Today. CS 188: Artificial Intelligence Fall Recap: Search. Example: Pancake Problem. Example: Pancake Problem. General Tree Search.

Transcription

1 CS 88: Artificil Intelligence Fll 00 Lecture : A* Serch 9//00 A* Serch rph Serch Tody Heuristic Design Dn Klein UC Berkeley Multiple slides from Sturt Russell or Andrew Moore Recp: Serch Exmple: Pncke Prolem Serch prolem: Sttes (configurtions of the world) Successor function: function from sttes to lists of (stte, ction, cost) triples; drwn s grph Strt stte nd gol test Serch tree: Nodes: represent plns for reching sttes Plns hve costs (sum of ction costs) Serch Algorithm: Systemticlly uilds serch tree Chooses n ordering of the fringe (unexplored nodes) Cost: Numer of pnckes flipped Exmple: Pncke Prolem enerl Tree Serch Stte spce grph with costs s weights Action: flip top two Cost: Action: Pth flip to ll rech fourgol: Flip Cost: four, flip three Totl cost: 7

2 Uniform Cost Serch Strtegy: expnd lowest pth cost The good: UCS is complete nd optiml! The d: Explores options in every direction No informtion out gol loction Strt ol c c c [demo: countours UCS] Exmple: Heuristic Function Heuristic: the lrgest pncke tht is still out of plce h(x) 0 Best First (reedy) Exmple: Heuristic Function Strtegy: expnd node tht you think is closest to gol stte Heuristic: estimte of distnce to nerest gol for ech stte A common cse: Best-first tkes you stright to the (wrong) gol Worst-cse: like dlyguided DFS [demo: countours greedy] h(x) Comining UCS nd reedy Uniform-cost orders y pth cost, or ckwrd cost g(n) Best-first orders y gol proximity, or forwrd cost h(n) S d h=6 h=5 h= c h=7 h=6 A* Serch orders y the sum: f(n) = g(n) + h(n) 5 e h= h=0 When should A* terminte? Should we stop when we enqueue gol? A S h = h = B h = No: only stop when we dequeue gol h = 0 Exmple: Teg renger

3 Is A* Optiml? Admissile Heuristics A h = 6 A heuristic h is dmissile (optimistic) if: S h = 7 h = 0 where is the true cost to nerest gol 5 Wht went wrong? Actul d gol cost < estimted good gol cost We need estimtes to e less thn ctul costs! Exmples: 5 Coming up with dmissile heuristics is most of wht s involved in using A* in prctice. Optimlity of A*: Blocking 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 *: lowest cost gol node : nother gol node Proof: Wht could go wrong? We d hve to hve to pop suoptiml gol off the fringe efore * This cn t hppen: Imgine suoptiml gol is on the queue Some node n which is supth of * must lso e on the fringe (why?) n will e popped efore Properties of A* UCS vs A* Contours Uniform-Cost A* Uniform-cost expnded in ll directions Strt ol A* expnds minly towrd the gol, ut does hedge its ets to ensure optimlity Strt ol [demo: countours UCS / A*]

4 Creting Admissile Heuristics Exmple: 8 Puzzle Most of the work in solving hrd serch prolems optimlly is in coming up with dmissile heuristics Often, dmissile heuristics re solutions to relxed prolems, where new ctions re ville 66 Indmissile heuristics re often useful too (why?) 5 Wht re the sttes? How mny sttes? Wht re the ctions? Wht sttes cn I rech from the strt stte? Wht should the costs e? Heuristic: Numer of tiles misplced Why is it dmissile? h(strt) = 8 This is relxedprolem heuristic 8 Puzzle I Averge nodes expnded when optiml pth hs length steps 8 steps steps UCS 6,00.6 x 0 6 TILES 9 7 Wht if we hd n esier 8-puzzle where ny tile could slide ny direction t ny time, ignoring other tiles? Totl Mnhttn distnce Why dmissile? h(strt) = = 8 8 Puzzle II Averge nodes expnded when optiml pth hs length steps 8 steps steps TILES 9 7 MANHATTAN Puzzle III How out using the ctul cost s heuristic? Would it e dmissile? Would we sve on nodes expnded? Wht s wrong with it? Trivil Heuristics, Dominnce Dominnce: h h c if Heuristics form semi-lttice: Mx of dmissile heuristics is dmissile With A*: trde-off etween qulity of estimte nd work per node! Trivil heuristics Bottom of lttice is the zero heuristic (wht does this give us?) Top of lttice is the exct heuristic

5 Other A* Applictions Pthing / routing prolems Resource plnning prolems Root motion plnning Lnguge nlysis Mchine trnsltion Speech recognition Tree Serch: Extr Work! Filure to detect repeted sttes cn cuse exponentilly more work. Why? [demo: pln tiny UCS / A*] rph Serch In BFS, for exmple, we shouldn t other expnding the circled nodes (why?) d c p h q e r q c f p S h q e r q c f p q rph Serch Ide: never expnd stte twice How to implement: Tree serch + list of expnded sttes ( closed set ) Expnd the serch tree node-y-node, ut Before expnding node, check to mke sure its stte is new Importnt: store the closed set s set, not list Cn grph serch wreck completeness? Why/why not? How out optimlity? Optimlity of A* rph Serch Proof: New possile prolem: some n on pth to * isn t in queue when we need it, ecuse some worse n for the sme stte dequeued nd expnded first (disster!) Tke the highest such n in tree Let p e the ncestor of n tht ws on the queue when n ws popped Assume f(p) < f(n) f(n) < f(n ) ecuse n is suoptiml p would hve een expnded efore n Contrdiction! Consistency Wit, how do we know prents hve etter f-vles thn their successors? Couldn t we pop some node n, nd find its child n to hve lower f vlue? YES: h = 0 h = 8 B g = 0 Wht cn we require to prevent these inversions? Consistency: A h = 0 Rel cost must lwys exceed reduction in heuristic Like dmissiility, ut etter! 5

6 Optimlity Tree serch: A* is optiml if heuristic is dmissile (nd non-negtive) UCS is specil cse (h = 0) rph serch: A* optiml if heuristic is consistent UCS optiml (h = 0 is consistent) Consistency implies dmissiility In generl, most nturl dmissile heuristics tend to e consistent, especilly if from relxed prolems Summry: A* A* uses oth ckwrd costs nd (estimtes of) forwrd costs A* is optiml with dmissile heuristics Heuristic design is key: often use relxed prolems Mzeworld Demos 6