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3 = Interleaving A sequence s is an interleaving of two sequences t and u if it can be split into a series of subsequences, with alternate subsequences extracted from t and u. 5 Processes Reversal If s is a sequence, s is formed by taking its elements in reverse order. For example 3, 5, 37 = 37 , 5, 3 Reversal is defined fully by the following laws L1 = L2 x = x L3 s t = t s Reversal enjoys a number of simple algebraic properties, including L4 s = s Exploration of other properties is left to the reader.

1 n } Laws In this section we show how to calculate the set of traces of any process defined using the notations introduced so far. As mentioned above, STOP has only one trace L1 traces(STOP ) = { t | t = }={ } A trace of (c → P ) may be empty, because is a trace of the behaviour of every process up to the moment that it engages in its very first action. 8 Traces of a process 29 These three laws are summarised in the single general law governing choice L4 traces(x : B → P (x)) = { t | t = ∨ (t0 ∈ B ∧ t ∈ traces(P (t0 ))) } To discover the set of traces of a recursively defined process is a bit more difficult.

In fact, any process defined solely by prefixing, choice, and guarded recursions will never stop. The only way to write a process that can stop is to include explicitly the process STOP , or the process (x : B → P (x)) where B is the empty set. By avoiding such elementary mistakes one can guarantee to write processes that never stop. However, after introduction of concurrency in the next chapter, such simple precautions are no longer adequate. 7.

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Communicating Sequential Processes Csp by C A R Hoare

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