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Specification of Reactive Hardware/Software Systems - Electronic ...

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9.5 A Computational Interleaving Semantics 277<br />

from which it can be reached. The latter two properties, which are proved in [Voe95b],<br />

justify the well-definedness <strong>of</strong> function Reset:<br />

Reset(C p (E1 ¥¡ ¡ ¡ ¥ En)) C p (E1 ¥¡ ¡ ¡ ¥ En)<br />

Reset(§ S p£ e C p E1£ ¡ ¡ ¡£ En¢ C p (E1 ¥¡ ¡ ¡ ¥ En)<br />

Reset(Cc E1 ¥¡ ¡ ¡ ¥ En ) C § c ¥¡ ¡ ¡ ¥ (E1 En)<br />

Cc ¥¡ ¡ ¡ ¥ (E1 En)<br />

Reset(§ BSpec e C c E1£ ¡ ¡ ¡£ En¢<br />

Reset(BSpece 1 ¢ BSpece2) Reset(BSpece ¢ 1) Reset(BSpece Reset(BSpec<br />

2)<br />

e L) Reset(BSpece ) L<br />

Reset(BSpece f ) § Reset(BSpec e )§ f <br />

Reset( BSpece ¥ envs¥ Sysp e<br />

<br />

)<br />

)¥<br />

Reset(BSpec Sys ¥<br />

¥ Sys p ¥ Sys<br />

We conclude this subsection by considering context condition (10’) <strong>of</strong> Section 9.4. We<br />

defined this condition as<br />

(10’) The set <strong>of</strong> communication channels as well as the message interface defined in<br />

a class definition conform to the corresponding instance methods or behaviour<br />

specification.<br />

Since the precise meaning <strong>of</strong> this definition is not clear at all, and since the condition is<br />

important for our semantics, we will formally rephrase it. We will do this in terms <strong>of</strong><br />

functions AASort and ChSort defined in this paragraph.<br />

(10’) Let CD p be a class definition <strong>of</strong> Sys p . Then<br />

(i) if CDp <br />

p process class C<br />

interface la 1 ¡ ¡ lam MD p ¡ ¡ i S p i , then<br />

¨<br />

£<br />

£<br />

<br />

¡ ¡ ¡ communication channels ch1 ¡ ¡ chp message<br />

¡<br />

¡ instance methods MDp1<br />

¡ ¡ MDpk<br />

¡ , such that for each i ¡ ¡ 1 k<br />

AASort(S p § § ¡ k¦ i ) 0 i = £<br />

ChSort(S p § § ¡ k¦ i ) 0 i = £<br />

l a 1 ¥¡ ¡ ¡ ¥ la m ¦<br />

ch1 ¥¡ ¡ ¡ ¥ chp ¦<br />

¨<br />

(ii) if CDp <br />

¡ ¡ ¡ ¡ communication channels ch1 ¡ ¡ chp message<br />

c cluster class C<br />

interface la 1 ¡ ¡ lam behaviour specification BSpecp , then<br />

AASort(BSpecp ¥ Sysp ) = £<br />

la 1 ¥¡ ¡ ¡ ¥ lam ¦<br />

ChSort(BSpecp ¥ Sysp ) = £<br />

¡ ¡ ¥ ¦ ¥¡ ch1 chp<br />

9.5.3 The Labeled-Transition System<br />

In the previous subsection we have defined the collection <strong>of</strong> configurations Conf p and<br />

the collection <strong>of</strong> atomic actions Act <strong>of</strong> the labeled transition system (Conf p £<br />

¥ £ ¡ a ¡ Act¥<br />

a<br />

Act¦ ). In this subsection we will take a closer look at the collection £<br />

Act¦ ¡ £ ¡ a )<br />

<strong>of</strong> labeled transition relations. Each a<br />

£ is a binary relation on Conf p . For configurations<br />

BSpece 1 ¥ envs1 ¥ Sysp ¥ Sys and BSpece 2 ¥ envs2 ¥ Sysp ¥ Sys<br />

<br />

¥ ¥<br />

, the intuitive meaning <strong>of</strong><br />

e BSpec1 envs1 Sysp <br />

¥ Sys<br />

¥ ¥ £<br />

a<br />

e BSpec2 envs2 Sysp ¥ Sys is that extended behaviour specifi-<br />

(with information<br />

cation BSpec e 1 (with information envs1, Sys p and Sys) leads to BSpec e 2<br />

a

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