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Problem Set 6: Predicate Transforms

Problem Set 6: Predicate Transforms. Hints and Notes. 1. Consider the assertion of weak correctness: {t=5 Λ z<0} s {y=z+1 Λ t=z}. Which of the following observations/facts would allow one to deduce that the assertion is FALSE and which would not? a. The wp (s, y=z) is z >- 5.

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Problem Set 6: Predicate Transforms

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  1. Problem Set 6: Predicate Transforms Hints and Notes

  2. 1. Consider the assertion of weak correctness: {t=5 Λ z<0} s {y=z+1 Λ t=z}. Which of the following observations/facts would allow one to deduce that the assertion is FALSE and which would not? a. The wp(s, y=z) is z>-5. b. The wlp(s, y=z) is z>-5.

  3. 1. Consider the assertion of weak correctness: {t=5 Λ z<0} s {y=z+1 Λ t=z}. Which of the following observations/facts would allow one to deduce that the assertion is FALSE and which would not? a. The wp(s, y=z) is z>-5. b. The wlp(s, y=z) is z>-5.

  4. 1. Consider the assertion of weak correctness: {t=5 Λ z<0} s {y=z+1 Λ t=z}. Which of the following observations/facts would allow one to deduce that the assertion is FALSE and which would not? a. The wp(s, y=z) is z>-5. Would. [wp(s, y=z) = z>-5] => [{t=5 Λ -5<z<0} s {y=z} strongly] => [{t=5 Λ z<0} s {y=z+1 Λ t=z} is false] since we know s will halt for at least four initial (integer) values of z<0 with post condition (y=z+1 Λ t=z) being false. b. The wlp(s, y=z) is z>-5.

  5. 1. Consider the assertion of weak correctness: {t=5 Λ z<0} s {y=z+1 Λ t=z}. Which of the following observations/facts would allow one to deduce that the assertion is FALSE and which would not? a. The wp(s, y=z) is z>-5. Would. [wp(s, y=z) = z>-5] => [{t=5 Λ -5<z<0} s {y=z} strongly] => [{t=5 Λ z<0} s {y=z+1 Λ t=z} is false] since we know s will halt for at least four initial (integer) values of z<0 with post condition (y=z+1 Λ t=z) being false. b. The wlp(s, y=z) is z>-5.

  6. 1. Consider the assertion of weak correctness: {t=5 Λ z<0} s {y=z+1 Λ t=z}. Which of the following observations/facts would allow one to deduce that the assertion is FALSE and which would not? a. The wp(s, y=z) is z>-5. Would. [wp(s, y=z) = z>-5] => [{t=5 Λ -5<z<0} s {y=z} strongly] => [{t=5 Λ z<0} s {y=z+1 Λ t=z} is false] since we know s will halt for at least four initial (integer) values of z<0 with post condition (y=z+1 Λ t=z) being false. b. The wlp(s, y=z) is z>-5. Would not. [wlp(s, y=z) = z>-5] => [{t=5 Λ z>-5} s {y=z}] weakly. But this does not imply that {t=5 Λ z<0} s {y=z+1 Λ t=z} is FALSE since s may not terminate for -5<z<0.

  7. 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if Under what circumstances will the program result in {x≤y<z}? (Hint: determine the wp of the program w.r.t. the desired result.)

  8. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if Under what circumstances will the program result in {x≤y<z}? (Hint: determine the wp of the program w.r.t. the desired result.) S2

  9. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if Under what circumstances will the program result in {x≤y<z}? (Hint: determine the wp of the program w.r.t. the desired result.) S2,1 S2 S2,2

  10. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if Under what circumstances will the program result in {x≤y<z}? (Hint: determine the wp of the program w.r.t. the desired result.) S2,1 S2 S2,2 S2,2,1

  11. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if wp(P, x≤y<z) = wp(S1; S2, x≤y<z) S2,1 S2 S2,2 S2,2,1

  12. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if wp(P, x≤y<z) = wp(S1; S2, x≤y<z) = wp(S1, wp(S2, x≤y<z)) S2,1 S2 S2,2 S2,2,1

  13. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if wp(P, x≤y<z) = wp(S1; S2, x≤y<z) = wp(S1, wp(S2, x≤y<z)) = wp(S1, wp(if temp>z then S2.1;S2.2, x≤y<z)) S2,1 S2 S2,2 S2,2,1

  14. S1 2. Consider the program, P, below: temp := x x := y y := temp if temp>z then y := z z := temp if x>y then temp := x x := y y := temp end_if end_if wp(P, x≤y<z) = wp(S1; S2, x≤y<z) = wp(S1, wp(S2, x≤y<z)) = wp(S1, wp(if temp>z then S2.1;S2.2, x≤y<z)) etc. S2,1 S2 S2,2 S2,2,1

  15. 3. We have learned that P => wp(s,Q) implies {P} s {Q}. However, {P} s {Q} does NOT imply P => wp(s,Q). Why?

  16. 3. We have learned that P => wp(s,Q) implies {P} s {Q}. However, {P} s {Q} does NOT imply P => wp(s,Q). Why? Because: P => wp(s,Q)<=>“{P} s {Q} strongly”

  17. 3. We have learned that P => wp(s,Q) implies {P} s {Q}. However, {P} s {Q} does NOT imply P => wp(s,Q). Why? Because: P => wp(s,Q)<=>“{P} s {Q} strongly” and “{P} s {Q} strongly” =>{P} s {Q}

  18. 3. We have learned that P => wp(s,Q) implies {P} s {Q}. However, {P} s {Q} does NOT imply P => wp(s,Q). Why? Because: P => wp(s,Q)<=>“{P} s {Q} strongly” and “{P} s {Q} strongly” =>{P} s {Q} but {P} s {Q} ≠> “{P} s {Q} strongly”.

  19. 4. a. Use the wlp rule for while-do statements given in Lecture Notes 20 to find the weakest liberal pre- condition of the following program with respect to the post-condition Z=XY. while J<>Y do Z := Z+X; J := J+1 end_while b. Consider the invariant, I: Z=XJ, used with the while- loop ROI to prove the assertion given on slide 11 of Lecture Notes 18. How does it compare in "strength“ to the weakest liberal pre-condition from part (a) above? (In particular, is one STRONGER than the other?) Briefly explain your answer.

  20. a. wlp rule for while-do statements: wlp(while b do S, Q) ≡ wp(while b do S, Q) V ¬wp(while b do S, true)

  21. a. wlp rule for while-do statements: wlp(while b do S, Q) ≡ wp(while b do S, Q) V ¬wp(while b do S, true) i. determining wp(while b do S, Q): H0: J=Y Л Z=XY while J<>Y do Z := Z+X; J := J+1 end_while

  22. a. wlp rule for while-do statements: wlp(while b do S, Q) ≡ wp(while b do S, Q) V ¬wp(while b do S, true) i. determining wp(while b do S, Q): H0: J=Y Л Z=XY H1: J<>Y Л wp(s, J=Y Л Z=XY) = J=Y-1 Л Z=X(Y-1) while J<>Y do Z := Z+X; J := J+1 end_while

  23. a. wlp rule for while-do statements: wlp(while b do S, Q) ≡ wp(while b do S, Q) V ¬wp(while b do S, true) i. determining wp(while b do S, Q): H0: J=Y Л Z=XY H1: J<>Y Л wp(s, J=Y Л Z=XY) = J=Y-1 Л Z=X(Y-1) H2: J<>y Л wp(s, J=Y-1 Л Z=X(Y-1)) = J=Y-2 Л Z=X(Y-2) while J<>Y do Z := Z+X; J := J+1 end_while

  24. a. wlp rule for while-do statements: wlp(while b do S, Q) ≡ wp(while b do S, Q) V ¬wp(while b do S, true) i. determining wp(while b do S, Q): H0: J=Y Л Z=XY H1: J<>Y Л wp(s, J=Y Л Z=XY) = J=Y-1 Л Z=X(Y-1) H2: J<>y Л wp(s, J=Y-1 Л Z=X(Y-1)) = J=Y-2 Л Z=X(Y-2) Hk: J=Y-k Л Z=X(Y-k) while J<>Y do Z := Z+X; J := J+1 end_while

  25. a. wlp rule for while-do statements: wlp(while b do S, Q) ≡ wp(while b do S, Q) V ¬wp(while b do S, true) i. determining wp(while b do S, Q): H0: J=Y Л Z=XY H1: J<>Y Л wp(s, J=Y Л Z=XY) = J=Y-1 Л Z=X(Y-1) H2: J<>y Л wp(s, J=Y-1 Л Z=X(Y-1)) = J=Y-2 Л Z=X(Y-2) Hk: J=Y-k Л Z=X(Y-k) Therefore, H0 V H1 V H2 V ... V Hk V ... simplifies to: (J=Y Л Z=XY) V (J<Y Л Z=XJ) = (J≤Y Л Z=XJ) while J<>Y do Z := Z+X; J := J+1 end_while

  26. ii. determining wp(while b do S, true):

  27. ii. determining wp(while b do S, true): H0: J=Y Л true while J<>Y do Z := Z+X; J := J+1 end_while

  28. ii. determining wp(while b do S, true): H0: J=Y Л true H1: J<>Y Л wp(s, J=Y) = J=Y-1 while J<>Y do Z := Z+X; J := J+1 end_while

  29. ii. determining wp(while b do S, true): H0: J=Y Л true H1: J<>Y Л wp(s, J=Y) = J=Y-1 H2: J<>y Л wp(s, J=Y-1) = J=Y-2 while J<>Y do Z := Z+X; J := J+1 end_while

  30. ii. determining wp(while b do S, true): H0: J=Y Л true H1: J<>Y Л wp(s, J=Y) = J=Y-1 H2: J<>y Л wp(s, J=Y-1) = J=Y-2 Hk: J=Y-k while J<>Y do Z := Z+X; J := J+1 end_while

  31. ii. determining wp(while b do S, true): H0: J=Y Л true H1: J<>Y Л wp(s, J=Y) = J=Y-1 H2: J<>y Л wp(s, J=Y-1) = J=Y-2 Hk: J=Y-k Therefore, H0 V H1 V H2 V ... V Hk V ... simplifies to: J≤Y while J<>Y do Z := Z+X; J := J+1 end_while

  32. ii. determining wp(while b do S, true): H0: J=Y Л true H1: J<>Y Л wp(s, J=Y) = J=Y-1 H2: J<>y Л wp(s, J=Y-1) = J=Y-2 Hk: J=Y-k Therefore, H0 V H1 V H2 V ... V Hk V ... simplifies to: J≤Y Thus, wlp(while b do S, Q) = (J≤Y Л Z=XJ) V J>Y = Z=XJ V J>Y while J<>Y do Z := Z+X; J := J+1 end_while

  33. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1.

  34. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} s T c F {Q}

  35. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = ? (s executes once only) s T c F {Q}

  36. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, ?) s T c F {Q}

  37. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) s T c F {Q}

  38. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = ? (s executes twice only) s T c F {Q}

  39. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = wp(s, ¬c Л?) s T c F {Q}

  40. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = wp(s, ¬c Лwp(s, c ЛQ)) s T c F {Q}

  41. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = wp(s, ¬c Лwp(s, c ЛQ)) s T c F {Q}

  42. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = wp(s, ¬c Лwp(s, c ЛQ)) = wp(s, ¬c ЛH1) s T c F {Q}

  43. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = wp(s, ¬c Лwp(s, c ЛQ)) = wp(s, ¬c ЛH1) H3 = wp(s, ¬c ЛH2) s T c F {Q}

  44. 6. a. Identify H1, H2, H3, and Hk such that wp(Repeat s Until c, Q) = H1 V H2 V H3 V...V Hk V... where Hi represents the necessary and sufficient condition that "Repeat s Until c“ terminates in state Q after i executions of s. For i>1, Hi should be expressed as a function of Hi-1. {wp} H1 = wp(s, c ЛQ) H2 = wp(s, ¬c Лwp(s, c ЛQ)) = wp(s, ¬c ЛH1) H3 = wp(s, ¬c ЛH2) --------------------------- Hk = wp(s, ¬c Л Hk-1) s T c F {Q}

  45. Problem Set 6: Predicate Transforms Hints and Notes

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