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And-Tree Search Modeling Example – Model Elimination

Explore formal modeling examples in CPSC 433 Artificial Intelligence course, focusing on propositional logic and modeling techniques involving clauses.

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And-Tree Search Modeling Example – Model Elimination

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  1. And-Tree Search ModelingExample – Model Elimination CPSC 433 T01 & T02 Andrew Kuipers CPSC 433 Artificial Intelligence

  2. The Formal Model C – the set of all clauses (our formal language) Prob,me 2C Solution Definition Erw,me((pr, ?), (pr, yes)) if P,P’pr, where  = mgu(P,P’) Branching Definition Erw,me((pr0,?),(pr0,?,pr1,…,prn)) where pr0 = Q  {L1…Ln} and pri = Q  {L1…Ln, Li}, 1 i  n and for some j s.t 1 j  n : Erw,me((prj, ?), (prj, yes)) CPSC 433 Artificial Intelligence

  3. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? CPSC 433 Artificial Intelligence

  4. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? CPSC 433 Artificial Intelligence

  5. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? CPSC 433 Artificial Intelligence

  6. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? CPSC 433 Artificial Intelligence

  7. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? CPSC 433 Artificial Intelligence

  8. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? CPSC 433 Artificial Intelligence

  9. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 6 {p,p},? {p,r},? CPSC 433 Artificial Intelligence

  10. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 6 {p,p},? {p,r},? CPSC 433 Artificial Intelligence

  11. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 6 {p,p},y {p,r},? CPSC 433 Artificial Intelligence

  12. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 6 {p,p},y {p,r},? CPSC 433 Artificial Intelligence

  13. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},? CPSC 433 Artificial Intelligence

  14. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y CPSC 433 Artificial Intelligence

  15. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y CPSC 433 Artificial Intelligence

  16. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},? {p,r,p},? CPSC 433 Artificial Intelligence

  17. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},y {p,r,p},? CPSC 433 Artificial Intelligence

  18. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},y {p,r,p},y CPSC 433 Artificial Intelligence

  19. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},y {p,r,p},y CPSC 433 Artificial Intelligence

  20. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},y {p,r,p},y 4 {p,r,s,s},? CPSC 433 Artificial Intelligence

  21. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},y {p,r,p},y 4 {p,r,s,s},y CPSC 433 Artificial Intelligence

  22. A Propositional Example 1) r  p  s 2) s  q 3) p  s 4) s 5) s  p 6) p  r { },? 3 {p},? {s},? 4 6 {p,p},y {p,r},? {s,s},y 1 {p,r,s},? {p,r,r},y {p,r,p},y 4 search complete! {p,r,s,s},y CPSC 433 Artificial Intelligence

  23. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? • = { } CPSC 433 Artificial Intelligence

  24. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? • = { } CPSC 433 Artificial Intelligence

  25. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},? {P(z),R(x)},? {P(z),Q(y)},? • = { } CPSC 433 Artificial Intelligence

  26. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? • = { z  x } CPSC 433 Artificial Intelligence

  27. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? 5 • = { z  x } {P(z),Q(y),Q(b)},? CPSC 433 Artificial Intelligence

  28. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? 5 • = { z  x, y  b } {P(z),Q(y),Q(b)},y CPSC 433 Artificial Intelligence

  29. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? 6 5 • = { z  x, y  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},? CPSC 433 Artificial Intelligence

  30. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? 6 5 • = { z  x, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y CPSC 433 Artificial Intelligence

  31. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? 6 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y substitute! CPSC 433 Artificial Intelligence

  32. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 4 {P(z),P(x)},y {P(z),R(x)},? {Q(z),Q(a)},? {P(z),Q(y)},? 6 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y CPSC 433 Artificial Intelligence

  33. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 4 {P(z),P(x)},y {P(z),R(x)},? {Q(z),Q(a)},? {P(z),Q(y)},? 6 can’t add z a! 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y CPSC 433 Artificial Intelligence

  34. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 backtrack! {P(z),P(x)},y {P(z),R(x)},? {P(z),Q(y)},? 6 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y CPSC 433 Artificial Intelligence

  35. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 5 {P(z),P(x)},y {P(z),R(x)},? {Q(z),Q(b)},? {P(z),Q(y)},? 6 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y CPSC 433 Artificial Intelligence

  36. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 5 {P(z),P(x)},y {P(z),R(x)},? {Q(z),Q(b)},y {P(z),Q(y)},? 6 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y CPSC 433 Artificial Intelligence

  37. A First-Order Example 1) P(x) Q(y)  R(x) 2) P(a) 3) P(z)  Q(z) 4) Q(a) 5) Q(b) 6) R(b) { }, ? 3 {P(z)},? {Q(z)},? 1 5 {P(z),P(x)},y {P(z),R(x)},? {Q(z),Q(b)},y {P(z),Q(y)},? 6 5 • = { z  b, y  b, x  b } {P(z),Q(y),Q(b)},y {P(z),R(x),R(b)},y search complete! CPSC 433 Artificial Intelligence

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