Contextual equivalences in .NET Connect Code 128 Code Set B in .NET Contextual equivalences

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2.5 Contextual equivalences using visual studio .net topaint ansi/aim code 128 in web,windows application Microsoft Official Website Formally we require t Visual Studio .NET barcode 128 hat if P, Q R and P P then there exists some process Q such that Q Q and P , Q R, and similarly for reductions from Q. We add this to our list of desirable properties, to give us our third, and nal property based semantic equivalence: De nition 2.

25 (reduction barbed congruence) Let = be the largest relation over processes that. preserves observations is contextual is reduction-closed. Proposition 2.26 The .net vs 2010 ANSI/AIM Code 128 relation = is an equivalence relation.

Proof: See Question 3 at the end of the chapter. With this new equivalence we can distinguish between the processes P2 , Q2 de ned above in (2.8).

We can prove that if R is any equivalence relation that has these desirable properties it is not possible for it to contain the pair P2 , Q2 . For if it did, it would also have to contain the pair Cbc [P2 ], Cbc [Q2 ] , where Cbc [ ] is the context [ ] . b! . c eureka!. But since Visual Studio .NET USS Code 128 R is reduction-closed and Cbc [Q2 ] Q , where Q is the deadlocked process Cbc [stop], this would require a reduction Cbc [P2 ] Q for some Q such that Q , Q R.

But it is straightforward to see that Q barb eureka for all Q such that Cbc [P2 ] Q ; there are in fact only three possibilities, up to structural equivalence, Cbc [P2 ] itself, c! . c eureka! and eureka !. Now, since R preserves observations, this would require Q barb eureka, which is obviously not true. We take = to be our touchstone equivalence, motivated by a minimal set of desirable properties one might require of a semantic equivalence.

The reasoning just carried out provides a general strategy, based on these properties, for showing that two processes are not equivalent. It is also possible to use the de nition of = in order to develop properties of it. For example structural equivalence has its three de ning properties, that is, it is contextual, reduction-closed and preserves observations.

Since = is the largest such relation it follows that P Q implies P = Q. However, as we have already demonstrated with , this form of de nition often makes it dif cult to show that a pair of processes are equivalent. In general it is necessary to construct a relation containing them that has the three de ning properties, something that is not always easy.

Another trivial example emphasises the point.. The asynchronous PI-CALCULUS Example 2.27 Let P3 , Q3 be the processes (b! b!) b! respectively, where the internal choice operator is de ned in (2.7) above.

To show P3 = Q3 we need to construct a relation R that contains the pair P3 , Q3 and has the three desirable properties. The obvious candidate. R1 = { P3 , Q3 } Id where Id is the ident ity relation, is reduction-closed (trivially) but fails to be contextual. Another possibility would be to let R2 be the closure of R1 under static contexts, as in the proof of Proposition 2.21.

But now one has to show that R2 is reduction-closed. This involves an inductive proof on the inductive closure, together with an analysis of b residuals of processes. The argument is not involved but it does seem overly complicated for such a simple pair of processes.

This provides a good raison d tre for bisimulation equivalence: Proposition 2.28 P bis Q implies P = Q. Proof: From Proposition 2.

21 and Proposition 2.20 we know that bis is contextual and it obviously preserves observations. Let us prove that it is also reduction-closed.

Suppose P bis Q and P P ; we must nd a matching move Q Q . By Proposition 2.14 we know P P for some P such that P P .

We can therefore nd a derivation Q Q for some Q such that P bis Q . Applying Proposition 2.15, we know Q Q and this is the required matching move; since is contained in bis (see Corollary 2.

17), it follows that P bis Q . So bisimulation equivalence has all the de ning properties of =. Since = is the largest such equivalence it follows that bis =.

Thus bisimulation equivalence may be viewed as providing a convenient methodology for proving processes semantically equivalent. For example to show P3 and Q3 , from Example 2.27 are equivalent, it is suf cient to show that R1 is a bisimulation (up to structural equivalence), a simple task.

Essentially we are replacing the need to analyse R2 , the closure of R1 under static contexts, with an analysis of R1 with respect to arbitrary actions. Can we always use this proof method to demonstrate the equivalence of processes The answer is no. Example 2.

29 It is trivial to see that c c! bis stop but it turns out that c c! = stop. To see this let R denote the set of all pairs P1 , P2 such that P1 (new n)(P . c c!) P2 (new n) P
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