S. Andova. Process algebra with probabilistic choice. Technical Report CSR 99-12, Eindhoven University of Technology, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....methods even more pressing. An important part of a formal approach is the semantical modeling of programs to give a precise description of a program and to create a setting in which one can reason about a program. Even when using a syntactical reasoning approach, such as e.g. process algebra [5, 3] or Hoare logic [20] one still needs a semantical model to check the correctness of reasoning rules. A metric approach to de ning semantics uses metric spaces as semantical domains. In reasoning about the processes in these domains, the properties of metric spaces can be used. Banach s theorem, ....

S. Andova. Process algebra with probabilistic choice. In J.-P. Katoen, editor, Proc. ARTS'99, pages 111-129. LNCS 1601, 1999.

....and Skou s [21] reactive probabilistic transition system is presented. This model considers three kinds of choices: action guarded probabilistic choice, external and internal choice. An operational preorder and a equivalence for processes based on testing are proposed. Based on ACP, we can cite [1]. In that paper, a probabilistic version of ACP is presented leading to a language that combines probability and nondeterminism. An operational semantics is de ned based on the alternating model [13] In the construction of the term models they use a probabilistic bisimulation showing soundness ....

....behaves like P with probability p, and like Q with probability 1 p, this decision been made internally. In order to deal with probabilities and nondeterminism we have to decide how to solve a situation where both choices appear. In the literature available we have, basically, two alternatives: in [1, 23, 31] probabilistic choices are solved rst, whereas in [22] the opposite approach is taken. Note that this decision is not at all meaningless, because, depending on the approach adopted, some properties of classical process algebras will be preserved or not. For example, idempotency of internal choice ....

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S. Andova. Process algebra with probabilistic choice. In Proceedings of ARTS'99, LNCS 1601, 1999.

....p Q behaves like P with probability p, and like Q with probability 1 p, and this decision is made internally. In order to deal with probabilities and nondeterminism we have to decide how to solve a situation where both choices appear. In the literature we have, basically, two alternatives: in [1, 12, 17] probabilistic choices are solved rstly, whereas in [11] the opposite approach is considered. Let us observe that this decision is not meaningless at all, because depending on the adopted approach, some properties of classical process algebras will be preserved or not. For example, idempotency ....

....p Q behaves like P with probability p and like Q with probability 1 p. The remaining rules establish the precedence of the probabilistic choice with respect to the other operators. Processes P and Q are supposed to be probabilistically independent processes in rules P2c, P3c, and P4c, like in [1, 6, 17]. The behaviour of the remaining operators is presented in Tables 2 and 3, where we assume that processes are now probabilistically stable. The second table de nes the rules for unlabelled transitions, i.e. those representing an internal evolution, P Q, with the usual meaning that P may evolve ....

S. Andova. Process algebra with probabilistic choice. In Proceedings of ARTS'99, LNCS 1601, 1999.

....Following the ideas of [18, 10] in this paper we will introduce a probabilistic extension that maintains nondeterminism. In order to deal with probabilities and nondeterminism we need to choose which of the two choices (probabilistic or internal choice) has to be resolved rst. Some models [1, 15, 18] consider that probabilistic choice is resolved rst, whereas other models [14] consider the opposite approach. A different point of view is considered in [12] where an actionguarded reactive probabilistic choice is introduced and the point at which probabilistic choice occurs has no ....

....process p Q behaves like P with probability p and like Q with probability 1 p. The remaining rules indicate that probabilistic choice is resolved before the other operators. In rules P2c, P3c, and P4c, it is assumend that P and Q are probabilistically independent processes; the same occurs in [1, 8, 18]. For rules in Tables p Q p P 0 (P1b) p Q 1 p Q P Q p P P Q q P Q P Q p q P P Q p P P Q q P Q P Q p q P P jQ p P P jQ q P jQ P jQ p q P 0 (P5)a PnA p P Table 1. Inference rules for transitions ....

S. Andova. Process algebra with probabilistic choice. In Proceedings of ARTS'99, LNCS 1601, 1999.

....in [14] where a denotational semantics of CSP is de ned by applying the probabilistic powerdomain construction of Jones and Plotkin [9] over a directed complete partial order. Probabilistic processes are considered to be probability distributions over processes of CSP. Based on ACP, we can cite [1]. In that paper, a probabilistic version of ACP is presented leading to a language that combines probability and nondeterminism. An operational semantics is de ned based on the alternating model [7] In the construction of the term models they use a probabilistic bisimulation showing soundness ....

....behaves like P with probability p, and like Q with probability 1 p, this decision been made internally. In order to deal with probabilities and nondeterminism we have to decide how to solve a situation where both choices appear. In the literature available we have, basically, two alternatives: in [1, 14, 21] probabilistic choices are solved rst, whereas in [13] the opposite approach is taken. Note that this decision is not at all meaningless, because, depending on the approach adopted, some properties of classical process algebras will be preserved or not. For example, idempotency of internal choice ....

[Article contains additional citation context not shown here]

S. Andova. Process algebra with probabilistic choice. In Proceedings of ARTS'99, LNCS 1601, 1999.

....in a model with a different kind of probabilistic choice, completing the independence proof. The key differences between the models also become apparent when looking at operational equalities that hold in the models. To demonstrate this, we look at the basic laws of process algebra (see e.g. [1,2,14]) and check which laws are satisfied in the different models. We write s i;j s 0 exactly when O i;j (s) O i;j (s 0 ) We simple write s s 0 when s i;j s 0 holds in each of the models (i.e. for i = loc; gdc; gdk and j = unc; cdc; cdk) Finally we use s glob s 0 to denote ....

S. Andova. Process algebra with probabilistic choice. In Proceedings ARTS'99, Bamberg, 1999. LNCS. to appear.

....resulting state transitions add up to a proper distribution. Furthermore, our argument leaves little freedom for the probability of the resulting transition (i.e. 1 2 in this example) so it does not make sense to require that these expressions are made explicit. Other authors, like Andova [And99] write state transitions with the probabilities omitted and compute those in a separate step along with the required well de nedness arguments) This change in notation agrees with the general observation that in the probabilistic setting the transitions to states depend on each other far more ....

S. Andova. Process algebra with probabilistic choice. Lecture Notes in Computer Science, 1601:111-129, 1999.

....of the mathematical model can already be problematic. A systematic approach to simplify the program, or obtain properties without having to actually calculate the semantics are useful. Approaches in this area are probabilistic process algebra and stochastic process algebra (see, for example, [5, 26, 3, 10, 9]) where equivalences of programs can be checked syntactically by equational reasoning. Another approach is to introduce a calculus or a proof system as a vehicle to reason about the probabilistic programs directly. Earlier work on the proof theory for probabilistic programs that has inspired the ....

S. Andova. Process algebra with probabilistic choice. In J.-P. Katoen, editor, Proceedings ARTS'99, pages 111-129, Bamberg, 1999. LNCS 1601.

....et al..1995) add two probability parameters. There are also different proposals with respect to the treatment of non determinism. In most models only the use of probabilistic choices is allowed (Glabbeek et al..1995; Baeten et al..1995; Nunez and Frutos 1995. Some models ( Hansson and Jonsson 1190; Andova 1999) support both non deterministic choices and probabilistic choices. Since the emphasis in this paper is on real time, probabilities and performance evaluation, we prefer a simple model of probabilities for the action part of the calculus. We therefore follow the approach of (Nunez and Frutos 1995) ....

....Since the timed part and the action part of the calculus are orthogonal, we foresee no difficulties when the simple model of actions and probabilities is replaced by a more advanced and useful one. The investigation of such models is still subject of active research (see for instance Andova 1999. 8 CONCLUSIONS AND FUTURE RESEARCH In this paper we developed a real time probabilistic calculus for performance evaluation. The calculus is called PRTCCS and is based on Milner s CCS. The calculus was developed to study a possible probabilistic extension of the realistic system level ....

Andova, S. 1999. "Process Algebra with Probabilistic Choice." In Proceedings of ARTS'99 (Bamberg, Germany May 26--28).

.... Gamma Gamma Gamma Gamma (i;j) P 0 Theta Q 0 P [fix X:P=X] a;p Gamma Gamma i P 0 fix X:P a;p Gamma Gamma i P 0 P a;p Gamma Gamma i P 0 PnA a; p (P;A) Gamma Gamma Gamma Gamma Gamma Gamma i P 0 nA (a = 2 A) Table 1 Operational semantics of PCCS Restriction Consider P = [ 1 6 ]a:0 [ 1 2 ]b:0 [ 1 3 ]c:0, a process that can either perform action a, b or c with probability 1 6 , 1 2 and 1 3 , respectively. The corresponding GPTS of P is depicted in Figure 1(a) For convenience we omit transition indices. Consider the transition P c; 1 3 Gamma Gamma 0. The ....

.... (i;j) P 0 Theta Q 0 P [fix X:P=X] a;p Gamma Gamma i P 0 fix X:P a;p Gamma Gamma i P 0 P a;p Gamma Gamma i P 0 PnA a; p (P;A) Gamma Gamma Gamma Gamma Gamma Gamma i P 0 nA (a = 2 A) Table 1 Operational semantics of PCCS Restriction Consider P = 1 6 ]a:0 [ 1 2 ]b:0 [ 1 3 ]c:0, a process that can either perform action a, b or c with probability 1 6 , 1 2 and 1 3 , respectively. The corresponding GPTS of P is depicted in Figure 1(a) For convenience we omit transition indices. Consider the transition P c; 1 3 Gamma Gamma 0. The value 1 3 ....

[Article contains additional citation context not shown here]

S. Andova. Process algebra with probabilistic choice. In J.-P. Katoen, editor, Formal Methods for Real-Time and Probabilistic Systems, LNCS 1601, pages 111--129, 1999.

....P p Q behaves like P with probability p, and like Q with probability 1 p, and this decision is made internally. In order to deal with probabilities and nondeterminism we have to decide how to solve a situation where both choices appear. In the literature we have, basically, two alternatives: in [1, 12, 15] probabilistic choices are solved rstly, whereas in [11] the opposite approach is considered. Also, a different point of view is considered in [9] where an actionguarded reactive probabilistic choice is introduced and the point at which probabilistic choice occurs has no signi cance. Let us ....

....P p Q behaves like P with probability p and like Q with probability 1 p. The remaining rules establish the precedence of the probabilistic choice with respect to the other operators. Processes P and Q are supposed to be probabilistically independent processes in rules P2c, P3c, and P4c, like in [1, 5, 15]. The behaviour of the remaining operators is presented in Tables 2 and 3, where we assume that processes are now probabilistically stable. The second table de nes the rules for unlabelled transitions, i.e. those representing an internal evolution, P Q, with the usual meaning that P may evolve ....

S. Andova. Process algebra with probabilistic choice. In Proceedings of ARTS'99, LNCS 1601, 1999.

....to be fixed, the values P(s, a, C) can be computed in time O(n 2 ) 27 independent action is executed next. During the last decade, these calculi have been extended in several ways in order to provide means to specify the probabilistic behaviour of parallel systems within the same formalism [32,50,54,37,35,3]. The essential ingredient is the incorporation of a probabilistic choice operator. Distinguishing fine points in this variety of approaches are (1) the interplay of probabilistic choice with nondeterministic choice, and (2) the semantics of parallel composition. 3 In the fully probabilistic ....

S. Andova. Process algebra with probabilistic choice. In J.-P. Katoen, editor, Formal Methods for Real-Time and Probabilistic Systems, LNCS 1601, pages 111--129, 1999.

....of [18, 10] in this paper we introduce a probabilistic extension that maintains nondeterminism. In order to deal with probabilities and nondeterminism we will consider that probabilistic choice is made first and later nondeterministic choice. This point of view is also taken in other models ([1], 15] 18] In [14] the opposite point of view is taken, leading to a model in which most of the properties of CSP are preserved. Finally, a different viewpoint is considered in [12] where an action guarded reactive probabilistic choice is introduced and the point at which probabilistic choice ....

....p Q behaves like P with probability p and like Q with probability 1 Gamma p. The remaining rules indicate that probabilistic choice is resolved before the other operators. In rules P2c, P3c, and P4c, it is considered that P and Q are probabilistically independent processes (the same occurs in [1], 8] 18] Table 1. Inference rules for transitions labelled by probabilities (P1a) P Omega p Q Gamma p P (P1b) P Omega p Q Gamma 1 Gammap Q (P2a) P Gamma p P 0 Q # P Phi Q Gamma p P 0 Phi Q (P2b) P # Q Gamma q Q 0 P Phi Q Gamma q P Phi Q 0 (P2c) P Gamma p P ....

S. Andova. Process algebra with probabilistic choice. In Proceedings of ARTS'99, LNCS 1601, 1999.

....typical for the work reported here (see also [12] In [6] taking the point of view of performance modeling and quantitative analysis (cf. e.g. 13, 7, 17] the parallel operator is studied. The induced nondeterminism, however, is resolved in favour of probability. In the recent work of Andova [1], in the alternating approach and incorporating the basic idea underlying [6] a probabilistic process algebra featuring both nondeterministic and probabilistic choice is considered. Furthermore, a sound and complete axiomatization with conditional equations is provided. A similar restriction on ....

....[6] a probabilistic process algebra featuring both nondeterministic and probabilistic choice is considered. Furthermore, a sound and complete axiomatization with conditional equations is provided. A similar restriction on contexts is found in the present work as well. The completeness result of [1] hints that this can not be circumvented. The transition system for the probabilistic parallel language as given in [12] has explicit labels for nondeterministic and probabilistic choice. This way it is ensured that for, e.g. the statement (a Phi 1=2 b)2(c Phi 1=2 d) no mixed distributions, ....

[Article contains additional citation context not shown here]

S. Andova. Process algebra with probabilistic choice. In Proceedings ARTS'99, Bamberg, 1999. to appear.

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S. Andova, Process algebra with probabilistic choice (extended abstract),Proc. ARTS'99, Bamberg, Germany, J.-P. Katoen, ed., LNCS 1601, Springer-Verlag, pp. 111-129, 1999. Full version report CSR 99-12, Eindhoven University of Technology, 1999.

....chain given in Figure 1a. In our theory the feature that relates these two processes can be easily expressed with the following verification rule: if X = a t b t ae i Delta X and i is an internal action, then Delta fig (X) Delta (a t = ae) b) For more details of the semantics see [1]. Someone familiar with process algebra can easily see the resemblance with the KFAR b 1 rule ( 3] with non deterministic choice replaced by probabilistic choice. a) 89: 0 1 Gamma Gammaae ss Phi Phi ae = 89: 1 1 JJ = 89: 1 1 TT b) 89: 0 1 Gamma ....

....77 ffl b) ONMLHIJKGFED ABC Y0 a; 1 Gamma 1 Gammaae Theta Theta Sigma Sigma Sigma Sigma Sigma Sigma Sigma Sigma Sigma Sigma Sigma Sigma b; 1 Gammaae) 1 Gammaae oeoe 8 8 8 8 8 8 8 8 8 8 8 8 ffl ffl Fig. 2. Fully (bisimilar) probabilistic process 2 Definitions and results In [1] a probabilistic process algebra containing both probabilistic choice and non deterministic choice is introduced. Our current work is based on a subalgebra of that one for which non deterministic choice has been excluded. Having both choices and abstraction at the same time leads to a more complex ....

[Article contains additional citation context not shown here]

S. Andova, Process algebra with probabilistic choice (extended abstract), Proc. ARTS'99, Bamberg, Germany, J.-P. Katoen, ed., LNCS 1601, Springer-Verlag, pp. 111-129,

No context found.

S. Andova. Process algebra with probabilistic choice. Technical Report CSR 99-12, Eindhoven University of Technology, 1999.

No context found.

S. Andova. Process algebra with probabilistic choice. In Proceedings of 5th AMAST Workshop on Real-Time and Probabilistic Systems (ARTS'99) Bamberg, Germany, May 1999, volume 1601 of Lecture Notes in Computer Science. Springer-Verlag, 1999.

No context found.

S. Andova, Process algebra with probabilistic choice, Proc. 5th International AMAST Workshop, ARTS'99, Bamberg, Germany (J.-P. Katoen, ed.), LNCS 1601, Springer-Verlag, 1999, pp. 111--129.

No context found.

S. Andova, Process algebra with probabilistic choice, Proc. ARTS'99, Bamberg, Germany, J.-P. Katoen, ed., LNCS 1601, Springer-Verlag, pp. 111-129, 1999.

No context found.

S. Andova. Process algebra with probabilistic choice. In Proceedings of 5th AMAST Workshop on Real-Time and Probabilistic Systems (ARTS'99) Bamberg, Germany, May 1999, volume 1601 of Lecture Notes in Computer Science. Springer-Verlag, 1999.

No context found.

Suzana Andova. Process algebra with probabilistic choice. In J.-P. Katoen, editor, Proc. ARTS'99, volume 1601 of Lecture Notes in Computer Science, pages 111--129. Springer Verlag, 1999.

No context found.

S. Andova. Process algebra with probabilistic choice. In Proceedings of 5th AMAST Workshop on Real-Time and Probabilistic Systems (ARTS'99) Bamberg, Germany, May 1999, volume 1601 of Lecture Notes in Computer Science. Springer-Verlag, 1999.

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