Water Jug & Missionaries

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Problem Solving and SearchIntroduction to Artificial Intelligence CS440/ECE448 Lecture 2

This Lecture Problem representation Problem solving through search

Reading Chapter 2

Announcements My office hours: Weds. From 2 to 3pm.

The 8-puzzle2 1 7 8 6 3 4 5 1 8 7 6 2 3 4 5

Start

Goal

How do we goal from start configuration to the goal configuration?

The corresponding search tree2 8 3 1 6 4 7 5

2 8 3 1 6 4 7 52 8 3 6 4 1 7 5 2 8 3 1 4 7 6 5

2 8 3 1 4 7 6 52 3 1 8 4 7 6 5

2 8 3 1 6 4 7 52 8 3 1 6 4 7 5

Toy 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

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

In 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.

Example: The 8-puzzle2 8 3 1 6 4 7 5 1 2 3 8 4 7 6 5

Start States: Operators: Goal Test: Path Cost: Constraints:

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

Example: The 8-puzzle2 8 3 1 6 4 7 5 1 2 3 8 4 7 6 5

Start States: Operators: Goal Test: Path Cost: Constraints:

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

Example: The 8-puzzle2 8 3 1 6 4 7 5 1 2 3 8 4 7 6 5

14 7

25 8

36 9

Start

Goal

Initial State = [4, 1, 3, 6, 9, 5, 7, 2] Goal State = [1, 2, 3, 6, 9, 8, 7, 4]

Missionaries 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.

Missionaries 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.

Missionaries and Cannibals(3,3,1)

Missionaries and CannibalsA missionary and cannibal cross

Missionaries and Cannibals(2,2,0)

Missionaries and CannibalsOne missionary returns

Missionaries and Cannibals(3,2,1)

Missionaries and CannibalsTwo cannibals cross

Missionaries and Cannibals(3,0,0)

Missionaries and CannibalsA cannibal returns

Missionaries and Cannibals(3,1,1)

Missionaries and CannibalsTwo missionaries cross

Missionaries and Cannibals(1,1,0)

Missionaries and CannibalsA missionary and cannibal return

Missionaries and Cannibals(2,2,1)

Missionaries and CannibalsTwo Missionaries cross

Missionaries and Cannibals(0,2,0)

Missionaries and CannibalsA cannibal returns

Missionaries and Cannibals(0,3,1)

Missionaries and CannibalsTwo cannibals cross

Missionaries and Cannibals(0,1,0)

Missionaries and CannibalsA cannibal returns

Missionaries and Cannibals(0,2,1)

Missionaries and CannibalsThe last two cannibals cross

Missionaries and Cannibals(0,0,0)

Water 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.

Water Jugs States: How full are the two jugs? State Representation: 4G = ? 3G = ? Initial State: 4G = 0 3G=0 Goal State: 4G=2

Constraints: 0 4G 4 0 3G 3

Operators 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 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.

Search Methods

A Toy Example: A Romanian Holiday 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

The state space

Search 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

General Search ExampleArad Zerind Sibiu Timisoara

Arad

Oradea

Fagaras

Rimnicu Vilcea

Sibiu

Bucharest

The solution

Tree search example

Tree search exampleExpanded node

Fringe

Tree search example

Implementation of Search AlgorithmsNodes: state, parent-node,operator, depth, path costFunction 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

States 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!

Search StrategiesA 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 memoryTime 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)

Uninformed Search StrategiesUninformed (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