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

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Presentation on theme: "CPSC 433 Artificial Intelligence And-Tree Search Modeling Example – Model Elimination CPSC 433 T01 & T02 Andrew Kuipers."— Presentation transcript:

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

2 CPSC 433 Artificial Intelligence The Formal Model C – the set of all clauses (our formal language) Prob ,me  2 C Solution Definition Erw ,me ((pr, ?), (pr, yes)) if P,  P’  pr, where  = mgu(P,P’) Branching Definition Erw ,me ((pr 0,?),(pr 0,?,pr 1,…,pr n )) where pr 0 = Q  {L 1  …  L n } and pr i = Q  {L 1  …  L n, L i }, 1  i  n and for some j s.t 1  j  n : Erw ,me ((pr j, ?), (pr j, yes))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

23 CPSC 433 Artificial Intelligence 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) { }, ?  = { }

24 CPSC 433 Artificial Intelligence 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)  = { } {P(z)},? {  Q(z)},? 3 { }, ?

25 CPSC 433 Artificial Intelligence 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)  = { } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},? {P(z),  Q(y)},? {P(z),  R(x)},? 1 { }, ?

26 CPSC 433 Artificial Intelligence 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)  = { z  x } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 { }, ?

27 CPSC 433 Artificial Intelligence 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)  = { z  x } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},? 5 { }, ?

28 CPSC 433 Artificial Intelligence 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)  = { z  x, y  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 { }, ?

29 CPSC 433 Artificial Intelligence 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)  = { z  x, y  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},? 6 { }, ?

30 CPSC 433 Artificial Intelligence 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)  = { z  x, y  b, x  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},y 6 { }, ?

31 CPSC 433 Artificial Intelligence 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)  = { z  b, y  b, x  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},y 6 substitute! { }, ?

32 CPSC 433 Artificial Intelligence 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)  = { z  b, y  b, x  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},y 6 {  Q(z),Q(a)},? 4 { }, ?

33 CPSC 433 Artificial Intelligence 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)  = { z  b, y  b, x  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},y 6 {  Q(z),Q(a)},? 4 can’t add z  a! { }, ?

34 CPSC 433 Artificial Intelligence 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)  = { z  b, y  b, x  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},y 6 backtrack! { }, ?

35 CPSC 433 Artificial Intelligence 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)  = { z  b, y  b, x  b } {P(z)},? {  Q(z)},? 3 {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? 1 {P(z),  Q(y),Q(b)},y 5 {P(z),  R(x),R(b)},y 6 {  Q(z),Q(b)},? 5 { }, ?

36 CPSC 433 Artificial Intelligence 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) {P(z)},? {  Q(z)},? {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? {P(z),  Q(y),Q(b)},y{P(z),  R(x),R(b)},y {  Q(z),Q(b)},y 3 51 6 5  = { z  b, y  b, x  b } { }, ?

37 CPSC 433 Artificial Intelligence 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) {P(z)},? {  Q(z)},? {P(z),  P(x)},y {P(z),  Q(y)},? {P(z),  R(x)},? {P(z),  Q(y),Q(b)},y{P(z),  R(x),R(b)},y {  Q(z),Q(b)},y 3 51 6 5  = { z  b, y  b, x  b } { }, ? search complete!

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