CS 188: Ar)ficial Intelligence

Transcription

1 CS 188: Ar)ficial Intelligence Search Instructors: Pieter Abbeel & Anca Dragan University of California, Berkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley (ai.berkeley.edu).]

2 Today Agents that Plan Ahead Search Problems Uninformed Search Methods Depth-First Search Breadth-First Search Uniform-Cost Search

3 Agents that Plan

4 Reflex Agents Reflex agents: Choose ac)on based on current percept (and maybe memory) May have memory or a model of the world s current state Do not consider the future consequences of their ac)ons Consider how the world IS Can a reflex agent be ra)onal? [Demo: reflex op)mal (L2D1)] [Demo: reflex op)mal (L2D2)]

5 Video of Demo Reflex Op)mal

6 Video of Demo Reflex Odd

7 Planning Agents Planning agents: Ask what if Decisions based on (hypothesized) consequences of ac)ons Must have a model of how the world evolves in response to ac)ons Must formulate a goal (test) Consider how the world WOULD BE Op)mal vs. complete planning Planning vs. replanning [Demo: re-planning (L2D3)] [Demo: mastermind (L2D4)]

8 Video of Demo Mastermind

9 Video of Demo Replanning

10 Search Problems

11 Search Problems A search problem consists of: A state space A successor func)on (with ac)ons, costs) A start state and a goal test N, 1.0 E, 1.0 A solu)on is a sequence of ac)ons (a plan) which transforms the start state to a goal state

12 Search Problems Are Models

13 Example: Traveling in Romania State space: Ci)es Successor func)on: Roads: Go to adjacent city with cost = distance Start state: Arad Goal test: Is state == Bucharest? Solu)on?

14 What s in a State Space? The world state includes every last detail of the environment A search state keeps only the details needed for planning (abstrac)on) Problem: Pathing States: (x,y) loca)on Ac)ons: NSEW Successor: update loca)on only Goal test: is (x,y)=end Problem: Eat-All-Dots States: {(x,y), dot booleans} Ac)ons: NSEW Successor: update loca)on and possibly a dot boolean Goal test: dots all false

15 State Space Sizes? World state: Agent posi)ons: 120 Food count: 30 Ghost posi)ons: 12 Agent facing: NSEW How many World states? 120x(2 30 )x(12 2 )x4 States for pathing? 120 States for eat-all-dots? 120x(2 30 )

16 Quiz: Safe Passage Problem: eat all dots while keeping the ghosts perma-scared What does the state space have to specify? (agent posi)on, dot booleans, power pellet booleans, remaining scared )me)

17 State Space Graphs and Search Trees

18 State Space Graphs State space graph: A mathema)cal representa)on of a search problem Nodes are (abstracted) world configura)ons Arcs represent successors (ac)on results) The goal test is a set of goal nodes (maybe only one) In a state space graph, each state occurs only once! We can rarely build this full graph in memory (it s too big), but it s a useful idea

19 State Space Graphs State space graph: A mathema)cal representa)on of a search problem Nodes are (abstracted) world configura)ons Arcs represent successors (ac)on results) The goal test is a set of goal nodes (maybe only one) b a d c e f G S h In a search graph, each state occurs only once! p q r We can rarely build this full graph in memory (it s too big), but it s a useful idea Tiny search graph for a 0ny search problem

20 Search Trees This is now / start N, 1.0 E, 1.0 Possible futures A search tree: A what if tree of plans and their outcomes The start state is the root node Children correspond to successors Nodes show states, but correspond to PLANS that achieve those states For most problems, we can never actually build the whole tree

21 State Space Graphs vs. Search Trees S State Space Graph a b c e d f h p q r G Each NODE in in the search tree is an en0re PATH in the state space graph. We construct both on demand and we construct as li:le as possible. b a d c a Search Tree S e e h r h r p q f p q f q c G q c G a a p q

22 Quiz: State Space Graphs vs. Search Trees Consider this 4-state graph: How big is its search tree (from S)? a S G b Important: Lots of repeated structure in the search tree!

23 Tree Search

24 Search Example: Romania

25 Searching with a Search Tree Search: Expand out poten)al plans (tree nodes) Maintain a fringe of par)al plans under considera)on Try to expand as few tree nodes as possible

26 General Tree Search Important ideas: Fringe Expansion Explora)on strategy Main ques)on: which fringe nodes to explore?

27 Example: Tree Search b a c G S d h e f p q r

28 Depth-First Search

29 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 Implementa0on: Fringe is a LIFO stack

30 Search Algorithm Proper)es

31 Search Algorithm Proper)es Complete: Guaranteed to find a solu)on if one exists? Op)mal: Guaranteed to find the least cost path? Time complexity? Space complexity? b 1 node b nodes Cartoon of search tree: b is the branching factor m tiers b 2 nodes m is the maximum depth solu)ons at various depths b m nodes Number of nodes in en)re tree? 1 + b + b 2 +. b m = O(b m )

32 Depth-First Search (DFS) Proper)es What nodes DFS expand? Some leo prefix of the tree. Could process the whole tree! b 1 node b nodes If m is finite, takes )me O(b m ) b 2 nodes How much space does the fringe take? Only has siblings on path to root, so O(bm) m tiers Is it complete? m could be infinite, so only if we prevent cycles (more later) b m nodes Is it op)mal? No, it finds the leomost solu)on, regardless of depth or cost

33 Breadth-First Search

34 Breadth-First Search Strategy: expand a shallowest node first Implementa0on: Fringe is a FIFO queue S b p a d q c h e r f G S Search Tiers b a d c a h e r p h q e r f p q p q f q c G q c G a a

35 Breadth-First Search (BFS) Proper)es What nodes does BFS expand? Processes all nodes above shallowest solu)on Let depth of shallowest solu)on be s Search takes )me O(b s ) s tiers b 1 node b nodes b 2 nodes How much space does the fringe take? b s nodes Has roughly the last )er, so O(b s ) Is it complete? s must be finite if a solu)on exists, so yes! b m nodes Is it op)mal? Only if costs are all 1 (more on costs later)

36 Quiz: DFS vs BFS

37 Quiz: DFS vs BFS When will BFS outperform DFS? When will DFS outperform BFS? [Demo: dfs/bfs maze water (L2D6)]

38 Video of Demo Maze Water DFS/BFS (part 1)

39 Video of Demo Maze Water DFS/BFS (part 2)

40 Itera)ve Deepening Idea: get DFS s space advantage with BFS s )me / shallow-solu)on advantages Run a DFS with depth limit 1. If no solu)on Run a DFS with depth limit 2. If no solu)on Run a DFS with depth limit 3... b Isn t that wastefully redundant? Generally most work happens in the lowest level searched, so not so bad!

41 Cost-Sensi)ve Search START 2 b 3 1 p 1 a d 2 c e h q r GOAL 2 f 2 BFS finds the shortest path in terms of number of ac)ons. It does not find the least-cost path. We will now cover a similar algorithm which does find the least-cost path.

42 Uniform Cost Search

43 Uniform Cost Search Strategy: expand a cheapest node first: Fringe is a priority queue (priority: cumulative cost) b 3 S 1 2 p 1 a d q 2 c h e 8 2 f r G 2 1 S 0 d 3 e 9 p 1 b 4 c 11 e 5 h 17 r 11 q 16 Cost contours a 6 a h 13 r 7 p q f p q f 8 q c G q 11 c G a 10 a

44 Uniform Cost Search (UCS) Proper)es What nodes does UCS expand? Processes all nodes with cost less than cheapest solu)on! If that solu)on costs C* and arcs cost at least ε, then the effec)ve depth is roughly C*/ε Takes )me O(b C*/ε ) (exponen)al in effec)ve depth) C*/ε tiers b c 1 c 2 c 3 How much space does the fringe take? Has roughly the last )er, so O(b C*/ε ) Is it complete? Assuming best solu)on has a finite cost and minimum arc cost is posi)ve, yes! Is it op)mal? Yes! (Proof next lecture via A*)

45 Uniform Cost Issues Remember: UCS explores increasing cost contours The good: UCS is complete and op)mal! c 1 c 2 c 3 The bad: Explores op)ons in every direc)on No informa)on about goal loca)on Start Goal We ll fix that soon! [Demo: empty grid UCS (L2D5)] [Demo: maze with deep/shallow water DFS/BFS/UCS (L2D7)]

46 Video of Demo Empty UCS

47 Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 1)

48 Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 2)

49 Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 3)

50 The One Queue All these search algorithms are the same except for fringe strategies Conceptually, all fringes are priority queues (i.e. collec)ons of nodes with asached priori)es) Prac)cally, 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 implementa)on that takes a variable queuing object

51 Search and Models Search operates over models of the world The agent doesn t actually try all the plans out in the real world! Planning is all in simula)on Your search is only as good as your models

52 Search Gone Wrong?