Μy f a f f f 1 y the composition of two changes of

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μY : f ( A ) f ( F ( f - 1 ( Y )))) The composition of two changes of symbol is defined by the composition of the two symbol-changing functions L5 f ( g ( P )) = ( f g )( P ) The traces of a process after change of symbol are obtained simply by changing the individual symbols in every trace of the original process L6 traces ( f ( P )) = { f * ( s ) | s traces ( P ) } The explanation of the next and final law is similar to that of L6. L7 f ( P ) /f * ( s ) = f ( P/s ) 30
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& $ % Process labelling A process P labelled by l is denoted by l : P It engages in the event l.x whenever P would have engaged in x . The function required to define l : P is f l ( x ) = l.x for all x in αP and the definition of labelling is l : P = f l ( P ) 31
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& $ % Example: (1) A pair of vending machines standing side by side ( left : V MS ) || ( right : V MS ) The alphabets of the two processes are disjoint, and every event that occurs is labelled by the name of the machine on which it occurred. Note: If we don’t use process labelling, what will happen next? V MS k V MS = V MS (???) 32
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& $ % Specification L1 If P sat S ( tr ) and Q sat T ( tr ) then ( P || Q ) sat ( S ( tr αP ) T ( tr αQ )) Example: Let αP = { a, c } , αQ = { b, c } and P = ( a c P ), Q = ( c b Q ) We wish to prove that ( P || Q ) sat 0 tr a - tr b 2 The proof of 1.10.2 X1 can obviously be adapted to show that P sat (0 tr a - tr c 1) and 33
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& $ % Q sat (0 tr c - tr b 1) By L1 it follows that ( P || Q ) sat (0 ( tr αP ) a - ( tr αP ) c 1 0 ( tr αQ ) c - ( tr αQ ) b 1) 0 tr a - tr b 2 34
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& $ % L2 If P and Q never stop and if ( αP αQ ) contains at most one event, then ( P || Q ) never stops. Example: The process ( P || Q ) defined in X1 will never stop, because αP αQ = c The proof rule for change of symbol is: L3 If P sat S ( tr ), then f ( P ) sat S ( f - 1 * ( tr )) 35
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& $ % Mathematical theory of deterministic processes In our description of processes, we have stated a large number of laws, and we have occasionally used them in proofs. The laws have been justified (if at all) by informal explanations of why we should expect and want them to be true. Question: Are these laws in fact true? Are you consistent? Are you complete? Could one manage with fewer and simpler laws? These are questions for which an answer must be sought in a deeper mathematical investigation. 36
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& $ % The basic definitions 1. From pair ( A, S ) to a process Consider now an arbitrary pair of sets ( A, S ) which satisfy these three laws (in 1.8.1, L6, L7, L8). This pair uniquely identifies a process P whose traces are S constructed according to the following definitions. Let P 0 = { x |h x i ∈ S } and, for all x in P 0 , let P ( x ) be the process whose traces are { t |h x i _ t S } Then αP = A and P = ( x : P 0 P ( x )) 37
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& $ % 2. From a process to pair ( A, S ) Furthermore, the pair ( A, S ) can be recovered by the equations A = αP S = traces ( x : P 0 P ( x )) 3. The one-one correspondence between a process and the corresponding pair Thus there is a one-one correspondence between each process P and the pairs of sets ( αP, traces ( P )).
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