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This document discusses various problem solving techniques through search. It begins with an introduction to problem representation, problem solving through search, and examples like the 8-puzzle and missionaries and cannibals problem. It then covers search methods and algorithms like breadth-first search, depth-first search, and A* search. Key concepts discussed include problem states, operators, initial states, goals, and search strategies. Real-world problems are abstracted and represented as states, operators, and paths for solving through search techniques.
![Example: The 8-puzzleExample: The 8-puzzle
2 8 3
1 6 4
7 5
Start
2
8
31
6
4
7 5
Goal
Initial State = [4, 1, 3, 6, 9, 5, 7, 2]
Goal State = [1, 2, 3, 6, 9, 8, 7, 4]
1 2 3
4 5 6
7 8 9](https://image.slidesharecdn.com/raipracticalpresentations-140805124655-phpapp01/85/Rai-practical-presentations-10-320.jpg)
![Implementation of Search AlgorithmsImplementation of Search Algorithms
Function GENERAL-SEARCH (problem, queing-fn)
returns a solution or failure
queue MAKE-QUEUE (MAKE-NODE(INITIAL-STATE[problem]))
loop do
if queue is empty, then return failure
node Remove-Front(queue)
if GOAL-TEST [problem] applied to STATE(node) succeeds
then return node
else queueQUEING-FN(queue,EXPAND(node,operators[problem]))
end
Function GENERAL-SEARCH (problem, queing-fn)
returns a solution or failure
queue MAKE-QUEUE (MAKE-NODE(INITIAL-STATE[problem]))
loop do
if queue is empty, then return failure
node Remove-Front(queue)
if GOAL-TEST [problem] applied to STATE(node) succeeds
then return node
else queueQUEING-FN(queue,EXPAND(node,operators[problem]))
end
Nodes: state, parent-node,operator, depth, path cost](https://image.slidesharecdn.com/raipracticalpresentations-140805124655-phpapp01/85/Rai-practical-presentations-49-320.jpg)
Rai practical presentations.
- 1. Problem SolvingProblem Solving and Searchand Search Introduction to Artificial Intelligence CS440/ECE448 Lecture 2
- 2. This LectureThis Lecture • Problem representation • Problem solving through search ReadingReading • Chapter 2 AnnouncementsAnnouncements • My office hours: Weds. From 2 to 3pm.
- 3. The 8-puzzleThe 8-puzzle 2 8 3 1 6 4 7 5 Start 2 8 31 6 4 7 5 Goal How do we goal from start configuration to the goal configuration?
- 4. The corresponding search treeThe corresponding search tree 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5 2 8 3 1 6 4 7 5
- 5. Toy Problems and Real ProblemsToy Problems and Real Problems • 8-puzzle • Vacuum World • Cryptarithmetic • 8-queens • The water jug problem • Missionaries and Cannibals • Towers of Hanoi • Traveling salesman • Robot navigation • Process or assembly planning • VLSI Layout
- 6. Problem Solving:Problem Solving: • World State – values of all attributes of interest in the world. • State Space – the set of all possible world states. • Operators – change one state into another; cost of applying operator. • Goal – An (often partial) world state or states; in an agent, often implemented as a function of state and current percept. • Initial State – The values of attributes that are in effect at the beginning of a problem before any operators have been applied. • Note: The states and the operators define a directed (possibly weighted) graph. • Solution (path) – a sequence of operators leading from the initial state to a goal state. • Path cost – e.g. sum of distances, number of operators executed…
- 7. In the real worldIn the real world • The real world is absurdly complex. – Real state space must be abstracted for problem solving. – An abstract state is equivalent to a set of real states. • Abstract operator is equivalent to a complex combination of real actions. – Robot operator: Move down hall – In practice, this might involve a complex set of sensor and motor activities. • An abstract solution is equivalent to a set of real paths that are solutions in the real world.
- 8. Example: The 8-puzzleExample: The 8-puzzle • States: • Operators: • Goal Test: • Path Cost: • Constraints: 2 8 3 1 6 4 7 5 Start 2 8 31 6 4 7 5 Goal 3£ 3 array of integer values Move tile number i left, right, up, down = goal state (given) 1 per move Can only move in a direction if that space is empty
- 9. Example: The 8-puzzleExample: The 8-puzzle • States: • Operators: • Goal Test: • Path Cost: • Constraints: 2 8 3 1 6 4 7 5 Start 2 8 31 6 4 7 5 Goal Integer location of tiles (ignore intermediate positions) Move blank left, right, up, down = goal state (given) 1 per move Can only move blank in a direction if it stays in puzzle
- 10. Example: The 8-puzzleExample: The 8-puzzle 2 8 3 1 6 4 7 5 Start 2 8 31 6 4 7 5 Goal Initial State = [4, 1, 3, 6, 9, 5, 7, 2] Goal State = [1, 2, 3, 6, 9, 8, 7, 4] 1 2 3 4 5 6 7 8 9
- 11. Missionaries and cannibalsMissionaries and cannibals • Three missionaries and three cannibals are on the left bank of a river. • There is one canoe which can hold one or two people. • Find a way to get everyone to the right bank, without ever leaving a group of missionaries in one place outnumbered by cannibals in that place.
- 12. Missionaries and cannibalsMissionaries and cannibals • States: three numbers (i,j,k) representing the number of missionaries, cannibals, and canoes on the left bank of the river. • Initial state: (3, 3, 1) • Operators: take one missionary, one cannibal, two missionaries, two cannibals, one missionary and one cannibal across the river in a given direction (I.e. ten operators). • Goal Test: reached state (0, 0, 0) • Path Cost: Number of crossings.
- 14. Missionaries and Cannibals A missionary and cannibal cross
- 16. Missionaries and Cannibals One missionary returns
- 18. Missionaries and Cannibals Two cannibals cross
- 20. Missionaries and Cannibals A cannibal returns
- 22. Missionaries and Cannibals Two missionaries cross
- 24. Missionaries and Cannibals A missionary and cannibal return
- 26. Missionaries and Cannibals Two Missionaries cross
- 28. Missionaries and Cannibals A cannibal returns
- 30. Missionaries and Cannibals Two cannibals cross
- 32. Missionaries and Cannibals A cannibal returns
- 34. Missionaries and Cannibals The last two cannibals cross
- 36. Water JugsWater Jugs • You are given: – a spigot, – a 3 Gallon jug, – a 4 Gallon jug. • The goal: Get 2 gallons of water in the 4 gallon jug. • Actions: Filling jugs from spigot, dumping water in jugs onto ground, dumping 4 gallon into 3 gallon jug until 3 gallon jug is full. Dumping 3 gallon jug into 4 gallon jug until empty or until 4 gallon is full, etc, etc.
- 37. Water JugsWater Jugs • States: How full are the two jugs? • State Representation: 4G = ? 3G = ? • Constraints: 0 ≤ 4G ≤ 4 0 ≤ 3G ≤ 3 • Initial State: 4G = 0 3G=0 • Goal State: 4G=2
- 38. OperatorsOperators • F3: Fill the 3 Gallon jug from the tap. • F4: Fill the 4 Gallon jug from the tap. • E4: Empty the 4-Gallon jug on the ground. • P43: Pour water from 4G jug into the 3G jug until 3G jug is full. • P34: Pour water from 3G jug into the 4G jug until 4G jug is full or 3G is empty.
- 39. • F3: Fill the 3 Gallon jug from the tap. • F4: Fill the 4 Gallon jug from the tap. • E4: Empty the 4-Gallon jug on the ground. • P43: Pour water from 4G jug into the 3G jug until 3G jug is full. • P34: Pour water from 3G jug into the 4G jug until 4G jug is full or 3G is empty. Partial State Graph And Solution Path
- 41. A Toy Example: A RomanianA Toy Example: A Romanian HolidayHoliday • State space: Cities in Romania • Initial state: Town of Arad • Goal: Airport in Bucharest • Operators: Drive between cities • Solution: Sequence of cities • Path cost: number of cities, distance, time, fuel
- 42. The state spaceThe state space
- 43. Search AlgorithmsSearch Algorithms • Basic Idea: Off-line exploration of state space by generating successors of already-explored states (also known as expanding states). Function GENERAL-SEARCH (problem, strategy) returns a solution or failure Initialize the search tree using the initial state of problem loop do if there are no candidates for expansion, then return failure Choose a leaf node for expansion according to strategy if node contains goal state then return solution else expand node and add resulting nodes to search tree. end Function GENERAL-SEARCH (problem, strategy) returns a solution or failure Initialize the search tree using the initial state of problem loop do if there are no candidates for expansion, then return failure Choose a leaf node for expansion according to strategy if node contains goal state then return solution else expand node and add resulting nodes to search tree. end
- 44. General Search ExampleGeneral Search Example Arad Zerind Sibiu Timisoara Arad Oradea Fagaras Rimnicu Vilcea Sibiu Bucharest
- 46. Tree search exampleTree search example
- 47. Tree search exampleTree search example Expanded node Fringe
- 48. Tree search exampleTree search example
- 49. Implementation of Search AlgorithmsImplementation of Search Algorithms Function GENERAL-SEARCH (problem, queing-fn) returns a solution or failure queue MAKE-QUEUE (MAKE-NODE(INITIAL-STATE[problem])) loop do if queue is empty, then return failure node Remove-Front(queue) if GOAL-TEST [problem] applied to STATE(node) succeeds then return node else queueQUEING-FN(queue,EXPAND(node,operators[problem])) end Function GENERAL-SEARCH (problem, queing-fn) returns a solution or failure queue MAKE-QUEUE (MAKE-NODE(INITIAL-STATE[problem])) loop do if queue is empty, then return failure node Remove-Front(queue) if GOAL-TEST [problem] applied to STATE(node) succeeds then return node else queueQUEING-FN(queue,EXPAND(node,operators[problem])) end Nodes: state, parent-node,operator, depth, path cost
- 50. States vs. nodesStates vs. nodes • A state is a (representation of a) physical configuration. • A node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(n). • States do not have parents, children, depth, or path cost!
- 51. Search StrategiesSearch Strategies A strategy is defined by picking the order of node expansion. Strategies are evaluated along the following dimensions: – completeness – does it always find a solution if one exists? – optimality – does it always find a least-cost solution? – time complexity – number of nodes generated/expanded – space complexity – maximum number of nodes in memory Time and space complexity are measured in terms of: b – maximum branching factor of the search tree d – depth of the least-cost solution m – maximum depth of the state space (may be infinite)
- 52. Uninformed Search StrategiesUninformed Search Strategies Uninformed (blind) strategies use only the information available in the problem definition. Informed search techniques which might have additional information (e.g. a compass). • Breadth-first search • Uniform-cost search • Depth-first search • Depth-limited search • Iterative deepening search