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# Course: Engineering Artificial Intelligence

Course: Engineering Artificial Intelligence. Dr. Radu Marinescu. Lecture 3. Classes of search. Any path search Uninformed search Informed search Optimal path search Uninformed search Informed search. Recap: Problem solving paradigm.

## Course: Engineering Artificial Intelligence

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1. Course:Engineering Artificial Intelligence Dr. Radu Marinescu Lecture 3

2. Classes of search • Any path search • Uninformed search • Informed search • Optimal path search • Uninformed search • Informed search

3. Recap: Problem solving paradigm • What are the states? (All relevant aspects of the problem) • Arrangement of parts (to plan an assembly) • Positions of trucks (to plan package distribution) • Cities (to plan a trip) • Set of facts (e.g., to prove a mathematical theorem) • What are the actions(operators)? (deterministic, discrete) • Assemble two parts • Move a truck to a new position • Fly to a new city • Apply a theorem to derive a new fact • What is the goaltest? (Condition for success) • All parts in place • All packages delivered • Reached destination city • Derived goal fact

4. Recap: Sliding tile puzzle state space

5. Recap: Search algorithms • Basic idea • Exploration of state space graph by generating successors of already-explored states (a.k.a. expanding states) • Every states is evaluated: is it a goal state?

6. Recap: Terminology • A state is a (representation of) a physical configuration • A node is a data structure constituting part of a search tree contains info such as: state, parent node, action, path costg(x), depth

7. Recap: Simple search algorithm • A search node is a path from some state X to the start state, e.g., (X B A S) • The state of a search node is the most recent state of the path, e.g., X • Let Q be a list of search nodes, e.g., ((X B A S) (C B A S) …) • Let S be the start state • Initialize Q with search node (S) as only entry, set Visited = (S) • If Q is empty, fail. Else, pick some node N from Q • If state(N) is goal, return N (we’ve reached the goal) • (Otherwise) Remove N from Q • Find all descendants of state(N) not in Visited and create all the one-step extensions of N to each descendant • Add the extended paths to Q; add children of state(N) to Visited • Go to step 2. Critical decisions: Step 2: picking N from Q Step 6: adding extensions of N to Q

8. Classes of search

9. Terminology • Heuristic – the word generally refers to a “rule of thumb”, something that may be helpful in some cases but not always. Generally held to be in contrast to “guaranteed” or “optimal” • Heuristic function – in search terms, a function that computes a value for a state (but does not depend on any path to that state) that may be helpful in guiding the search • Estimated distance to goal – this type of heuristic function depends on the state and the goal. The classic example is straight-line distance used as an estimate for actual distance in a road network. This type of information can help increase the efficiency of search.

10. Implementing the search strategies • Depth-first • Pick first element of Q • Add path extensions to front of Q • Breadth-first • Pick first element of Q • Add path extensions to end of Q • Best-first (Greedy) • Pick “best” (measured by heuristic value of state) element of Q • Add path extensions anywhere in Q (it may be more efficient to keep Q ordered in some way to make it easier to find the “best”)

11. Implementation issue: finding the best node • There are many possible approaches to finding the best node in Q • Scanning Q to find lowest value • Sorting Q and picking the first element • Keeping Q sorted by doing “sorted” insertions • Keeping Q as a priority queue • Which of these is best will depend among other things on how many children nodes have on average.

12. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q C G A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

13. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q 1 C G A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

14. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q 2 1 C G A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

15. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q 3 2 1 C G A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

16. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q 3 2 1 C G 4 A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

17. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q 3 5 2 1 C G 4 A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

18. Best-First Pick best (by heuristic value) element of Q; Add path extensions anywhere in Q 3 5 2 1 C G 4 A Heuristic Values A=2 C=1 S=10 B=3 D=4 G=0 D S B Added paths in blue; heuristic value of node’s state is in front. We show the paths in reversedorder; the node’s state is the first entry.

19. Classes of search

20. Simple search algorithm • A search node is a path from some state X to the start state, e.g., (X B A S) • The state of a search node is the most recent state of the path, e.g., X • Let Q be a list of search nodes, e.g., ((X B A S) (C B A S) …) • Let S be the start state • Initialize Q with search node (S) as only entry, set Visited = (S) • If Q is empty, fail. Else, pick some node N from Q • If state(N) is goal, return N (we’ve reached the goal) • (Otherwise) Remove N from Q • Find all descendants of state(N) not in Visited and create all the one-step extensions of N to each descendant • Add the extended paths to Q; add children of state(N) to Visited • Go to step 2. Critical decisions: Step 2: picking N from Q Step 6: adding extensions of N to Q

21. Simple search algorithm • A search node is a path from some state X to the start state, e.g., (X B A S) • The state of a search node is the most recent state of the path, e.g., X • Let Q be a list of search nodes, e.g., ((X B A S) (C B A S) …) • Let S be the start state • Initialize Q with search node (S) as only entry, set Visited = (S) • If Q is empty, fail. Else, pick some node N from Q • If state(N) is goal, return N (we’ve reached the goal) • (Otherwise) Remove N from Q • Find all descendants of state(N) not in Visited and create all the one-step extensions of N to each descendant • Add the extended paths to Q; add children of state(N) to Visited • Go to step 2. Don’t use Visited for optimal search Critical decisions: Step 2: picking N from Q Step 6: adding extensions of N to Q

22. Why not a Visited list? • For the any-path algorithms, the Visited list would not cause us to fail to find a path when one existed, since the path to a state did not matter • However, the Visited list in connection with optimal searches can cause us to miss the best path

23. Why not a Visited list? • For the any-path algorithms, the Visited list would not cause us to fail to find a path when one existed, since the path to a state did not matter • However, the Visited list in connection with optimal searches can cause us to miss the best path • The shortest path from S to G is • (S A D G) 1 A 2 G D 4 S

24. Why not a Visited list? • For the any-path algorithms, the Visited list would not cause us to fail to find a path when one existed, since the path to a state did not matter • However, the Visited list in connection with optimal searches can cause us to miss the best path • The shortest path from S to G is • (S A D G) • But, when extending (S), A and D • would be added to the Visited list and • so (S A) would not be extended to • (S A D) 1 A 2 G D 4 S

25. Implementing optimal search strategies • Uniform cost • Pick best (measured by path length) element of Q • Add path extensions anywhere in Q

26. Uniform Cost • Like best-first except that is uses the “total length (cost)” of a path instead of a heuristic value of the state • Each link has a “length” or “cost” (which is always > 0) • We want “shortest” or “least cost” path C G 2 A 3 2 4 D 2 5 S 1 B 5

27. Uniform Cost • Like best-first except that is uses the “total length (cost)” of a path instead of a heuristic value of the state • Each link has a “length” or “cost” (which is always > 0) • We want “shortest” or “least cost” path Total path cost: (S A C) 4 C G 2 A 3 2 4 D 2 5 S 1 B 5

28. Uniform Cost • Like best-first except that is uses the “total length (cost)” of a path instead of a heuristic value of the state • Each link has a “length” or “cost” (which is always > 0) • We want “shortest” or “least cost” path Total path cost: (S A C) 4 (S B D G) 8 C G 2 A 3 2 4 D 2 5 S 1 B 5

29. Uniform Cost • Like best-first except that is uses the “total length (cost)” of a path instead of a heuristic value of the state • Each link has a “length” or “cost” (which is always > 0) • We want “shortest” or “least cost” path Total path cost: (S A C) 4 (S B D G) 8 (S A D C) 9 C G 2 A 3 2 4 D 2 5 S 1 B 5

30. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 1 C G 2 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

31. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 2 1 C G 2 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

32. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 3 2 1 C G 2 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

33. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 3 2 1 C G 2 4 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

34. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 3 2 5 1 C G 2 4 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

35. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 3 2 5,6 1 C G 2 4 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

36. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 3 7 2 5,6 1 C G 2 4 A 3 2 4 D 2 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

37. Uniform Cost Pick best (by path length) element of Q; Add path extensions anywhere in Q 3 7 2 5,6 1 C G 2 4 A 3 2 4 D 2 Shortest path (S A D G) with length 8 5 S 1 B 5 Added paths in blue; underlined paths are chosen for extension. We show the paths in reversedorder; the node’s state is the first entry.

38. Why not stop on first generating a goal? • When doing Uniform Cost, it is not correct to stop the search when the first path to a goal is generated, that is, when a node whose state is a goal is added to Q

39. Why not stop on first generating a goal? • When doing Uniform Cost, it is not correct to stop the search when the first path to a goal is generated, that is, when a node whose state is a goal is added to Q • We must wait until such a path is pulled off the Q and tested in step 3. It is only at this point that we are sure it is the shortest path to a goal since there are no other shortest paths that remain unexpanded

40. Why not stop on first generating a goal? • When doing Uniform Cost, it is not correct to stop the search when the first path to a goal is generated, that is, when a node whose state is a goal is added to Q • We must wait until such a path is pulled off the Q and tested in step 3. It is only at this point that we are sure it is the shortest path to a goal since there are no other shortest paths that remain unexpanded • This contrasts with the non-optimal searches where the choice of whether to test for a goal was a matter of convenience and efficiency, not correctness

41. Why not stop on first generating a goal? • When doing Uniform Cost, it is not correct to stop the search when the first path to a goal is generated, that is, when a node whose state is a goal is added to Q • We must wait until such a path is pulled off the Q and tested in step 3. It is only at this point that we are sure it is the shortest path to a goal since there are no other shortest paths that remain unexpanded • This contrasts with the non-optimal searches where the choice of whether to test for a goal was a matter of convenience and efficiency, not correctness • In the previous example, a path to G was generated at step 5, but it was a different, shorter, path at step 7 that we accepted

42. Uniform Cost: another (easier?) way to see it S B 2 5 A 6 4 6 10 D G D C 9 8 9 8 C G C G C G Total path cost 2 A 3 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

43. Uniform Cost: another (easier?) way to see it 1 S B 2 5 A 6 4 6 10 D G D C 1 9 8 9 8 C G C G C G Total path cost 2 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

44. Uniform Cost: another (easier?) way to see it 1 S 2 B 2 5 A 2 6 4 6 10 D G D C 1 9 8 9 8 C G C G C G Total path cost 2 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

45. Uniform Cost: another (easier?) way to see it 1 S 3 2 B 2 5 A 2 3 6 4 6 10 D G D C 1 9 8 9 8 C G C G C G Total path cost 2 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

46. Uniform Cost: another (easier?) way to see it 1 S 3 4 2 B 2 5 A 2 3 6 4 6 10 D G D C 1 9 8 9 8 C G C G C G Total path cost 2 4 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

47. Uniform Cost: another (easier?) way to see it 1 S 3 4 2 B 2 5 A 2 5 3 6 4 6 10 D G D C 5 1 9 8 9 8 C G C G C G Total path cost 2 4 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

48. Uniform Cost: another (easier?) way to see it 1 S 3 4 2 B 2 5 A 2 6 5 3 6 4 6 10 D G D C 5,6 1 9 8 9 8 C G C G C G Total path cost 2 4 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

49. Uniform Cost: another (easier?) way to see it 1 S 3 4 2 7 B 2 5 A 2 6 5 3 6 4 6 10 D G D C 5,6 7 1 9 8 9 8 C G C G C G Total path cost 2 4 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

50. Uniform Cost: another (easier?) way to see it 1 S 3 4 2 7 B 2 5 A 2 6 5 3 6 4 6 10 D G D C 5,6 7 1 9 8 9 8 C G C G C G Total path cost 2 4 A 3 Order pulled off of Q (expanded) 2 4 D 2 5 S 1 B 5 UC enumerates paths in order of total path cost!

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