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2 definition of p n if p is a process and n is a

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guarded recursion has only one solution. 2. Definition of P n If P is a process and n is a natural number, we define ( P n ) as a process which behaves like P for its first n events, and then stops; more formally ( A, S ) n = ( A, { s | s S # s n } ) 48
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& $ % 3. Properties of P n L1 P 0 = STOP L2 P n v P ( n + 1) v P L3 P = t n 0 P n L4 t n 0 P n = t n 0 ( P n n ) 4. Constructive Function Let F be a monotonic function from processes to processes. F is said to be constructive if F ( X ) ( n + 1) = F ( X n ) ( n + 1) for all X This means that the behaviour of F ( X ) on its first n + 1 steps is determined by the behaviour of X on its first n steps only; so if s 6 = hi s traces ( F ( X )) s traces ( F ( X (# s - 1))) 49
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& $ % 5. Constructive Properties of Processes: (a) Prefixing is the primary example of a constructive function, since ( c P ) ( n + 1) = ( c ( P n )) ( n + 1) (b) General choice is also constructive ( x : B P ( x )) ( n + 1) = ( x : B ( P ( x ) n )) ( n + 1) (c) The identity function I is not constructive, since I ( c P ) 1 = c STOP 6 = STOP = I (( c P ) 0) 1 50
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& $ % 6. Main Result: Let F be a constructive function. The equation X = F ( X ) has only one solution for X . Proof Let X be an arbitrary solution. First by induction we prove the lemma that X n = F n ( STOP ) n Base case X 0 = STOP = STOP 0 = F 0 ( STOP ) 0 Induction step X ( n + 1) { since X = F ( X ) } = F ( X ) ( n + 1) { F is constructive } = F ( X n ) ( n + 1) { hypothesis } = F ( F n ( STOP ) n ) ( n + 1) { F is constructuve } = F ( F n ( STOP )) ( n + 1) { Def of F n } = F n +1 ( STOP ) ( n + 1) 51
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& $ % Now we go back to the main theorem X { L3 } = t n 0 ( X n ) { just proved } = t n 0 F n ( STOP ) n { L4 } = t n 0 F n ( STOP ) { 2.8 L7 } = μX F ( X ) Thus all solutions of X = F ( X ) are equal to μX F ( X ); or in other words, μX F ( X ) is the only solution of the equation. 52
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& $ % 7. Nondestructive Function A function G is said nondestructive if G ( P ) n = G ( X n ) n for all n Example: (1) Alphabet transformation is nondestructive in this sense, since f ( P ) n = f ( P n ) n (2) Identity function is nondestructive. (3) Any monotonic function which is constructive is also nondestructive. (4) But the after operator is destructive. For the reason, please see page 78. 53
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& $ % 8. Guarded Expression Let E be an expression containing the process variable X . Then E is said to be guarded in X if every occurrence of X in E has a constructive function applied to it, and no destructive function. 9. Syntactically Defining Constructive Function D1 Expressions constructed solely by means of the operators concurrency, symbol change, and general choice are said to be guard-preserving. D2 An expression which does not contain X is said to be guarded in X . D3 A general choice ( x : B P ( X, x )) is guarded in X if P ( X, x ) is guard-preserving for all x . D4 A symbol change f ( P ( X )) is guarded in X if P ( X ) is guarded in X . D5 A concurrent system P ( X ) || Q ( X ) is guarded in X if both P ( X ) and Q ( X ) are guarded in X . 54
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& $ % 10. Main Result L6 If E is guarded in X , then the equation X = E has a unique solution.
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