Searching for Solutions

1. Artificial Intelligence CMSC 25000 January 11, 2007 Searching for Solutions

2. Agenda • Search – Motivation • Problem-solving agents • Rigorous problem definitions • Blind exhaustive search: • Breadth-first search • Uniform search • Depth-first search • Iterative deepening search • Search analysis • Computational cost, limitations

3. Problem-Solving Agents • Goal-based agents • Identify goal, sequence of actions that satisfy • Goal: set of satisfying world states • Precise specification of what to achieve • Problem formulation: • Identify states and actions to consider in achieving goal • Given a set of actions, consider sequence of actions leading to a state with some value • Search: Process of looking for sequence • Problem -> action sequence solution

4. Agent Environment Specification • Dimensions • Fully observable vs partially observable: Fully • Deterministic vs stochastic: Deterministic • Static vs dynamic: Static • Discrete vs continuous: Discrete • Issues?

5. Closer to Reality • Sensorless agents (conformant problems) • Replace state with “belief state” • Multiple physical states, successors:sets of successors • Partial observability (contigency problems) • Solution is tree, branch chosen based on percepts

6. Example: vacuum world • Single-state, start in #5. Solution?

7. Example: vacuum world • Single-state, start in #5. Solution?[Right, Vacuum] • Sensorless, start in {1,2,3,4,5,6,7,8}e.g., Right goes to {2,4,6,8} Solution?

8. Example: vacuum world • Sensorless, start in {1,2,3,4,5,6,7,8}e.g., Right goes to {2,4,6,8} Solution?[Right,Vacuum,Left,Vacuum] • Contingency • Nondeterministic: Vacuum may dirty a clean carpet • Partially observable: location, dirt at current location. • Percept: [L, Clean], i.e., start in #5 or #7Solution?

9. Example: vacuum world • Sensorless, start in {1,2,3,4,5,6,7,8}e.g., Right goes to {2,4,6,8} Solution?[Right,Vacuum,Left,Vacuum] • Contingency • Nondeterministic: Vacuum may dirty a clean carpet • Partially observable: location, dirt at current location. • Percept: [L, Clean], i.e., start in #5 or #7Solution?[Right, if dirt then Vacuum]

10. Formal Problem Definitions • Key components: • Initial state: • E.g. First location • Available actions: • Successor function: reachable states • Goal test: • Conditions for goal satisfaction • Path cost: • Cost of sequence from initial state to reachable state • Solution: Path from initial state to goal • Optimal if lowest cost

11. Selecting a state space • Real world is absurdly complex  state space must be abstracted for problem solving • (Abstract) state = set of real states • (Abstract) action = complex combination of real actions • e.g., "Arad  Zerind" represents a complex set of possible routes, detours, rest stops, etc. • For guaranteed realizability, any real state "in Arad“ must get to some real state "in Zerind" • (Abstract) solution = • set of real paths that are solutions in the real world • Each abstract action should be "easier" than the original problem

12. Why Search? • Not just city route search • Many AI problems can be posed as search • Planning: • Vertices: World states; Edges: Actions • Game-playing: • Vertices: Board configurations; Edges: Moves • Speech Recognition: • Vertices: Phonemes; Edges: Phone transitions

13. Vacuum world state space graph • states? • actions? • goal test? • path cost?

14. Vacuum world state space graph • states?integer dirt and robot location • actions?Left, Right, Vacuum • goal test?no dirt at all locations • path cost?1 per action

15. Example: The 8-puzzle • states? • actions? • goal test? • path cost?

16. Example: The 8-puzzle • states?locations of tiles • actions?move blank left, right, up, down • goal test?= goal state (given) • path cost? 1 per move [Note: optimal solution of n-Puzzle family is NP-hard]

17. Example: robotic assembly • states?: real-valued coordinates of robot joint angles, parts of the object to be assembled • actions?: continuous motions of robot joints • goal test?: complete assembly • path cost?: time to execute

18. Basic Search Algorithm • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Remove 1st path from queue; extend path one step • Reject all paths with loops • Add new paths to queue • If goal found=>success; else, failure Paths to extend Order of paths added Position new paths added

20. A B C S G D E F Basic Search Problem • Vertices: Cities; Edges: Steps to next, distance • Find route from S(tart) to G(oal) 4 4 3 5 5 4 3 2 4

21. Formal Statement • Initial State: in(S) • Successor function: • Go to all neighboring nodes • Goal state: in(G) • Path cost: • Sum of edge costs

25. Blind Search • Need SOME route from S to G • Assume no information known • Depth-first search, breadth-first search, uniform-cost search, iterative deepening search • Convert search problem to search tree • Root=Zero length path at Start • Node=Path: label by terminal node • Child one-step extension of parent path

26. S A D B D A E E B B F C E D F B F D E A C G G C G F G Search Tree

27. S D A A E B D B B F C E E D F B F D E A C G Breadth-first Search • Explore all paths to a given depth

28. Breadth-first Search Algorithm • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Remove 1st path from queue; extend path one step • Reject all paths with loops • Add new paths to BACK of queue • If goal found=>success; else, failure

29. Analyzing Search Algorithms • Criteria: • Completeness: Finds a solution if one exists • Optimal: Find the best (least cost) solution • Time complexity: Order of growth of running time • Space complexity: Order of growth of space needs • BFS: • Complete: yes; Optimal: only if # steps= cost • Time complexity: O(b^d+1); Space: O(b^d+1)

30. Uniform-cost Search • BFS: • Extends path with fewest steps • UCS: • Extends path with least cost • Analysis: • Complete?: Yes; Optimal?: Yes • Time: O(b^(C*/e)); Space: O(b^(C*/e))

31. Uniform-cost Search Algorithm • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Remove 1st path from queue; extend path one step • Reject all paths with loops • Add new paths to queue • Sort paths in order of increasing length • If goal found=>success; else, failure

32. S A B C E F D G Depth-first Search • Pick a child of each node visited, go forward • Ignore alternatives until exhaust path w/o goal

33. Depth-first Search Algorithm • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Remove 1st path from queue; extend path one step • Reject all paths with loops • Add new paths to FRONT of queue • If goal found=>success; else, failure

34. Search Issues • Breadth-first search: • Good if many (effectively) infinite paths, b<< • Bad if many end at same short depth, b>> • Uniform-cost search: • Tries paths with many short steps first • Identical to BFS if all steps same cost • Depth-first search: • Good if: most partial=>complete, not too long • Bad if many (effectively) infinite paths

35. Iterative Deepening • Problem: • DFS good space behavior • Could go down blind path, or sub-optimal • Solution: • Search at progressively greater depths: • 1,2,3,4,5…..

36. Progressive Deepening • Question: Aren’t we wasting a lot of work? • E.g. cost of intermediate depths • Answer: (surprisingly) No! • Most nodes at fringe • Fringe (depth d): Cost = b^d • Preceding depths: b^0 + b^1+…b^(d-1) • (b^d - 1)/(b -1) • Ratio of last ply cost/all preceding ~ b - 1 • For large branching factors, prior work small relative to maximum depth

37. Heuristic Search • A little knowledge is a powerful thing • Order choices to explore better options first • More knowledge => less search • Better search alg?? Better search space • Measure of remaining cost to goal-heuristic • E.g. actual distance => straight-line distance A B C 10.4 6.7 4.0 11.0 S G 3.0 8.9 6.9 D E F

38. Hill-climbing Search • Select child to expand that is closest to goal S 8.9 A D 10.4 6.9 E 10.4 A B F 3.0 6.7 G

39. Hill-climbing Search Algorithm • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Remove 1st path from queue; extend path one step • Reject all paths with loops • Sort new paths by estimated distance to goal • Add new paths to FRONT of queue • If goal found=>success; else, failure

40. S 10.4 8.9 D A 8.9 10.4 6.9 6.7 A E B D 6.7 3 4.0 6.9 B F C E A C G Beam Search • Breadth-first search of fixed width - top w • Guarantees limited branching factor, E.g. w=2

41. Beam Search Algorithm • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Extend all paths one step • Reject all paths with loops • Sort all paths in queue by estimated distance to goal • Put top w in queue • If goal found=>success; else, failure

42. Best-first Search • Expand best open node ANYWHERE in tree • Form a 1-element queue of 0 cost=root node • Until first path in queue ends at goal or no paths • Remove 1st path from queue; extend path one step • Reject all paths with loops • Put in queue • Sort all paths by estimated distance to goal • If goal found=>success; else, failure

43. Heuristic Search Issues • Parameter-oriented hill climbing • Make one step adjustments to all parameters • E.g. tuning brightness, contrast, r, g, b on TV • Test effect on performance measure • Problems: • Foothill problem: aka local maximum • All one-step changes - worse!, but not global max • Plateau problem: one-step changes, no FOM + • Ridge problem: all one-steps down, but not even local max • Solution (local max): Randomize!!

44. Summary • Blind search: • Find some path to goal • Depth-first, Breadth-first • Iterative Deepening • Analyzing search algorithms • Completeness, Optimality, Time, Space • Next time: • A little knowledge is a useful thing...

45. Search Costs