CSP-chapter2

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

• Notes
• 55

This preview shows pages 48–55. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

& \$ % 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
& \$ % 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

This preview has intentionally blurred sections. Sign up to view the full version.

& \$ % 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
& \$ % 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

This preview has intentionally blurred sections. Sign up to view the full version.

& \$ % 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
& \$ % 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

This preview has intentionally blurred sections. Sign up to view the full version.

& \$ % 10. Main Result L6 If E is guarded in X , then the equation X = E has a unique solution.
This is the end of the preview. Sign up to access the rest of the document.
• Fall '12
• ZhuHuibiao
• Continuous function, ST OP

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern