CS 343H: Artificial Intelligence

1. CS 343H: Artificial Intelligence Week 2b: Informed Search

2. Quiz : review • Which are true about DFS? (b is branching factor, m is depth of search tree.) • At any given time during the search, the number of nodes on the fringe can be no larger than bm. • At any given time in the search, the number of nodes on the fringe can be as large as b^m. • The number of nodes expanded throughout the entire search can be no larger than bm. • The number of nodes expanded throughout the entire search can be as large as b^m.

3. Quiz : review • Which are true about BFS? (b is the branching factor, s is the depth of the shallowest solution) • At any given time during the search, the number of nodes on the fringe can be no larger than bs. • At any given time during the search, the number of nodes on the fringe can be as large as b^s. • The number of nodes considered throughout the entire search can be no larger than bs. • The number of nodes considered throughout the entire search can be as large as b^s.

4. Quiz: search execution • Run DFS, BFS, and UCS:

5. Today • Informed search • Heuristics • Greedy search • A* search • Graph search

6. Recap: Search • Search problem: • States (configurations of the world) • Actions and costs • Successor function: a function from states to lists of (state, action, cost) triples (world dynamics) • Start state and goal test • Search tree: • Nodes: represent plans for reaching states • Plans have costs (sum of action costs) • Search algorithm: • Systematically builds a search tree • Chooses an ordering of the fringe (unexplored nodes) • Optimal: finds least-cost plans

7. Example: Pancake Problem Cost: Number of pancakes flipped

8. Example: Pancake Problem State space graph with costs as weights 4 2 3 2 3 4 3 4 2 3 2 2 4 3

9. General Tree Search Action: flip top twoCost: 2 Action: flip all fourCost: 4 Path to reach goal: Flip four, flip three Total cost: 7

10. Recall: Uniform Cost Search • Strategy: expand lowest path cost • The good: UCS is complete and optimal! • The bad: • Explores options in every “direction” • No information about goal location c  1 … c  2 c  3 Start Goal [demo: countours UCS]

11. Example: Uniform cost search

12. Another example

13. Informed search: main idea

14. Search heuristic • A heuristic is: • A function that estimates how close a state is to a goal • Designed for a particular search problem 10 5 11.2

15. Example: Heuristic Function h(x)

16. Example: Heuristic Function Heuristic: the largest pancake that is still out of place 3 h(x) 4 3 4 3 0 4 4 3 4 4 2 3

17. How to use the heuristic? • What about following the “arrow” of the heuristic?.... Greedy search

18. Example: Heuristic Function h(x)

19. Best First / Greedy Search • Expand the node that seems closest… • What can go wrong?

20. Greedy search b • Strategy: expand a node that you think is closest to a goal state • Heuristic: estimate of distance to nearest goal for each state • A common case: • Best-first takes you straight to the (wrong) goal • Worst-case: like a badly-guided DFS … b …

21. Quiz: greedy search • Which solution would greedy search find if run on this graph?

22. Enter: A* search

23. Combining UCS and Greedy • Uniform-costorders by path cost, or backward cost g(n) • Greedyorders by goal proximity, or forward cost h(n) • A* Search orders by the sum: f(n) = g(n) + h(n) 5 e h=1 1 1 3 2 S a d G h=6 h=5 1 h=2 h=0 1 c b h=7 h=6 Example: Teg Grenager

24. When should A* terminate? • Should we stop when we enqueue a goal? • No: only stop when we dequeue a goal A 2 2 h = 2 G S h = 3 h = 0 B 3 2 h = 1

25. Quiz: A* Tree search

26. Quiz: A* Tree search • When running A*, the first node expanded is S (like all search algorithms). After expanding S, two nodes will be on the fringe: S->A and S->D. Fill in: • g((S->A)) = • h((S->A)) = • f((S->A)) = • g((S->D))= • h((S->D)) = • f((S->D)) = • Which node will be expanded next?

27. Is A* Optimal? 1 • What went wrong? • Actual bad goal cost < estimated good goal cost • We need estimates to be less than actual costs! A 3 h = 6 h = 0 S G h = 7 5

28. Idea: admissibility Inadmissible (pessimistic): break optimality by trapping good plans on the fringe Admissible (optimistic): slows down bad plans but never outweigh true costs

29. Admissible Heuristics • A heuristic h is admissible(optimistic) if: where is the true cost to a nearest goal • Examples: • Coming up with admissible heuristics is most of what’s involved in using A* in practice. 15 4

30. Optimality of A* Notation: • g(n) = cost to node n • h(n) = estimated cost from n to the nearest goal (heuristic) • f(n) = g(n) + h(n) =estimated total cost via n • A: a lowest cost goal node • B: another goal node … A B Claim: A will exit the fringe before B.

31. Claim: A will exit the fringe before B. Optimality of A* • Imagine B is on the fringe. • Some ancestor n of A must be on the fringe too (maybe n is A) • Claim: n will be expanded before B. • f(n) <= f(A) … A B • f(n) = g(n) + h(n) // by definition • f(n) <= g(A) // by admissibility of h • g(A) = f(A) // because h=0 at goal

32. Claim: A will exit the fringe before B. Optimality of A* • Imagine B is on the fringe. • Some ancestor n of A must be on the fringe too (maybe n is A) • Claim: n will be expanded before B. • f(n) <= f(A) • f(A) < f(B) … A B • g(A) < g(B) // B is suboptimal • f(A) < f(B) // h=0 at goals

33. Claim: A will exit the fringe before B. Optimality of A* • Imagine B is on the fringe. • Some ancestor n of A must be on the fringe too (maybe n is A) • Claim: n will be expanded before B. • f(n) <= f(A) • f(A) < f(B) • n will expand before B … A B • f(n) <= f(A) < f(B) // from above • f(n) < f(B)

34. Claim: A will exit the fringe before B. Optimality of A* • Imagine B is on the fringe. • Some ancestor n of A must be on the fringe too (maybe n is A) • Claim: n will be expanded before B. • f(n) <= f(A) • f(A) < f(B) • n will expand before B • All ancestors of A expand before B • A expands before B … A B

35. Properties of A* Uniform-Cost A* b b … …

36. UCS vs A* Contours • Uniform-cost expands equally in all directions • A* expands mainly toward the goal, but does hedge its bets to ensure optimality Start Goal Start Goal

37. Recall: greedy

38. Uniform cost search

39. A*

40. Quiz: A* tree search

41. Quiz

42. A* applications • Pathing / routing problems • Video games • Resource planning problems • Robot motion planning • Language analysis • Machine translation • Speech recognition • …

43. Creating Admissible Heuristics • Most of the work in solving hard search problems optimally is in coming up with admissible heuristics • Often, admissible heuristics are solutions to relaxed problems, where new actions are available • Inadmissible heuristics are often useful too (why?) 366 15

44. Example: 8 Puzzle • What are the states? • How many states? • What are the actions? • What states can I reach from the start state? • What should the costs be?

45. 8 Puzzle I • Heuristic: Number of tiles misplaced • Why is it admissible? • h(start) = • This is a relaxed-problem heuristic 8

46. 8 Puzzle II • What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles? • Total Manhattan distance • Why admissible? • h(start) = 3 + 1 + 2 + … = 18

47. 8 Puzzle II • What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles? • Total Manhattan distance • Why admissible? • h(start) = 3 + 1 + 2 + … = 18

48. Quiz 1 • Consider the problem of Pacman eating all dots in a maze. Every movement action has a cost of 1. Select all heuristics that are admissible, if any. • the total number of dots left • the distance to the closest dot • the distance to the furthest dot • the distance to the closest dot plus distance to the furthest dot • none of the above

49. 8 Puzzle III • How about using the actual cost as a heuristic? • Would it be admissible? • Would we save on nodes expanded? • What’s wrong with it? • With A*: a trade-off between quality of estimate and work per node!

50. S e e p d q h h r r b c p p q q f f a a q q c c G G a a Graph Search • In BFS, for example, we shouldn’t bother expanding the circled nodes (why?)