition of simulation between programs. In: 2nd International Joint Conference on Artificial Intelligence, pp. 481–489. British Computer Society, London (1971)

    Google Scholar 

  70. Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

  71. Milner, R.: Communicating and Mobile Systems: the π-Calculus. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  72. Monniaux, D.: Abstract interpretation of programs as Markov decision processes. Technical report, Départment d’Informatique,École Normale Supérieure, 45, rue d’Ulm, 75230 Paris cedex 5, France (2001)

    Google Scholar 

  73. Monniaux, D.: Backwards abstract interpretation of probabilistic programs. In: Sands, D. (ed.) ESOP 2001. LNCS, vol. 2028, p. 367. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  74. Monniaux, D.: Abstract interpretation of probabilistic semantics. In: Palsberg, J. (ed.) SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  75. Monniaux, D.: An abstract analysis of the probabilistic termination of programs. In: Cousot, P. (ed.) SAS 2001. LNCS, vol. 2126, Springer, Heidelberg (2001)

    Google Scholar 

  76. Monniaux, D.: An abstract Monte-Carlo method for the analysis of probabilistic programs (extended abstract). In: 28th Symposium on Principles of Programming Languages POPL 2001, pp. 93–101. Association for Computer Machinery (2001)

    Google Scholar 

  77. Monniaux, D.: Analyse de programmes probabilistes par interprétation abstraite. Résumé étendu en français. Contents in English,Résumé étendu en français. Contents in English (2001)

    Google Scholar 

  78. Morgan, C., McIver, A., Seidel, K., Sanders, J.W.: Refinement-oriented probability for CSP. Formal Aspects of Computing 8(6), 617–647 (1996)

    Article  MATH  Google Scholar 

  79. Morgan, C., McIver, A.: An expectation-based model for probabilistic temporal logic. Logic Journal of the IGPL 7(6), 779–804 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  80. Morgan, C., McIver, A., Seidel, K.: Probabilistic predicate transformers. ACM Transactions on Programming Languages and Systems 18(3), 325–353 (1996)

    Article  Google Scholar 

  81. Motvani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  82. Nielson, F., Nielson, H.R., Hankin, C.: Principles of Program Analysis. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  83. Papadimitriou, C.H.: Games against nature. Journal of Computer and System Sciences 31, 288–301 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  84. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  85. Park, D.M.R.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104, pp. 167–183. Springer, Heidelberg (1989)

    Google Scholar 

  86. Pasareanu, C.S.: DEOS kernel: Environment modeling using LTL assumptions. Technical Report #NASA-ARC-IC-2000-196, NASA Ames (July 2000)

    Google Scholar 

  87. Phillipou, A., Lee, I., Sokolsky, O.: Weak Bisimulation for Probabilistic Systems. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 334–349. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  88. Di Pierro, A., Hankin, C., Wiklicky, H.: Approximate Non-interference. In: CSFW 2002 15th IEEE Computer Security Foundation Workshop, June 2002, pp. 1–15 (2002)

    Google Scholar 

  89. Di Pierro, A., Wicklicky, H.: Probabilistic Abstract Interpretation and Statistical Testing. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, pp. 211–212. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  90. Plotkin, G.D.: A Structural Approach to Operational Semantics. Technical Report FN-19, DAIMI, Computer Science Department, Aarhus University, Ny Munkegade, Building 540, DK-8000 Aarhus, Denmark (September 1981); Reprinted April 1991

    Google Scholar 

  91. Puterman, M.L.: Markov decision processes: discrete stochastic dynamic programming. In: Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Chichester (1994)

    Google Scholar 

  92. Bahar, R.I., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., Somenzi, F.: Algebraic Decision Diagrams and Their Applications. In: IEEE /ACM International Conference on CAD, Santa Clara, California, pp. 188–191. IEEE Computer Society Press, Los Alamitos (1993)

    Google Scholar 

  93. Sagiv, M., Reps, T., Wilhelm, R.: Parametric Shape Analysis via 3-Valued Logic. In: Proceedings of the 26th ACM SIGPLAN-SIGACT Symposium on Principles of programming languages, San Antonio, Texas, January 20-22, pp. 105–118 (1999)

    Google Scholar 

  94. Scott, D.: Continuous lattices. In: Lawvere, F.W. (ed.) Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics, vol. 274, pp. 97–136. Springer, Heidelberg (1972)

    Chapter  Google Scholar 

  95. Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Laboratory for Computer Science, Massachusetts Institute of Technology, Available as Technical Report MIT/LCS/TR-676 (June 1995)

    Google Scholar 

  96. Segala, R., Lynch, N.: Probabilistic Simulations for Probabilistic Processes. Nordic Journal of Computing 2(2), 250–273 (Summer 1995)

    MATH  MathSciNet  Google Scholar 

  97. Seidel, K., Morgan, C., McIver, A.: An introduction to probabilistic predicate transformers. Technical Report PRG-TR-6-96, Programming Research Group, Oxford Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD (1996)

    Google Scholar 

  98. Selic, B.: Physical programming: Beyond mere logic. In: ETAPS 2001 (2001) (Invited Talk)

    Google Scholar 

  99. van Glabbeek, R.J., Weijland, W.P.: Branching Time and Abstraction in Bisimulation Semantics. Journal of the ACM 43(3), 555–600 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  100. Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. 26th IEEE Symp. on Foundations of Computer Science, Portland, Oregon, October 1985, pp. 327–338 (1985)

    Google Scholar 

  101. Volpano, D.: Provably secure programming languages for remote evaluation. ACM Computing Surveys: electronic 28A(2) (December 1996)

    Google Scholar 

  102. Waszkiewicz, P.: Quantitative Continuous Domains. PhD thesis, School of Computer Science, University of Birmingham, United Kingdom (July 2002)

    Google Scholar 

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Huth, M. (2004). An Abstraction Framework for Mixed Non-deterministic and Probabilistic Systems. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, JP., Siegle, M. (eds) Validation of Stochastic Systems. Lecture Notes in Computer Science, vol 2925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24611-4_12

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