J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In Concurrency Theory, LNCS 1664, pp. 258--273. Springer, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....ultra metric spaces and thus ts in the AmericaRutten approach to solving domain equations. This is illustrated in section 4 by building a simple probabilistic branching domain using domain equations. Van Breugel and Worrell, inspired by work done by Desharnais, Gupta, Jagadeesan and Panangaden [16], introduce in [14, 13] a metric on a domain of measures. A key di erence with their approach is that they use the Hutchinson metric to provide quantitative information about the probabilities and in this way a pseudometric rather than an ultrametric is obtained. The processes in their distance ....

....balls are either disjoint or one is contained in the other. In [14] a real valued logic is introduced which can be used to characterize the distance between processes. In this paper a comparison is also given with the pseudo metric introduced by Desharnais, Gupta, Jagadeesan and Panangaden in [16], the metric on compact support measures from [33] also used here and with the pseudometric introduced by Norman in his thesis [25, section 6.1] In [6] Baier and Kwiatkowska present a metric space of valuations and compare it with spaces de ned using a set theoretic and a complete partial order ....

J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In J.C.M. Baeten and S. Mauw, editors, Proc. CONCUR'99, pages 258-273. LNCS 1664, 1999.

....system. However, as we illustrate below, the truth of a given formula is not continuous with respect to the obvious notion of approximation on transition systems. This phenomena is discussed extensively by Desharnais et al. in [12] In a similar vein, Giacalone et al. 14] and Desharnais et al. [11] have criticized all or nothing notions of behavioural equivalence for probabilistic transition systems such as Larsen and Skou s probabilistic bisimulation [24] Recall that a probabilistic bisimulation is an equivalence relation on the state space of a transition system such that related states ....

....s 0 a; 1 2 a; 1 2 s ffl 0 a; 1 2 ffl a; 1 2 Gammaffl s 2 are only probabilistic bisimilar if ffl is 0. However, the two states behave almost the same for very small ffl different from 0. In the words of [11], behavioural equivalences like probabilistic bisimilarity are not robust, since they are too sensitive to the exact probabilities of the various transitions. The lack of robustness of these logics and behavioural equivalences has a number of implications. First of all, the logics and behavioural ....

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In J.C.M. Baeten and S. Mauw, editors, Proceedings of the 10th International Conference on Concurrency Theory, volume 1664 of Lecture Notes in Computer Science, pages 258--273, Eindhoven, August 1999. SpringerVerlag.

.... a computational interpretation following the approach of [AR00] In another direction, since sometimes covert channels cannot be avoided completely, one should consider relaxing the conditions towards accepting small in uences of high behaviour on low probabilities, possibly along the lines of [DGJP99] 8 Acknowledgements Many thanks for interesting comments go to Samson Abramsky, Bruno Dutertre (also for making [Dut99] available) Riccardo Focardi, Dusko Pavlovi c, and furthermore to Peter Ryan for providing the opportunity to attend the workshop on Information Flow (London, Dec. 1999) ....

J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panagaden. Metrics for labeled markov systems. In CONCUR, 1999.

....page 130] 2 Secondly, in some cases the metric structure can be used other than for modelling infinite computations as limits. It may be exploited to capture the difference of computations quantitatively. For example, metrics have been used in this way for probabilistic systems (see, e.g. [GJS90, DGJP99]) In those cases, we don t want any halves in the equations as they blur the quantitative picture. We assume that the reader is familiar with metric spaces, coalgebras and categories. For more details we refer the reader to, e.g. BV96, JR97, Mac71] Acknowledgements The author has benefited ....

J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for Labeled Markov Systems. In J.C.M. Baeten and S. Mauw, editors, Proceedings of the 10th International Conference on Concurrency Theory, volume 1664 of Lecture Notes in Computer Science, Eindhoven, August 1999. Springer-Verlag. 6

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. Theoretical Computer Science, 318:323--354, 2004.

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In Proceedings of CONCUR99, Lecture Notes in Computer Science. Springer-Verlag, 1999.

....this result is also true and was proven in [DGJP00] 3 Approximating Labelled Markov Processes In a recent paper [DGJP00, DGJP03] we developed a theory of approximation of labelled Markov processes. Using this theory we can extend our results about the metric for the discrete case reported in [DGJP99] to the continuous case. De ning the metrics for processes that have continuous state spaces does not require the approximation theory, however showing certain properties of a class of functions (that they characterize bisimulation) does use the approximation theory to lift the result from the ....

....occurs. It is useful to emphasis concisely the semantics we want to use, since we never work with functionals that have di erent parameters. The role of c is to discount the e ect of future actions. For c = 1 all transitions are counted equally even if they are far into the future. Note that in [DGJP99] f q was written bfc q and that we had an additional functional written dfe = min(f; q) The latter is not necessary since it can be represented by using the functional min and the constant function q : 1 (1 q) We use hai f to represent hai haif where hai appears n times. One ....

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In Proceedings of CONCUR99, Lecture Notes in Computer Science. Springer-Verlag, 1999.

....by a distribution over states. Traditionally, the satisfaction relation is a pairing between formulas and states yielding a truth value; s #; the analogue is given by the integral fd yielding a real number between 0 and 1. In a related paper on metrics for probabilistic processes [DGJP99] we use these ideas very literally. Here we use the spirit of these same ideas. Essentially we take the point of view that such integrals define the quantities of interest. We provide a technique for systematically approximating such integrals by showing that there are enough finite ....

....of techniques that permit the approximation of integrals [Par67] Intuitively, in our context, given that LMP s form a Polish space, these theories permit the approximation of integrals by finite sums evaluated at finite trees. In previous work we had developed a theory of metrics between LMPs [DGJP99] for robust reasoning on probabilistic processes. Intuitively, the metrics of [DGJP99] measure the closeness of the transition probabilities of processes. The metric distance in this paper is unrelated to the closeness of probability numbers and have very di#erent convergence properties, e.g. ....

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In Proceedings of CONCUR99, Lecture Notes in Computer Science. Springer-Verlag, 1999.

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In Concurrency Theory, LNCS 1664, pp. 258--273. Springer, 1999.

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Josee Desharnais, Vineet Gupta, Radha Jagadeesan, and Prakash Panangaden. Metrics for labeled markov systems. In Proceeding of 10th CONCUR, volume 1664 of LNCS, pages 258--273, 1999.

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In International Conference on Concurrency Theory, pages 258--273, 1999. full version available via http://wwwacaps. cs.mcgill.ca/ prakash/.

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J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In International Conference on Concurrency Theory, pages 258--273, 1999. full version available via http://wwwacaps. cs.mcgill.ca/ prakash.

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