CS 221: Artificial Intelligence Fall 2011

Transcription

1 CS 221: Artificil Intelligence Fll 2011 Lecture 2: Serch (Slides from Dn Klein, with help from Sturt Russell, Andrew Moore, Teg Grenger, Peter Norvig)

2 Problem types! Fully observble, deterministic! single-belief-stte problem! Non-observble! sensorless (conformnt) problem! Prtilly observble/non-deterministic! contingency problem! interleve serch nd execution! Unknown stte spce! explortion problem! execution first 9

3 Serch Problems! A serch problem consists of:! A stte spce! A trnsition model N, 1 E, 1! A strt stte, gol test, nd pth cost function! A solution is sequence of ctions ( pln) which trnsforms the strt stte to gol stte

4 Trnsition Models! Successor function! Successors( ) = {(N, 1, ), (E, 1, )}! Actions nd Results! Actions( ) = {N, E}! Result(, N) = ; Result(, E) =! Cost(, N, ) = 1; Cost(, E, ) = 1

5 Exmple: Romni! Stte spce:! Cities! Successor function:! Go to dj city with cost = dist! Strt stte:! Ard! Gol test:! Is stte == Buchrest?! Solution?

6 Stte Spce Grphs! Stte spce grph: A mthemticl representtion of serch problem! For every serch problem, there s corresponding stte spce grph S b d c h e f G! The successor function is represented by rcs p q r! This cn be lrge or infinite, so we won t crete it in memory Ridiculously tiny serch grph for tiny serch problem

7 Exponentil Stte Spce Sizes! Serch Problem: Et ll of the food! Pcmn positions: 10 x 12 = 120! Food count: 30! Ghost positions: 12! Pcmn fcing: up, down, left, right

8 Serch Trees N, 1 E, 1! A serch tree:! This is wht if tree of plns nd outcomes! Strt stte t the root node! Children correspond to successors! Nodes contin sttes, correspond to pths to those sttes! For most problems, we cn never ctully build the whole tree

9 Another Serch Tree! Serch:! Expnd out possible plns! Mintin frontier of unexpnded plns! Try to expnd s few tree nodes s possible

10 Generl Tree Serch! Importnt ides:! Frontier (k fringe)! Expnsion! Explortion strtegy Detiled pseudocode is in the book!! Min question: which frontier nodes to explore?

11 Stte Spce vs. Serch Tree S b d c h e f G Ech NODE in in the serch tree is n entire PATH in the stte spce. p q r S d e p We construct both on demnd nd we construct s little s possible. b c p h e q r f h r p q f q c G q q c G

12 Sttes vs. Nodes! Nodes in stte spce grphs re problem sttes! Represent n bstrcted stte of the world! Hve successors, cn be gol / non-gol, hve multiple predecessors! Nodes in serch trees re pths! Represent pth (sequence of ctions) which results in the node s stte! Hve problem stte nd one prent, pth length, ( depth) & cost! The sme problem stte my be chieved by multiple serch tree nodes Stte Spce Grph Serch Tree Prent Depth 5 Node Action Depth 6

13 Depth First Serch S b d p c e p h f r q q c G q e p h f r q q c G S G d b p q c e h f r q p h f d b c e r Strtegy: expnd deepest node first Implementtion: Frontier is LIFO stck [demo: dfs]

14 Bredth First Serch Strtegy: expnd shllowest node first Implementtion: Fringe is FIFO queue S b p d q c h e r f G S Serch Tiers b d c h e r p h q e r f p q p q f q c G q c G [demo: bfs]

15 Sntyn s Wrning! Those who cnnot remember the pst re condemned to repet it. George Sntyn! Filure to detect repeted sttes cn cuse exponentilly more work (why?)

16 Grph Serch! In BFS, for exmple, we shouldn t bother expnding the circled nodes (why?) S d e p b c e h r q h r p q f p q f q c G q c G

17 Grph Serch! Very simple fix: never expnd stte twice! Cn this wreck completeness? Lowest cost?

18 Grph Serch Hints! Grph serch is lmost lwys better thn tree serch (when not?)! Implement explored s dict or set! Implement frontier s priority Q nd set

19 Costs on Actions b d 2 c e GOAL 2 f START 1 p 4 15 q 4 h r 1 Notice tht BFS finds the shortest pth in terms of number of trnsitions. It does not find the lest-cost pth. We will quickly cover n lgorithm which does find the lest-cost pth.

20 Uniform Cost Serch Expnd chepest node first: Frontier is priority queue b 3 S 1 2 p 1 d q 2 c h e 8 1 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 Cost contours 6 p h 13 q r f 7 8 p q f q c G q 11 c G 10

21 Uniform Cost Issues! Remember: explores incresing cost contours! The good: UCS is complete nd optiml! c 1 c 2 c 3! The bd:! Explores options in every direction! No informtion bout gol loction Strt Gol [demos: ucs, ucs2]

22 Uniform Cost Serch! Wht will UCS do for this grph? 0 b START GOAL! Wht does this men for completeness?

23 AI Lesson To do more, Know more

24 Serch Heuristics! Any estimte of how close stte is to gol! Designed for prticulr serch problem! Exmples: Mnhttn distnce, Eucliden distnce

25 Heuristics

26 Greedy Best First Serch! Expnd the node tht seems closest to gol! Wht cn go wrong? [demos: gbf1, gbf2]

27 Greedy goes wrong S G

28 Best First / Greedy Serch! Strtegy: expnd the closest node to the gol 2 b h=8 h= d h=8 S 1 h=12 p 15 h=11 2 c 8 h= h 4 h=6 q 4 3 h=9 G h=0 h=4 2 5 e f h=4 5 r h=6 [demos: gbf1, gbf2]

29 Combining UCS nd Greedy! Uniform-cost orders by pth cost, or bckwrd cost g(n)! Best-first orders by distnce to gol, or forwrd cost h(n) 5 1 S 1 3 d h=6 1 h=5 h=2 1 c b h=7 h=6 e h=1 2 G h=0! A* Serch orders by the sum: f(n) = g(n) + h(n)

30 A* Serch Progress source: wikipedi pge for A* Algorithm; by Subh83

31 When should A* terminte?! Should we stop when we enqueue gol? 2 A 2 S h = 3 h = 2 G 2 B h = 1 3 h = 0! No: only stop when we dequeue gol

32 Is A* Optiml? 1 A h = 6 3 S h = 7 h = 0 G 5! Wht went wrong?! Actul bd pth cost (5) < estimte good pth cost (1+6)! We need estimtes (h=7) to be less thn ctul (5) costs!

33 Admissible Heuristics! A heuristic h is dmissible (optimistic) if: where is the true cost to nerest gol Never overestimte!

34 Creting Admissible Heuristics! Most of the work in solving hrd serch problems optimlly is in coming up with dmissible heuristics! Often, dmissible heuristics re solutions to relxed problems, where new ctions re vilble ! Indmissible heuristics re often useful too (why?)

35 Optimlity of A*: Blocking Nottion:! g(n) = cost to node n! h(n) = estimted cost from n to the nerest gol (heuristic)! f(n) = g(n) + h(n) = estimted totl cost vi n! G*: lowest cost gol node! G: nother gol node

36 Optimlity of A*: Blocking Proof:! Wht could go wrong?! We d hve to hve to pop suboptiml gol G off the frontier before G*! This cn t hppen:! Imgine suboptiml gol G is on the queue! Some node n which is subpth of G* must lso be on the frontier (why?)! n will be popped before G

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

38 UCS vs A* Contours! Uniform-cost expnded in ll directions Strt Gol! A* expnds minly towrd the gol, but does hedge its bets to ensure optimlity Strt Gol [demos: conu, con]

39 Exmple: 8 Puzzle! Wht re the sttes?! How mny sttes?! Wht re the ctions?! Wht sttes cn I rech from the strt stte?! Wht should the costs be?

40 8 Puzzle! Heuristic: Number tiles misplced! Why is it dmissible?! h(strt) =! 8! This is relxed-problem heuristic: Averge nodes expnded when optiml pth hs length 4 steps 8 steps 12 steps UCS 112 6, x 10 6 TILES Move A to B if djcent(a,b) nd empty(b)

41 8 Puzzle! Wht if we hd n esier 8-puzzle where ny tile could slide one step t ny time, ignoring other tiles?! Totl Mnhttn distnce! Why dmissible?! h(strt) =! = 18! Relxed problem: Averge nodes expnded when optiml pth hs length 4 steps 8 steps 12 steps TILES MANHATTAN Move A to B if djcent(a,b) nd empty(b)

42 Trivil Heuristics, Dominnce! Dominnce: h h c if! Heuristics form semi-lttice:! Mx of dmissible heuristics is dmissible! Trivil heuristics! Bottom of lttice is the zero heuristic (wht does this give us?)! Top of lttice is the exct heuristic

43 Other A* Applictions! Pth finding / routing problems! Resource plnning problems! Robot motion plnning! Lnguge nlysis! Mchine trnsltion! Speech recognition!

44 Summry: A*! A* uses both bckwrd costs, g(n), nd (estimtes of) forwrd costs, h(n)! A* is optiml with dmissible heuristics! Heuristic design is key: often use relxed problems! A* is not the finl word in serch lgorithms (but it does get the finl word for tody)