Problem Solving as State Space Search

Transcription

1 Problem olving as tate pace earch rian. Williams pril 5 th, 00 lides adapted from: 6.0 Tomas Lozano Perez, Russell and Norvig IM rian Williams, pring 0

2 elf-iagnosing Explorers courtesy of JPL rian Williams, pring 0

3 In pace The Exception is the Rule courtesy of N POLLO Quintuple fault occurs (three shorts, tank-line and pressure jacket burst, panel flies off). Power limitations too severe to perform new mission.. Novel reconfiguration identified, exploiting LEM batteries for power. waggert & Lovell work on pollo emergency rig lithium hydroxide unit. rian Williams, pring 0

4 omplex missions must carefully: Plan complex sequences of actions chedule tight resources Monitor and diagnose behavior Repair or reconfigure hardware. Most utonomy problems, search through a space of options. We formulate as state space search. rian Williams, pring 0

5 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search rian Williams, pring 0 5

6 stronaut oose rain Fox Rover an the astronaut get its produce safely across the Martian canal? stronaut + item allowed in the rover. oose alone eats rain Fox alone eats oose Early I: What are the universal problem solving methods? imple Trivial rian Williams, pring 0 6

7 Problem olving as tate pace earch Formulate oal Formulate Problem tates Operators enerate olution equence of states rian Williams, pring 0 7

8 Problem olving as tate pace earch Formulate oal stronaut, Fox, oose & rain across river Formulate Problem tates Location of stronaut, Fox, oose & rain at top or bottom river bank Operators Move rover with astronaut & or 0 items to other bank. rian Williams, pring 0 8

9 rian Williams, pring 0 9 stronaut oose rain Fox rain Fox stronaut oose stronaut rain Fox oose oose stronaut Fox rain stronaut oose rain Fox stronaut oose rain Fox rain stronaut oose Fox stronaut oose rain Fox Fox stronaut oose rain stronaut oose Fox rain oose Fox stronaut rain oose rain stronaut Fox stronaut rain oose Fox stronaut Fox oose rain oose rain Fox stronaut stronaut oose rain Fox

10 Example: 8-Puzzle tart oal tates: Operators: oal Test: integer location for each tile N move empty square up, down, left, right goal state as given rian Williams, pring 0 0

11 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search rian Williams, pring 0

12 Problem Formulation: raph Operator Link (edge) tate Node (vertex) irected raph (one-way streets) Undirected raph (two-way streets) rian Williams, pring 0 5

13 Examples of raphs oston an Fran irline Routes Wash L allas Planning ctions (graph of possible states of the world) Put on Put on Put on Put on Put on rian Williams, pring 0 6

14 olution is a tate equence: Problem olving earches Paths Represent searched paths using a tree. rian Williams, pring 0 7

15 olution is a tate equence: Problem olving earches Paths Represent searched paths using a tree. rian Williams, pring 0 8

16 olution is a tate equence: Problem olving earches Paths Represent searched paths using a tree. rian Williams, pring 0 9

17 earch Trees Root Link (edge) Node (vertex) rian Williams, pring 0 0

18 earch Trees iblings Parent (ncestor) hild (escendant) rian Williams, pring 0

19 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search epth first in lecture readth first at home rian Williams, pring 0

20 lasses of earch lind epth-first ystematic exploration of whole tree (uninformed) readth-first until the goal is found. Iterative-eepening Heuristic Hill-limbing Uses heuristic measure of goodness est-first of a node,e.g. estimated distance to. eam goal. Optimal ranch&ound Uses path length measure. Finds * shortest path. * also uses heuristic rian Williams, pring 0

21 lasses of earch lind epth-first ystematic exploration of whole tree (uninformed) readth-first until the goal is found. Iterative-eepening rian Williams, pring 0

22 epth First earch (F) Idea: Explore descendants before siblings Explore siblings left to right rian Williams, pring 0 5

23 rian Williams, pring 0 6 readth First earch (F) Idea: Explore relatives at same level before their children Explore relatives left to right

24 Elements of lgorithm esign escription: stylized pseudo code, sufficient to analyze and implement the algorithm. nalysis: oundness: is a solution returned by the algorithm guaranteed to be correct? ompleteness: is the algorithm guaranteed to find a solution when there is one? Optimality: is the algorithm guaranteed to find a best solution when there is one? Time complexity: how long does it take to find a solution? pace complexity: how much memory does it need to perform search? rian Williams, pring 0 7

25 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search generic search algorithm epth-first search example Handling cycles readth-first search example (do at home) rian Williams, pring 0 8

26 imple earch lgorithm How do we maintain the earch tate? set of partial paths explored thus far. n ordering on which partial path to expand next (called a queue Q). earch repeatedly: elects next partial path Expands it. Terminates when goal found. rian Williams, pring 0 9

27 imple earch lgorithm Let denote the start node and a goal node. partial path is a path from to some node, e.g., ( ) The head of a partial path is the most recent node of the path, e.g.,. The Q is a list of partial paths, e.g. (( ) ( ) ). rian Williams, pring 0 0

28 imple earch lgorithm Let Q be a list of partial paths, Let be the start node and Let be the oal node.. Initialize Q with partial path (). If Q is empty, fail. Else, pick a partial path N from Q. If head(n) =, return N (goal reached!). Else: a) Remove N from Q b) Find all children of head(n) and create all the one-step extensions of N to each child. c) dd all extended paths to Q d) o to step. rian Williams, pring 0

29 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search generic search algorithm epth-first search example Handling cycles readth-first search example (do at home) rian Williams, pring 0

30 epth First earch (F) epth-first: dd path extensions to front of Q Pick first element of Q For each search type, where do we place the children on the queue? rian Williams, pring 0

31 epth-first Pick first element of Q; dd path extensions to front of Q Q () 5 rian Williams, pring 0

32 epth-first Pick first element of Q; dd path extensions to front of Q Q () 5 rian Williams, pring 0 5

33 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) 5 dded paths in blue rian Williams, pring 0 6

34 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) 5 dded paths in blue rian Williams, pring 0 7

35 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) 5 dded paths in blue rian Williams, pring 0 8

36 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) 5 dded paths in blue rian Williams, pring 0 9

37 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 dded paths in blue rian Williams, pring 0 0

38 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 dded paths in blue rian Williams, pring 0

39 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 dded paths in blue rian Williams, pring 0

40 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) dded paths in blue rian Williams, pring 0

41 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) dded paths in blue rian Williams, pring 0

42 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( )( ) ( ) rian Williams, pring 0 5

43 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( )( ) ( ) rian Williams, pring 0 6

44 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( )( ) ( ) 6 ( )( ) rian Williams, pring 0 7

45 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( )( ) ( ) 6 ( )( ) rian Williams, pring 0 8

46 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search generic search algorithm epth-first search example Handling cycles readth-first search example (do at home) rian Williams, pring 0 9

47 Issue: tarting at and moving top to bottom, will depth-first search ever reach? an effort be wasted in more mild cases? rian Williams, pring 0 50

48 epth-first Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) 5 6 ( ) ( ) ( )( ) ( ) ( )( ) visited multiple times Multiple paths to, & How much wasted effort can be incurred in the worst case? rian Williams, pring 0 5

49 How o We void Repeat Visits? Idea: Keep track of nodes already visited. o not place visited nodes on Q. oes it maintain correctness? ny goal reachable from a node that was visited a second time would be reachable from that node the first time. oes it always improve efficiency? uarantees each node appears at most once at the head of a path in Q. rian Williams, pring 0 5

50 imple earch lgorithm Let Q be a list of partial paths, Let be the start node and Let be the oal node.. Initialize Q with partial path () as only entry; set Visited = ( ). If Q is empty, fail. Else, pick some partial path N from Q. If head(n) =, return N (goal reached!). Else a) Remove N from Q b) Find all children of head(n) not in Visited and create all the one-step extensions of N to each child. c) dd to Q all the extended paths; d) dd children of head(n) to Visited e) o to step. rian Williams, pring 0 5

51 Outline Problem encoding as state space search raphs and search trees epth and breadth-first search generic search algorithm epth-first search example Handling cycles readth-first search example (do at home) rian Williams, pring 0 55

52 epth First earch (F) epth-first: dd path extensions to front of Q Pick first element of Q readth First earch (F) readth-first: dd path extensions to back of Q Pick first element of Q For each search type, where do we place the children on the queue? rian Williams, pring 0 56

53 readth-first Pick first element of Q; dd path extensions to end of Q Q () Visited 5 6 rian Williams, pring 0 58

54 readth-first Pick first element of Q; dd path extensions to end of Q Q () Visited 5 6 rian Williams, pring 0 59

55 readth-first Pick first element of Q; dd path extensions to end of Q Q () ( ) ( ) Visited,, 5 6 rian Williams, pring 0 60

56 readth-first Pick first element of Q; dd path extensions to end of Q Q () ( ) ( ) Visited,, 5 6 rian Williams, pring 0 6

57 readth-first Pick first element of Q; dd path extensions to end of Q Q () ( ) ( ) Visited,, ( ) ( ) ( ),,,, 5 6 rian Williams, pring 0 6

58 readth-first Pick first element of Q; dd path extensions to end of Q Q () ( ) ( ) Visited,, ( ) ( ) ( ),,,, 5 6 rian Williams, pring 0 6

59 readth-first Pick first element of Q; dd path extensions to end of Q Q () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* Visited,,,,,,,,,,, ( ) ( ) ( ),,,,,,,,,, rian Williams, pring 0 68

60 epth-first with visited list Pick first element of Q; dd path extensions to front of Q Q () ( ) ( ) ( ) ( ) ( ) Visited,,,,,, 5 5 ( ) ( ) ( ) ( ),,,,,,,,, rian Williams, pring 0 69

61 ummary Most problem solving tasks may be encoded as state space search. asic data structures for search are graphs and search trees. epth-first and breadth-first search may be framed, among others, as instances of a generic search strategy. ycle detection is required to achieve efficiency and completeness. rian Williams, pring 0 70

62 ppendix rian Williams, pring 0 7

63 readth-first (without Visited list) Pick first element of Q; dd path extensions to end of Q Q () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* ( ) ( ) ( ) ( ) ( ) ( ) ( ) 7 ( ) ( ) ( ) rian Williams, pring 0 78