521495A: Artificial Intelligence

Transcription

1 521495A: Artificial Intelligence Informed Search Lectured by Abdenour Hadid Adjunct Professor, CMVS, University of Oulu Slides adopted from

2 Today Informed Search Heuristics Greedy Search A* Search Graph Search First Programming Exercises out!

3 Recap: Search

4 Search Example: Romania

5 Search Example: Pacman

6 Example: Traveling in Romania State space: Cities Successor function: Roads: Go to adjacent city Cost = distance Start state: Arad Goal test: Is state == Bucharest? Solution? Sequence of cities

7 Recap: General Tree Search The simplest approach to problem solving using search algorithms is a tree search. Basic idea: Offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states)

8 Searching with a Search Tree

9 Recap:Searching with a Search Tree Search: Expand out potential plans (tree nodes) Maintain a fringe of partial plans under consideration Try to expand as few tree nodes as possible

10 Recap: General Tree Search Fringe Expand Which nodes to explore next?

11 Data Structure and Algorithms - Stack A stack is an Abstract Data Type (ADT), commonly used in most programming languages. Stack allows all data operations at one end only. At any given time, we can only access the top element of a stack.

12 Data Structure and Algorithms - Queue Queue is an abstract data structure, somewhat similar to Stacks. Unlike stacks, a queue is open at both its ends. One end is always used to insert data (enqueue) and the other is used to remove data (dequeue). Queue follows First-In-First-Out methodology, i.e., the data item stored first will be accessed first.

13 Recap: Uninformed search strategies Uninformed (blind) strategies use only the information available in the problem definition. Uninformed Search Methods: Depth-First Search Breadth-First Search Uniform-Cost Search

14 Uninformed search strategies b c 1 c 2 b b c 3 Uniform-Cost Search Breadth-First Search Depth-First Search

15 Recap: General Tree Search

16 Depth-First Search S a b d p a c e p h f r q q c G a q e p h f r q q c G a S G d b p q c e h a f r q p h f d b a c e r Strategy: expand a deepest node first Implementation: Fringe is a LIFO stack i.e., put successors at front

17 Breadth-First Search S a b d p a c e p h f r q q c G a q e p h f r q q c G a S G d b p q c e h a f r Strategy: expand a shallowest node first Implementation: Fringe is a FIFO queue (i.e., new successors go at end)

18 Uniform Cost Search Strategy: expand a cheapest node first (least-cost unexpanded node) Fringe = queue ordered by path cost, lowest first 2 b 1 3 S 1 p a d q 2 c h e 8 2 r G 2 f 1 S 0 d 3 e 9 p 1 b 4 c 11 e 5 h 17 r 11 q 16 a 6 a h 13 r 7 p q f p q f 8 q 11 c G a 10 q c G a

19 The One Queue All these search algorithms are the same except for fringe strategies Conceptually, all fringes are priority queues (i.e. collections of nodes with attached priorities) Practically, for DFS and BFS, you can avoid the log(n) overhead from an actual priority queue, by using stacks and queues Can even code one implementation that takes a variable queuing object

20 Uninformed Search

21 Uniform Cost Search Strategy: expand lowest path cost The good: UCS is complete and optimal! c 1 c 2 c 3 The bad: Explores options in every direction No information about goal location Start Goal

22 Uniform Cost Search Start Goal

23 Uniform Cost Search

24 Informed Search

25 Today Informed Search Heuristics Greedy Search A* Search Graph Search First Programming Exercises out!

26 Informed Search

27 Start Goal

28 Search Heuristics A heuristic is: A function that estimates how close a state is to a goal Designed for a particular search problem Examples: Manhattan distance, Euclidean distance for pathing

29 Example: Heuristic Function h(x)

30 Greedy Search

31 Greedy Search Start Goal

32 Example: Heuristic Function h(x)

33 Greedy Search Expand the node that seems closest What can go wrong?

34 Greedy Search Strategy: expand a node that you think is closest to a goal state Heuristic: estimate of distance to nearest goal for each state b A common case: Best-first takes you straight to the (wrong) goal b Worst-case: like a badly-guided DFS [Demo: contours greedy empty (L3D1)] [Demo: contours greedy pacman small maze (L3D4)]

35 Video of Demo Contours Greedy (Empty)

36 Video of Demo Contours Greedy (Pacman Small Maze)

37 A* Search

38 Combining UCS and Greedy Start Goal

39 A* Search UCS Greedy A*

40 Combining UCS and Greedy h=6 Uniform-cost orders by path cost, or backward cost g(n) Greedy orders by goal proximity, or forward cost h(n) 1 3 S a d h=6 1 h=5 h=2 1 c b h=7 8 1 e h=1 A* Search orders by the sum: f(n) = g(n) + h(n) 2 G h=0 g = 2 h=6 g = 3 h=7 g = 1 h=5 b c a d G S g = 4 h=2 g = 6 h=0 g = 0 h=6 e d G g = 9 h=1 g = 10 h=2 g = 12 h=0 Example: Teg Grenager

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

42 Admissible Heuristics

43 Idea: Admissibility Inadmissible (pessimistic) heuristics break optimality by trapping good plans on the fringe Admissible (optimistic) heuristics slow down bad plans but never outweigh true costs

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

45 Optimality of A* Tree Search

46 Optimality of A* Tree Search Assume: A is an optimal goal node B is a suboptimal goal node h is admissible Claim: A will exit the fringe before B

47 Properties of A*

48 Properties of A* Uniform-Cost A* b b

49 UCS vs A* Contours Uniform-cost expands equally in all directions Start Goal A* expands mainly toward the goal, but does hedge its bets to ensure optimality Start Goal [Demo: contours UCS / greedy / A* empty (L3D1)] [Demo: contours A* pacman small maze (L3D5)]

50 Video of Demo Contours (Empty) -- UCS

51 Video of Demo Contours (Empty) -- Greedy

52 Video of Demo Contours (Empty) A*

53 Video of Demo Contours (Pacman Small Maze) A*

54 Comparison Greedy Uniform Cost A*

55 Creating Heuristics

56 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

57 Semi-Lattice of Heuristics

58 Trivial Heuristics, Dominance Dominance: h a h c if Heuristics form a semi-lattice: Max of admissible heuristics is admissible Trivial heuristics Bottom of lattice is the zero heuristic Top of lattice is the exact heuristic

59 Graph Search

60 Tree Search: Extra Work! Failure to detect repeated states can cause exponentially more work. State Graph Search Tree

61 Graph Search Idea: never expand a state twice How to implement: Tree search + set of expanded states ( closed set ) Expand the search tree node-by-node, but Before expanding a node, check to make sure its state has never been expanded before If not new, skip it, if new add to closed set Important: store the closed set as a set, not a list

62 Consistency of Heuristics Main idea: estimated heuristic costs actual costs A h=4 1 C h=1 h=2 3 G Admissibility: heuristic cost actual cost to goal h(a) actual cost from A to G Consistency: heuristic arc cost actual cost for each arc h(a) h(c) cost(a to C) Consequences of consistency: The f value along a path never decreases h(a) cost(a to C) + h(c) A* graph search is optimal

63 Optimality of A* Graph Search

64 Optimality Tree search: A* is optimal if heuristic is admissible UCS is a special case (h = 0) Graph search: A* optimal if heuristic is consistent UCS optimal (h = 0 is consistent) Consistency implies admissibility In general, most natural admissible heuristics tend to be consistent, especially if from relaxed problems

65 A*: Summary

66 A*: Summary A* uses both backward costs and (estimates of) forward costs A* is optimal with admissible / consistent heuristics Heuristic design is key: often use relaxed problems

67 Thanks