unit6 - Hardegree, Intermediate Logic ; Unit 6: Derivations...

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Unformatted text preview: Hardegree, Intermediate Logic ; Unit 6: Derivations in Description Logic 1 of 9 Unit 6: Derivations in Description Logic 1. EXERCISES Directions: For each of the following, construct a formal derivation of the conclusion (indicated by "/") from the premises (if any). In cases in which two formulas are separated by '//', construct a derivation of each formula from the other. 1. EXERCISE SET A 1. 2200 xFx ; 5 x[x = t] / F[t] 2. 2200 xFx ; 5 x[x = x] / 5 xFx 3. F[t] ; 5 x[x = t] / 5 xFx 4. / 2200 x(Fx 5 yFy) 5. / 2200 x( 2200 yFy Fx) 6. / 2200 x 5 y[x = y] 7. / 2200 x 2200 y{x = y (Fx Fy)} 8. ~ 5 x[x = x] // 2200 x(Fx & ~ Fx) 9. R[s,t] ; 5 x[x = s] / 5 xR[x,t] 10. ~ 5 x 5 y[x = y] // 2200 x[x & x] 2. EXERCISE SET B 11. / 2200 x 5 y[x = y] 12. / 2200 x(x = yFy 2200 y(Fy y = x)) 13. 5 x[x = xFx] / 2200 x(Fx x = xFx) 14. 5 x[x = xFx] / F[ xFx] 15. 5 x[x = xFx] // 5 x 2200 y(Fy y = x) 16. / 2200 x[x = y[y = x]] 17. / 2200 x 2200 y { x=y z [z =x ] = z [ z=y ] } 18. / 2200 x{Fx F[ y[y = x]]} 19. / 2200 x 2200 y{x = y [ zRxz = zRyz]} 20. 5 x 2200 y(Fy y = x) / 2200 x(Fx x = xFx) 21. G[ xFx]; 5 x[x = xFx] / 5 x(Fx & Gx) 22. G[ xFx] ; 5 x[x = xFx] / 5 x( 2200 y(Fy y = x) & Gx) 23. G[ xFx] ; ~ 5 xGx / ~ 5 x[x = xFx] 24. 5 x 2200 y{Fy y = x} / 5 x(Fx & x = xFx) Hardegree, Intermediate Logic ; Unit 6: Derivations in Description Logic 2 of 9 3. EXERCISE SET C 25. 5 x{ 2200 y(Fy y = x) & Gx} / G[ xFx] 26. 2200 x{Fx x = xFx} / 5 xFx F[ xFx] 27. t = xFx ; 5 x[x = t] / 5 x{Fx & 2200 y[Fy y = x]} 28. Fa ; 2200 x 2200 y{(Fx & Fy) x = y} / a = xFx 29. t = xFx ; 5 x 5 y{x & y & Fx & Fy} / ~ 5 x[x = t] 30. 5 x 2200 y([y & t] y = x) / 2200 x([x = t] [x = x[x & t]]) 31. 2200 x 5 y 2200 z(Rxz z = y) ; 2200 x 2200 y(Rxy Ryx) / 2200 x[ yRxy = yRyx] 32. 5 x 2200 y(Fy y = x) / 2200 x(Fx x = xFx) 33. ~ F[ xFx] ; 5 xFx / 5 x 5 y{x & y & Fx & Fy} 34. 2200 x(Fx Gx) ; 5 x 2200 y(Gy y = x) / 2200 x(Fx [x = xGx]) 35. 5 x 2200 y{Fy y = x} / 2200 x(Fx F[ xFx]) Hardegree, Intermediate Logic ; Unit 6: Derivations in Description Logic 3 of 9 2. ANSWERS TO UNIT 6 EXERCISES #1: (1) 2200 xFx Pr (2) 5 x[x = t] Pr (3) : F[t] DD (4) | a = t 2, 5 O (5) | Fa 1, 2200 O (6) | F[t] 4,5,LL #2: (1) 2200 xFx Pr (2) 5 x[x = x] Pr (3) : 5 xFx 5, 5 I (4) | a = a 2, 5 O (5) | Fa 1, 2200 O #3: (1) F[t] Pr (2) 5 x[x = t] Pr (3) : 5 xFx 5, 5 I (4) | a = t 2, 5 O (5) | Fa 1,4,LL #4: (1) : 2200 x(Fx 5 yFy) UCD (2) | Fa As (3) | : 5 yFy 2, 5 I #5: (1) : 2200 x( 2200 yFy...
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unit6 - Hardegree, Intermediate Logic ; Unit 6: Derivations...

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