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Artificial Intelligence Tutorials

Learn how to design heuristic functions to mimic different search algorithms behavior such as Uniform Cost Search, BFS, and more. Understand the principles behind A* search and its variations.

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Artificial Intelligence Tutorials

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  1. Artificial Intelligence Tutorials

  2. Tutorial # 3

  3. Q1.1 Describe a heuristic estimating function that will make A* search behave exactly like uniform-cost search for a given cost function.

  4. Q1.1 – When A* search Like UCS • We know: • A* expands nodes with minimal f(n) = g(n) + h(n). • UCS expands nodes with minimal path cost g(n). • Then: • For h(n) = 0 or constant  A* Expands nodes with minimal path cost g(n)  i.e. A* behave like UCS.

  5. Q1.2 Describe a heuristic estimating function that will make greedy search behave like breadth-first search.

  6. Q1.2 – When Greedy search Like BFS • We know: • Greedy search expand nodes with minimal h(n). • Then, If h(n) = d(n) where d(n) is the depth of node n: • Nodes will be expanded according to increasing values of depth  i.e. the shallowest before the deepest  i.e. like BFS.

  7. Q2 proof that If h(n) is consistent, then the values of f(n) along any path are non-decreasing.

  8. Q2 - h(n) consistent : f(n) non-decreasing. • We know in A*: • F(n) = g(n) + h(n) • If n’ successor of n  g(n’) = g(n) + c(n,a,n’) • if h(n) consistent  h(n) < c(n,a,n’) + h(n’). • Then: • F(n’) = g(n’) + h(n’) • F(n’) = g(n) + c(n,a,n’) + h(n’) • F(n’) = g(n) + c(n,a,n’) + h(n’) >= g(n) + h(n) • F(n’) >= F(n)  i.e. nodes expanding is non-decreasing order of [f(n)].

  9. Q3 Trace BFS, DFS, IDS and UCS …

  10. BFS • Final Closed List [A B C D E] • Solution path A B D G • Order of node expansion A B C A C D A B D E B C G

  11. DFS • Final Closed List [A B C D] • Solution path A B C D G • Order of node expansion A B A C A B D B C G

  12. IDS • D=0 • Final Closed List [A] • Solution Failure • Order of node expansion A • D=1 • Final Closed List [ABC] • Solution Failure • Order of node expansion A B C

  13. IDS cont • D=2 • Final Closed List [ABCDE] • Solution Failure • Order of node expansion A B A C D C A B D E • D=3 • Final Closed List [ABCD] • Solution pathA B C D G • Order of node expansion A B A C A B D B C G

  14. UCS • Final Closed List [A0 C2 B5 D5 E5] • Solution path [A0 C2 D5 G6] • Order of node expansion A0 C2 A4 B5 B5 D5 E5 G6

  15. Q4 Prove…

  16. Q4.1 – BFS special of UCS • We know: • When step costs are the same  UCS expands with minimal path cost. • Then: • Path costs will increase in depth. • Nodes of the same level have equal path cost. • Path cost of nodes at level I is less than path cost of nodes at level i+1. • Then: • UCS will expand shallowest nodes before deepest, i.e. like the case of BFS.

  17. Q4.2.1 – BFS special of Best First search • We know: • Best first search expands nodes with minimal f(n). • Using Q1.2  Greedy search behave like BFS when h(n) = d(n)  f(n) = d(n)  BFS is a special case of Greedy search. • Greedy search is a best first search. • Then: • BFS is a special case of Best First search.

  18. Q4.2.2 – DFS special of Best First search • We know: • Best first search expands nodes with minimal f(n). • If h(n) = 1/d(n)  Greedy search will expand deepest nodes before shallowest  behave like DFS  i.e. DFS is a special case of Greedy search. • Greedy search is a best first search. • Then: • BFS is a special case of Best First search.

  19. Q4.2.3 – UCS special of Best First search • We know: • Best first search expands nodes with minimal f(n). • Using Q1.1  A* behave like UCS when h(n) = 0  f(n) = g(n)  UCS is a special case of A*. • A* is a best first search. • Then: • UCS is a special case of Best First search.

  20. Q4.3 – UCS special of A* search • Same as Q4.2.3.

  21. G e f d c b a Q5 Answer questions on the figure…

  22. Q5.1 – Complete States G e 4 f 3 d c 2 1 b a

  23. G e f d c b a Q5.2 – Which better (DFS or BFS) • DFS… a b d 1 a 2 3 4 1 b 2 c 1 . . . Repeated States  Infinite Loop!

  24. G e f d c b a Q5.2 – Which better (DFS or BFS) • BFS… a b d 1 a a 3 4 . . . . . . e d f c d e b 2 2 2 3 4 1 . . . . . . . . . . . . . . . 1 3 G 3 1 4 3 G G . . . . 2 e f d • Complete!

  25. G e 4 f 3 d 5.3 – More Suitable Technique c 2 1 b a Try A* technique

  26. Example…g(n) 1 5 G e 4 2 2 1 2 f 3 d 5 2 5 c 3 1 2 1 1 b a 6

  27. Example…h(n) 6 5 0 G e 4 6 2 f 3 d 5 3 c 6 4 2 1 8 b a 5

  28. 6 6 5 2 1 D 4 E 5 0 1 2 G 3 2 5 2 5 3 F 8 A 2 6 2 6 5 4 A*: Final Closed List [A D 3 B 4 E F 1] Solution path A D 3 E G Order of node expansion A D 3 B 4 D E F 1 E G 1 B 1 5 1 C 3

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