C. Baier and M. Kwiatkowska, Domain Equations for Probabilistic Processes, MSCS, Vol. 10, No. 6, pp. 665--717, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 approach. The valuations introduced in [6] are similar to the compact support measures used here. Slightly di erent are the valuations used in the ....

....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 approach. The valuations introduced in [6] are similar to the compact support measures used here. Slightly di erent are the valuations used in the thesis work of Jones [21] to introduce probabilistic semantical domains. These valuations are a slight simpli cation of measures. Not every measure necessarily has a corresponding valuation. ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes (extended abstract). In M. Mislove, M. Nivat, C. Papadimitriou, C. Palamidessi, and J. Parrow, editors, Proc. Express'97. Electronic Notes in Theoretical Computer Science 7, 1997.

....LMP with least upper bounds of # chains (w.r.t. the simulation ordering) This shows that LMP can be used as the target of interpretation of a syntax that includes recursion. The internal logic of Proc is a logic complete for reasoning about simulation of LMPs. Baier and Kwiatkowska [BK96]show that bisimulation and simulation equivalence agree for discrete space systems. Our results show that this coincidence of bisimulation and simulation equivalence extends to continuous space systems. 1.1 Summary of paper In Section 2 we recall background material on labelled Markov ....

....this is certainly a very interesting development. The work is not really comparable to ours since we are not attempting to use our formalism for verification in the traditional sense but it might well influence us in the future. In terms of the domain theory, the work of Baier and Kwiatkowska [BK96] is closest to this paper. They study the solution of a similar recursive domain equation in a variety of categories. The focus of their paper is the interaction of nondeterminism and probability in the discrete state space setting they do not explicitly consider the issues of continuous ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

....to probability. See [7,17,5] for example. The mathematical background is sketched in the next section. For our study we have used a construction for obtaining a metric space of measures from a given metric space that is borrowed from [23,11] A similar construction is available for valuations, cf. [3]. The paper contributes a first step towards a systematic comparison of the various ways to combine probabilistic choice with more classical process algebra constructions. For the treatment of the parallel operator in the context of failure we propose a notion of take over. The metric techniques ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes (extended abstract). In Proc. Express'97. Electronic Notes in Theoretical Computer Science 7, 1997. 22

....if they assign exactly the same probability to processes which are close to each other. As far as the probabilistic information is concerned, the approach of this paper is qualitative; one can check whether or not two probabilities are the same, but not how far apart two probabilities are. In [BK97] a metric denotational semantics for an extension of CCS with action guarded probabilistic choice is shown to be fully abstract with respect to probabilistic bisimulation. The present paper establishes full abstraction with respect to an operational model. Identi cation of bisimilar processes ....

C. Baier and M.Z. Kwiatkowska. Domain equations for probabilistic processes (extended abstract). In C. Palamidessi and J. Parrow, editors, Proc. Express'97. ENTCS 7, 1997.

....Petri nets [Mar89, VN92] add Markov chains to the underlying Petri net model. Probabilistic extensions of IO Automata [Seg95, WSS97] have also been developed. The veri cation community has been active in developing model checking tools for probabilistic systems, for example [BLL 96, BdA95, BK97, CY95, HK97] By and large, the above work focuses on discrete state systems. The other investigation that we are aware of apart from our earlier papers [BDEP97, DEP98] into continuous state spaces was by deVink and Rutten [dVR97] They de ne a probabilistic transition system as a coalgebra ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. ENTCS, 7, July 1997.

....this is certainly a very interesting development. The work is not really comparable to ours since we are not attempting to use our formalism for veri cation in the traditional sense but it might well in uence us in the future. In terms of the domain theory, the work of Baier and Kwiatkowska [BK96] is closest to this paper. They study the solution of a similar recursive domain equation in a variety of categories. The focus of their paper is the interaction of nondeterminism and probability in the discrete state space setting they do not explicitly consider the issues of continuous state ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

.... following themes: formulating probabilistic extensions of modal temporal logics; constructing algorithms for automatic veri cation of probabilistic systems against probabilistic temporal logic speci cations; and ecient data structures for probabilistic veri cation [HK97, BCHG 97, BKa, BK97a, BK97b, BKb] Lastly, our work on the logical foundations of multi agent systems is also related to features, since we are studying mechanisms for building highly expressive logics by adding o the shelf features to simple logics [LR98d, LR98b, LR97, LR98c, LR98a] 3.3 FUNDP: Namur, Belgium The ....

Christel Baier and Marta Kwiatkowska. Domain equations for probabilistic processes (extended abstract). In Proceedings, EXPRESS'97, volume 7 of Electronic Notes in TCS. Elsevier, 1997. Full version submitted to Information and Computation, Technical Report CSR-97-7.

.... worth some investigation is the combination F (D# (X) of probabilistic and ordinary nondeterminism; there is natural distributive law # : D# (F (X) # F (D# (X) which makes this a monad; this way to combine the two forms of nondeterminism is used in a domain theoretic context in [4] modulo actions and mentioned in [19] where an interesting idea of restricting to a#ne sets of evaluations is advocated. In so far as we are successful with such examples, the question of how to treat other monads and their operations is the more pressing; exceptions, state and continuations ....

C. Baier and M. Kwiatkowska, Domain Equations for Probabilistic Processes, MSCS, Vol. 10, No. 6, pp. 665--717, 2000.

....10 As for D in the previous section, M 1 can be regarded as a functor, now on the category UMS of ultrametric spaces and nonexpansive mappings. For this purpose, M 1 (M) should first of all be endowed with a metric structure. For a similar result in the context of evaluations, see, e.g. [BK97]. Lemma 5.3 Let (M, d) be an ultrametric space. Put O # = O # O #x # O: B # (x) # O , for # 0. Let the distance d on M 1 (M) induced by d on M , be given by d( #) inf # 0 #O # O # : O) #(O) Then (M 1 (M) d) is an ultrametric space. Proof Since M is an ....

....Further investigations of the proposed notion of Borel decomposition should clarify how the latter relates to the use of Polish spaces as in [BDEP97] We expect that the technical result obtained there, on the existence of weak pullbacks, applies also to our setting. Other related work includes [BK97] where discrete probabilistic bisimulation is discussed in the context of CCS extended with probabilistic choice, and [Mos97] where coalgebraic logics also based on functors, including our functor D, are studied. Connections with modal logics are considered by Larsen and Skou and by Desharnais ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Technical Report CSR--97--7, University of Birmingham, 1997. Also appeared in Proc. EXPRESS'97, Electronic Notes in Theoretical Computer Science, volume 7, Elsevier Science 1997.

....In order not to clutter up the picture, we have not labeled the dotted lines that represent a transitions with probability 3=4. S [0; 2] a[3=4] ## # # # # # (2; 3] b ### # # # fs; tg f0g ## (0; 1 3 ] ## ( 1 3 ; 2 3 ) ## a[ 1 12 ] ## # # # # # # ( 2 3 ; 1] ## a[ 1 6 ] ## (1; 2] a ## # # # # # # (2; 3] b ## fs; tg The initial state is ( 2=3; 1] 2) and satisfies the following formulas: hai 11 12 T; hai 3=4 hai 0 T; and hai 1=6 hbi 0 T where 0. Note that the set (1; 2] will never be split in any approximations, since it contains ....

....and Hermanns [5] have used approximate reasoning for continuous time Markov chains for verifying PCTL formulas. While they point out the problems in using approximation for logical reasoning, they do not provide an explicit example. In terms of the domain theory, the work of Baier and Kwiatkowska [6] is closest to this paper. They study the solution of a similar recursive domain equation in a variety of categories. The focus of their paper is the interaction of nondeterminism and probability in the discrete state space setting they do not consider the issues of continuous state spaces and ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

....if statement are equivalent for the low observer. Thus we should factor in this (probabilistic) equivalence. We will develop probabilistic noninterference criteria based on probabilistic bisimulation. The idea is based on the definition of bisimulation on probabilistic transition systems [LS91] [BK97]. 3.2 Relational Definitions and Notation Let us first introduce some standard definitions and notation for relations. Let Rel(A) denote the set of binary relations on A. If R 2 Rel(A) is an equivalence relation then A=R is the set of R Gammaequivalence classes, A=R (A) Id A denotes the ....

....a partial probabilistic bisimulation on configurations. We write c d (resp. C D) iff there exists a partial probabilistic bisimulation on configurations (commands) relating c and d (C and D) It is possible to show that both relations are themselves (partial) probabilistic bisimulations (c.f. [BK97]) We sketch a proof of this fact in Appendix A. The partial probabilistic bisimulation on thread pools could form the basis of a compositional semantics although one must at least restrict it to relations between equal length vectors to obtain compositionality. We will not pursue this ....

Christel Baier and Marta Z Kwiatkowska. Domain equations for probabilistic processes. Technical Report CSR-97-7, University of Birmingham, School of Computer Science, July 1997. 12

....fashion by a finite multiset. An example is given in subsection 5.3 were the functor is used in specifying a linear domain. The functor is that same as the functor M 0 1 introduced in [30] restricted to CUMS. A more detailed description and more general results can be found there. Also in [4] a functor yielding distributions (called evaluations in [4] is given on the category CUMS. A comparison with set theoretic and complete partial order approaches can be found there. Definition 4.14 Let X be a metric space. a) A oe algebra A over X is a collection of subsets which is closed ....

....5.3 were the functor is used in specifying a linear domain. The functor is that same as the functor M 0 1 introduced in [30] restricted to CUMS. A more detailed description and more general results can be found there. Also in [4] a functor yielding distributions (called evaluations in [4]) is given on the category CUMS. A comparison with set theoretic and complete partial order approaches can be found there. Definition 4.14 Let X be a metric space. a) A oe algebra A over X is a collection of subsets which is closed under complement and countable union. b) The collection B(X) of ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. In M. Mislove, M. Nivat, C. Papadimitriou, C. Palamidessi, and J. Parrow, editors, ENTCS, volume 7. Elsevier, 1997.

....probability. See [6, 16, 4] for example. The mathematical background is sketched in the next section. For our study we have used a construction for obtaining a metric space of measures from a given metric space that is borrowed from [22, 10] A similar construction is available for valuations, cf. [2]. The paper contributes a first step towards a systematic comparison of the various ways to combine probabilistic choice with more classical process algebra constructions. For the treatment of the parallel operator in the context of failure we propose a notion of takeover. The metric techniques to ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes (extended abstract). In Proc. Express'97. Electronic Notes in Theoretical Computer Science 7, 1997.

....There are several papers now on probabilistic analysis, modeling and verification. There are even several papers on probabilistic process algebra analyzing notions of testing and simulation, investigating model checking and exploring various other ideas [SL94, vGSST90, JL91, JY95, JS90, CSZ92, BK96, HK96] One of the most stimulating recent results on applications is the work of Hillston [Hil94] on PEPA and its uses in compositional performance evaluation. There are also several other interesting practical developments which are worthy of attention. In particular telecommunication [RAvO97] ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

....a very natural notion of probabilistic synchronization trees yields a model of a probabilistic version of CCS with both probabilistic branching and nondeterministic branching. If, in that model, one looks only at probabilistic branching, equality is precisely the Larsen Skou notion of bisimulation [BK96]. The paper by Larsen and Skou does much more than just define bisimulation. They introduce the notion of testing a probabilistic process and associating probabilities with the possible outcomes. They then introduce a notion of testable properties. The link with probabilistic bisimulation is that ....

....stochastic kernels play an important role. There are several papers now on probabilistic analysis, modeling and verification. There are even several papers on probabilistic process algebra analyzing notions of testing and simulation, investigating model checking and exploring various other ideas [SL94, vGSST90, JL91, JY95, JS90, CSZ92, BK96, HK96]. There are several interesting practical developments, other than PEPA, which are worthy of attention. In particular telecommunication [AJKvO97] real time systems [BLFG95] and modeling physical systems [GSS95] are areas where probabilistic systems are very important. It is particularly for the ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

....There are several papers now on probabilistic analysis, modeling and verification. There are even several papers on probabilistic process algebra analyzing notions of testing and simulation, investigating model checking and exploring various other ideas [SL94, vGSST90, JL91, JY95, JS90, CSZ92, BK96, HK96] There are several interesting practical developments, other than PEPA, which are worthy of attention. In particular telecommunication [AJKvO97] real time systems [BLFG95] and modeling physical systems [GSS95] are areas where probabilistic systems are very important. It is particularly ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

....the algorithm above. 6 Related Work There are several papers now on probabilistic analysis, modeling and verification. There are even several papers on probabilistic process algebra analyzing notions of testing and simulation, investigating model checking and exploring various other ideas [SL94, vGSST90, JL91, JY95, JS90, CSZ92, BK96, HK96]. One of the most stimulating recent results on applications is the work of Hillston [Hil94] on PEPA and its uses in compositional performance evaluation. There are also several other interesting practical developments which are worthy of attention. In particular telecommunication [RAvO97] ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

....very natural notion of probabilistic synchronization trees yields a model of a probabilistic version of CCS with both probabilistic branching and nondeterministic branching. If, in that model, one looks only at probabilistic branching, equality is precisely the Larsen Skou notion of bisimulation [BK96] The paper by Larsen and Skou does much more than just define bisimulation. They introduce the notion of testing a probabilistic process and associating probabilities with the possible outcomes. They then introduce a notion of testable properties. The link with probabilistic bisimulation is that ....

C. Baier and M. Kwiatkowska. Domain equations for probabilistic processes. Available from URL http://www.cs.bham.ac.uk/ mzk/, March 1996.

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C.Baier and M.Kwiatkowska. Domain equations for probabilistic processes, preprint.

.... computations, p[k] 2 D k , is defined inductively on k 2 IN by putting p[0] p 0 for all p and p[k 1] ae p 0 if p = p 0 (a; f [k] if p = a; f) for some a 2 A and f 2 (D ) ae ae ae ae= Z Z Z Z ae ae ae ae= Z Z Z Z ae ae ae ae= Z Z Z Z r r r r r r r r r r r r r r r p[3] p[2] p[1] p[0] a a 1 2 1 2 b b 2 3 1 3 a a 1 2 1 2 1 c 1 c 1 b 1 c a 1 Fig. 1. An illustration of truncations. The truncation of probabilistic distributions (and respectively of probabilistic computations, which we omit for reasons of space) satisfies the following properties useful ....

....d(p; q) d(q; r) d(p; r) follows from Lemma 8 in all cases except p = a; f ) q = a; g) and r = a; h) in which case it holds since d is a metric. ut It should be noted that our metric is not an ultra metric. An ultrametric can be defined in terms of truncations in the standard way, see [3], but it results in different convergence as demonstrated in the example below. p q r a 1 b a 1 b 1 ae ae ae ae= Z Z Z Z 1 c 1 c 1 a a r r r r r r r r r r r (1 Gamma ffl) ffl Fig. 2. Smooth metric d (this paper) vs discrete ultrametric of [3] Example 1. Consider the ....

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C.Baier and M.Kwiatkowska. Domain equations for probabilistic processes, preprint.

....that probabilistic bisimulation is a congruence This is a preliminary version. The final version can be accessed at URL: http: www.elsevier.nl locate entcs volume16.html Kwiatkowska and Norman over their calculus PCCS (which contains all the usual SCCS operators) and Baier and Kwiatkowska [3] show congruence properties of full CCS extended with action guarded probabilistic choice. Generally, if one works with a fine (or strong) equivalence such as probabilistic bisimulation then almost all CCS or CSP operators can be adapted to the probabilistic setting. However, there are cases when ....

C. Baier and M.Z. Kwiatkowska. Domain equations for probabilistic processes (Extended Abstract). In Proc. EXPRESS Workshop, volume 7, Electronic Notes in Theoretical Computer Science, Elsevier, 1997.

....as probabilistic bisimulation then almost all CCS or CSP operators can be adapted to the probabilistic setting. For example, van Glabbeek et al. 6] show that probabilistic bisimulation is a congruence over their calculus PCCS (which contains all the usual SCCS operators) and Baier and Kwiatkowska [2] show congruence properties of full CCS extended with action guarded probabilistic choice. However, there are cases when probabilistic bisimulation is too fine, as it discriminates between processes that cannot be distinguished under a realistic testing scenario. One alternative is to work with a ....

.... to these problems have been investigated in [18] As far as metric space denotational models are concerned, we should mention the metric model of [12] where a deterministic subcalculus of CCS extended with action guarded probabilistic choice was considered, and that of Baier and Kwiatkowska [2] for full CCS extended with action guarded probabilistic choice. 2 The Model and Testing Equivalence In this section we overview the definitions necessary for the technical development included in the remainder of this paper; for detailed justification of the constructions see [13,20] First, we ....

[Article contains additional citation context not shown here]

C. Baier and M.Z. Kwiatkowska. Domain equations for probabilistic processes (Extended Abstract). In C. Palamidessi and J. Parrow , editors, Proc. EXPRESS'97, volume 7, Electronic Notes in Theoretical Computer Science, Elsevier, 1997.

No context found.

C. Baier and M. Kwiatkowska, Domain Equations for Probabilistic Processes, MSCS, Vol. 10, No. 6, pp. 665--717, 2000.

No context found.

C. Baier and M.Z. Kwiatkowska. Domain equations for probabilistic processes (Extended Abstract). In Proc. EXPRESS Workshop, volume 7, Electronic Notes in Theoretical Computer Science, Elsevier, 1997.

No context found.

C. Baier and M.Z. Kwiatkowska. Domain equations for probabilistic processes (Extended Abstract). In C. Palamidessi and J. Parrow , editors, Proc. EXPRESS'97, volume 7, Electronic Notes in Theoretical Computer Science, Elsevier, 1997.

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