J.W. de Bakker and J.-J.Ch. Meyer. Metric semantics for concurrency. BIT, 28:504--529, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....programs in Set is to use guarded terms . This is a rather informal notion and the aim of this section is to use the categorical modelling of the rules to make it formal and general. Typically, a term is called guarded if it is of the form a ; t, that is, if it starts with an action. Cf [5,3]. For instance, following [3] the guarded terms corresponding to the language in (1) are: g : a j g ; t j g k g : 21) The idea is that this can be generalized by expressing, in a formal way, that a term is guarded if its behaviour does not depend on the behaviour of its variables. First note ....

.... guarded terms . This is a rather informal notion and the aim of this section is to use the categorical modelling of the rules to make it formal and general. Typically, a term is called guarded if it is of the form a ; t, that is, if it starts with an action. Cf [5,3] For instance, following [3], the guarded terms corresponding to the language in (1) are: g : a j g ; t j g k g : 21) The idea is that this can be generalized by expressing, in a formal way, that a term is guarded if its behaviour does not depend on the behaviour of its variables. First note that the functor T ae : ....

J.W. de Bakker and J.-J.Ch. Meyer. Metric semantics for concurrency. BIT, 28:504--529, 1988.

....can be found in the equivalence proof itself. The equivalence proof does not split as usual into O D and D O. Rather, both the operational and denotational semantics are represented as least upperbounds of chains and we prove equality of the approximating elements. See also [KR] [BM] where although not made explicit in the setting of complete metric spaces operational and denotational semantics can be represented as limits of Cauchy sequences. The denotational semantics makes use of a fixed point construction with respect to environments. The environment transformation ....

J.W. de Bakker and J.-J.Ch. Meyer, "Metric Semantics for Concurrency," BIT 28, pp. 504-529 (1988).

....Banach s theorem can be applied. In [KR] unicity of the fixed point of the operational higher order operator F has been exploited successfully to derive compact equivalence proofs for operational and denotational meanings along the lines sketched above. A similar line of reasoning has been used in [BM]. The fact that our operator F admits more than one fixed point seems to be an essential consequence due to the fact that we abstracted away from the intermediate configurations. For instance, it is not possible to define in a straightforward way a contraction on metric spaces which yields the ....

....point of this paper is [KR]a where for the first time the observation was made that compact equivalence proofs could be realized using higher order transformations. To a great extent, we also benefit from the work on metric semantics of concurrency performed by De Bakker e.a. e.g. BZ] BKMOZ] [BM]a, Ba2] It is a pleasure to thank the forum formed by the members of the working group on concurrency, Jaco de Bakker, Frank de Boer, Joost Kok, Jan Rutten and others for their comments and the good scientific atmosphere they provided. We are also indebted to Philippe Darondeau for his ....

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J.W. de Bakker and J.-J.Ch. Meyer, "Metric Semantics for Concurrency," BIT 28, pp. 504529 (1988).

....deterministic action behaviour rather than the usual non deterministic behaviour. We have not discussed here the possibility of working in categories of complete metric spaces. There, we expect that working with the closed and compact (non empty) metric powerdomains should yield the metric version [9] of the semantics in (10) and its finitary restriction respectively. On a more speculative level, the idea of having a basic notion of observation given by the endofunctor B and then adding non determinism by lifting B to the Kleisli category of a monad for non determinism seems to bring the ....

J.W. de Bakker and J.-J. Meyer. Metric semantics for concurrency. BIT, 28(3):504--529, 1988.

....3 y = a(x y ) Define on P (A st ) as the extended and lifted version of on A f . Having now available semantical interpretations for our syntactical operators we can proceed 9 with the definition of the denotational semantics for Prog. It follows the same general scheme as in, e.g. [BM], Vi1] 4.2) DEFINITION Define the collection of environments Env, with typical element h, by Env = X P (A st ) Define the statement evaluator S: Stat Env P (A st ) by S(a ) h) a , S(x ) h) h(x ) S(s 1 s 2 ) h) S(s 1 ) h) S(s 2 ) h) for , Let for ....

....5.4. ii) Directly from H d r = r H d (where r = lh .lx.r(h (x ) and the Fixed Point Transformation Lemma. iii) Immediate consequence of (i) and (ii) 5 The last step to take is proving the equivalence of O d and D d . For this we follow the approach as proposed in [KR]a and [BM]a in the setting of complete metric spaces. First we introduce the higher order transformation F d of which both O and D will be fixed points. 5.7) DEFINITION Let d Decl. Define the higher order transformation F d : Stat E P (A t st ) Stat E P (A t st ) ....

J.W. de Bakker and J.-J.Ch. Meyer, "Metric Semantics for Concurrency," BIT 28, pp. 504-529 (1988).

....call. Metric spaces The development of the semantics of B takes place in a metrical setting. The use of metric topology for the formulation of denotational semantics of programming languages has been advocated over the years by De Bakker and co workers. See, e.g. de Bakker and Zucker, 1982] [de Bakker et al., 1988], America et al., 1989] de Boer et al., 1990] Here, following the method proposed in [Kok and Rutten, 1990] not only the denotational semantics but also the operational semantics, and moreover, the equivalence of the operational and (a projection of) the denotational semantics is established ....

....in verifying the equality of operational and denotational semantics. See [de Bakker, 1991] Jacquet and Monteiro, 1990] Rutten, 1990] America and Rutten, 1991] van Breugel, 1991] for several applications in imperative, logical, object oriented and real time programming languages. See also [de Bakker and Meyer, 1988], de Bakker, 1989] de Bakker and de Vink, 1991] for some more introductory overviews of the method for the several programming paradigms. In short, when we succeed in proving that both the operational semantics and the denotational semantics are a fixed point of a higher order contraction, ....

J.W. de Bakker and J.-J.Ch. Meyer, Metric semantics for concurrency, BIT 28 (1988) pp. 504-529

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