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20071020 verification konev_lecture02

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20071020 verification konev_lecture02

1. 1. þ ¾ ý ÓÒ ÚÐ Ú ÖÔÓÓÐº ºÙ Ä Ú ÖÔÓÓÐ ÍÒ Ú Ö× ØÝ ¹ ¾¼¼
2. 2. ´Ö Ø Ú ×Ý×Ø Ñ×µ þ ¸ ¸ ¸ ººº ¸ ¸ Ò× ¸ ¸ þ ¸
3. 3. þ þ
4. 4. þ þ » º ¸ ¸ ¸ ººº ü
5. 5. ü » º º º º þ º
6. 6. ü » º º º º þ º
7. 7. ¸
8. 8. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, E , T , q0 , L) L1 Q S1 S2 E L1 L1 L2 T ⊆ Q×E × Q L1 S4 S3 q0 L : Q → Prop L2
9. 9. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
10. 10. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
11. 11. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
12. 12. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
13. 13. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
14. 14. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
15. 15. ´ØÖ Ò× Ø ÓÒ ×Ý×Ø Ñµ (Q, T , q0 , L) S1 S2 Q T ⊆Q ×Q q0 S4 S3 L : Q → Prop
16. 16. Prop = {On, Fault} Q = {1, 2, 3} press 2 q0 = 1 1 ~On On press T = {(1, press, 2), ~Fault ~Fault (2, press, 1), press (2, press, 3), (1, repair , 1)} repair L = {1 → {} ~On Fault 2 → {On} 3 3 → {Fault}}
17. 17. ¸ þ ´ µ ´ µ
18. 18. S1 = (Q1 , E1 , T1 , q0,1 , L1 ) S2 = (Q2 , E2 , T2 , q0,2 , L2 ) S1 × S2 = (Q, E , T , q0 , L) Q = Q1 × Q2 E = (E1 ∪ {−}) × (E2 ∪ {−}) T = ′ ′ i = 1, 2 ei =′ −′ qi′ = qi ((q1 , q2 ), (e1 , e2 ), (q1 , q2 )) ei = −′ (qi , ei , qi′ ) ∈ Ti ′ q0 = (q0,1 , q0,2 ) L((q1 , q2 )) = L1 (q2 ) ∪ L2 (q2 ) ′ −′
19. 19. S1 = (Q1 , E1 , T1 , q0,1 , L1 ) S2 = (Q2 , E2 , T2 , q0,2 , L2 ) X ⊆ (E1 ∪ {−}) × (E2 ∪ {−}) S1 × S2 = (Q, E , T , q0 , L) Q = Q1 × Q2 E = (E1 ∪ {−}) × (E2 ∪ {−}) T = (e1 , e2 ) ∈ X i = 1, 2 ei =′ −′ ′ ′ ((q1 , q2 ), (e1 , e2 ), (q1 , q2 )) qi′ = qi ei = −′ (qi , ei , qi′ ) ∈ ′ Ti q0 = (q0,1 , q0,2 ) L((q1 , q2 )) = L1 (q2 ) ∪ L2 (q2 )
20. 20. repair repair Zero One Two A B C þ · X = {(press, −), (repair , repair )} (press,-) (press,-) (repair,repair) (3,A) (1,A) (2,A) (press,-) (press,-) (1,B) … (press,-)
21. 21. þ 3 ¸ ´ ³−³µ ¿ ¸ ¾ ´ ³−³µ 3×3=9 ¸ 4×2 =8
22. 22. v := 0 v := v + 1 ¸ ÒØ¸ Ó Ø¸º º º (PC= 1, 0) v= (PC= 1, 1) v= (PC= 1, 143) v= ………………. v: 0 = v: 0 = v: 0 = (PC= 2,v= 0) (PC= 2,v= 1) v: v+ 1 = (PC= 3,v= 1) …..
23. 23. ½¼ Û Ð ÌÖÙ Ó ½½ Û Ø ´ØÙÖÒ ¼µ ½¾ ØÙÖÒ ½ ½¿ Ò Û Ð || turn ¾¼ Û Ð ÌÖÙ Ó ¾½ Û Ø ´ØÙÖÒ ½µ ¾¾ ØÙÖÒ ¼ ¾¿ Ò Û Ð
24. 24. t=0 t=1 10,20 10,20 t=0 t=0 t=1 t=1 10,21 11,20 10,21 11,20 t=0 t=0 t=1 t=1 11,21 12,20 10,22 11,21 t=0 t=1 12,21 11,22
25. 25. ´½µ ¸ ´ µ ÈÖÓ ×× ÈÖÓ ×× x := x + y ; y := y + x; x = 2¸ y = 3 x y ÈÖÓ ×× ÈÖÓ ×× Ö½¸ Ö¾ Ö¾¸ Ö½ þ Ü Ö½ ¸ Ý Ö¾ Ü Ö½ ¸ Ý Ö¾
26. 26. ´¾µ ÈÖÓ ×× ÈÖÓ ×× x := x + y ; y := y + x; x = 2¸ y = 3 x y ÈÖÓ ×× ÈÖÓ ×× ÐÓ Ö½¸ Ñ½ ÐÓ Ö¾¸ Ñ¾ Ö½¸ Ñ¾ Ö¾¸ Ñ½ ×ØÓÖ Ö½¸ Ñ½ ×ØÓÖ Ö¾¸ Ñ¾ þ Ü Ñ½ ¸ Ý Ñ¾ ¸ Ö½ ¸ Ö¾ Ü Ñ½ ¸ Ý Ñ¾ ¸ Ö½ ¸ Ö¾ Ü Ñ½ ¸ Ý Ñ¾ ¸ Ö½ ¸ Ö¾
27. 27. ü ´ µ ü
28. 28. t=0 t=1 10,20 10,20 t=0 t=0 t=1 t=1 10,21 11,20 10,21 11,20 t=0 t=0 t=1 t=1 11,21 12,20 10,22 11,21 t=0 t=1 12,21 11,22 ¸ È ½ ½¾ È ¾ ¾¾
29. 29. A B C G (Q, E , T , q0 , L) q ¸ q0 D F Ê (S) = {q|q S} E
30. 30. ü (Q, E , T , q0 , L) P ⊆Q ÈÖ (P) = {q ∈ Q | ∃p ∈ P, ∃e ∈ E : (q, e, p) ∈ T } ÈÓ×Ø(P) = {q ∈ Q | ∃p ∈ P, ∃e ∈ E : (p, e, q) ∈ T } ¸ press 2 1 ~On On press ÈÖ ({1, 2}) = {1, 2, 3} ~Fault ~Fault press ÈÓ×Ø({2}) = {1, 3} repair ~On Fault 3
31. 31. ü new ¸ þ S new ¸ þ Ê (S) new ← {q0 } R ← {} Û Ð new = {} Ó n q new ¸ q m new q∈R Ø Ò / O(n + m) q R ÈÓ×Ø(q) new Ò Ò Û Ð Ö ØÙÖÒ R
32. 32. ü ü ¸ ¸ ¸ ÈÖ (P)º ¸ ¸ ø
33. 33. þ ÄÌÄ ÆÙËÅÎ