Artificial IntelligenceArtificial IntelligenceArtificial IntelligenceArtificial Intelligence

Module Module –– 11

Part Part -- 33

Solving problems by searching

Simple Vacuum Reflex Agent

function Vacuum-Agent([location,status])

returns Action

if status = Dirty then return Suck

else if location = A then return Right

else if location = B then return Left

It is based on reflex actions, i.e., actions having a direct

mapping on states. A generalized version of above

algorithm is as follows:

Simple Vacuum Reflex Agent- defined by percepts

function Simple-Reflex-Agent(percept) returns an action

Static: rules, a set of conditions-action rules

state ← Interpret-Input(percept)

rule ← Rule-Match(state, rules)

action ← Rule-Action[rule]action ← Rule-Action[rule]

return action

A simple reflex agent, i.e., it acts according to a rule whose

condition matches the current state based on the percept

A model based Reflex Agent

function Reflex-Agent-State(percept) returns an action

Static: state, a description of the current world state

rules, a set of conditions-action rules

action, the most recent action, initially none

It keeps track of the current states using an internal

model, it chooses an action as a reflex agent

state ← Update-State(state, action, percept)

rule ← Rule-Match(state, rules)

action ← Rule-Action[rule]

return action

As the current percept is mixed with the internal state

hence this model is an Intelligent agent having learning.

Goal based Agents• They consider future actions, the desirability of their outcomes

as goals help organize behaviour by limiting the objectives that

the agent is trying to achieve.

• Goal based agents are also called problem solving agents.

Problem solving agents decide what to do by finding

sequences of actions that lead to the desirable states.sequences of actions that lead to the desirable states.

• Goal formulation is based on the current situation and the

agent performance measure is the first step in problem solving.

• Problem formulation is the process of deciding what actions

and states to consider, given a goal.

Problem-solving agents

Example: Romania

� On holiday in Romania; currently in Arad.

� Flight leaves tomorrow from Bucharest

� Formulate goal:� be in Bucharest

� Formulate problem:� states: various cities� states: various cities

� actions: drive between cities

� Find solution:� sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

Example: RomaniaSource

Destination

Problem types

� Deterministic, fully observable � single-state problem� Agent knows exactly which state it will be in; solution is a sequence

� Non-observable � sensorless problem (conformant)� Agent may have no idea where it is; solution is a sequence

Nondeterministic and/or partially observable contingency � Nondeterministic and/or partially observable � contingency problem� percepts provide new information about current state

� often interleave search, execution

� Unknown state space � exploration problem

Single-state problem formulation

A problem is defined by four items:

1. initial state e.g., "at Arad“

2. actions or successor function S(x) = set of action–state pairs � e.g., S(Arad) = {<Arad � Zerind, Zerind>, … }

3. goal test, can be3. goal test, can be� explicit, e.g., x = "at Bucharest"� implicit, e.g., Checkmate(x)

4. path cost (additive)� e.g., sum of distances, number of actions executed, etc.� c(x,a,y) is the step cost, assumed to be ≥ 0

� A solution is a sequence of actions leading from the initial state to a goal state

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 � 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

Example: Water Pouring

• Given a 4 gallon bucket and a 3 gallon bucket, there is unlimited amount of water supply

• How can we measure exactly 2 gallons into the 4 gallon bucket?gallons into the 4 gallon bucket?

– There are no markings on the bucket

– You must fill each bucket completely

Example: Water Pouring

• State: (x, y)

x = 0, 1, 2, 3, or 4 y = 0, 1, 2, 3

• Initial state:

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• Initial state:

The buckets are empty, represented by ( 0 0 )

• Goal state:• One of the buckets has two gallons of water in it

• Represented by either ( x 2 ) or ( 2 x )

• Path cost:• 1 per unit step

Example: Water Pouring

• Actions and Successor Function– Fill a bucket

• (x y) -> (3 y)

• (x y) -> (x 4)

– Empty a bucket

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– Empty a bucket• (x y) -> (0 y)

• (x y) -> (x 0)

– Pour contents of one bucket into another• (x y) -> (0, x+y) or (x+y-4, 4)

• (x y) -> (x+y, 0) or (3, x+y-3)

Example: Water Pouring

Water Pouring : Puzzle-solving as Search

• State representation: (x, y)

– x: Contents of four gallon

– y: Contents of three gallon

• Start state: (0, 0)

• Goal state: (2, n)• Goal state: (2, n)

• Operators

– Fill 3-gallon from reservoir, fill 4-gallon from reservoir

– Fill 3-gallon from 4-gallon , fill 4-gallon from 3-gallon

– Empty 3-gallon into 4-gallon, empty 4-gallon into 3-gallon

– Dump 3-gallon down drain, dump 4-gallon down drain

Production Rules for the Water Jug Problem

1. Fill the 4-gallon jug

1. Fill the 3-gallon jug

1. Pour some water out of the 4-gallon jug

1. Pour some water out of the 3-gallon jug

1. Empty the 4-gallon jug on the ground1. Empty the 4-gallon jug on the ground

1. Empty the 3-gallon jug on the ground

1. Pour water from the 3-gallon jug into the 4-gallon jug until the 4-gallon jug is full

2. Pour water from the 4-gallon jug into the 3-gallon jug until the 3-gallon jug is full

3. Pour all the water from the 3-gallon jug into the 4-gallon jug

4. Pour all the water from the 4-gallon jug into the 3-gallon jug

Production rules for Water Jug ProblemThe operators to be used to solve the problem can be described as follows:

Sl No Current state Next State Descripion

1 (x,y) if x < 4 (4,y) Fill the 4 gallon jug

2 (x,y) if y <3 (x,3) Fill the 3 gallon jug

3 (x,y) if x > 0 (x-d, y) Pour some water out of the 4 gallon jug

4 (x,y) if y > 0 (x, y-d) Pour some water out of the 3-gallon jug

5 (x,y) if x>0 (0, y) Empty the 4 gallon jug

6 (x,y) if y >0 (x,0) Empty the 3 gallon jug on the ground

7 (x,y) if x+y >= 4 and y >0

(4, y-(4-x)) Pour water from the 3 –gallon jug into the 4 –gallon jug until the 4-gallon jug is full

8 (x, y) if x+y >= 3 and x>0

(x-(3-y), 3) Pour water from the 4-gallon jug into the 3-gallon jug until the 3-gallon jug is full

9 (x, y) if x+y <=4 and y>0

(x+y, 0) Pour all the water from the 3-gallon jug into the 4-gallon jug

Production rules for Water Jug Problem

10 (x, y) if x+y <= 3 and x>0

(0, x+y) Pour all the water from the 4-gallon jug into the 3-gallon jug

11 (0,2) (2,0) Pour the 2 gallons from 3-gallon jug into the 4-gallon jug

12 (2,y) (0,y) Empty the 2 gallons in the 4-gallon jug on the ground

One Solution to the Water Jug Problem

Gallons in the 4-Gallon Jug

Gallons in the 3-Gallon Jug

Rule Applied

0 0 2

0 3 9

3 0 23 0 2

3 3 7

4 2 5 or 12

0 2 9 or 11

2 0

Vacuum world state space graph

� states?� actions?� goal test?

� path cost?

Vacuum world state space graph

� states? integer dirt and robot location

� actions? Left, Right, Suck

� goal test? no dirt at all locations

� path cost? 1 per action

Example: The 8-puzzle

� states?

� actions?

� goal test?

� path cost?

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

Example: 8-queens problem

State-Space problem formulation

• states? -any arrangement of n<=8 queens

-or arrangements of n<=8 queens in leftmost n

columns, 1 per column, such that no queen

attacks any other.

• initial state? no queens on the board

• actions? -add queen to any empty square• actions? -add queen to any empty square

-or add queen to leftmost empty square such that it is not attacked by other

queens.

• goal test? 8 queens on the board, none attacked.

• path cost? 1 per move

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

Tree search algorithms

� Basic idea:

� simulated exploration of state space by generating successors of already-explored states (expanding states)

This “strategy” is

what differentiates

different search

algorithms

Tree search example

Tree search example

Tree search example

Implementation: general tree search

Search Tree for the 8 puzzle problem

Implementation: 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 state, parent node, action, path cost g(x), depth

� The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.

Search strategies

� A search 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?

� time complexity: number of nodes generated� time complexity: number of nodes generated

� space complexity: maximum number of nodes in memory

� optimality: does it always find a least-cost solution?

� Time, 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 ∞)

Uninformed search strategies

� Uninformed search strategies use only the information available in the problem definition

� Breadth-first search

� Uniform-cost search� Uniform-cost search

�Depth-first search

�Depth-limited search

� Iterative deepening search

� Bidirectional search

Breadth-first search

� Expand shallowest unexpanded node

� Implementation:

� fringe is a FIFO queue, i.e., new successors go at end

Breadth-first search

� Expand shallowest unexpanded node

� Implementation:

� fringe is a FIFO queue, i.e., new successors go at end

Breadth-first search

� Expand shallowest unexpanded node

� Implementation:

� fringe is a FIFO queue, i.e., new successors go at end

Breadth-first search

� Expand shallowest unexpanded node

� Implementation:

� fringe is a FIFO queue, i.e., new successors go at end

Properties of breadth-first search

� Complete? Yes (if b is finite)

� Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1)

� Space? O(bd+1) (keeps every node in memory)

�Optimal? Yes (if cost = 1 per step)

� Space is the bigger problem (more than time)

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Depth-first search

� Expand deepest unexpanded node

� Implementation:

� fringe = LIFO queue, i.e., put successors at front

Properties of depth-first search

� Complete? No: fails in infinite-depth spaces, spaces with loops

�Modify to avoid repeated states along path

� complete in finite spaces

� Time? O(bm): terrible if m is much larger than d� Time? O(bm): terrible if m is much larger than d� but if solutions are dense, may be much faster than

breadth-first

� Space? O(bm), i.e., linear space!

�Optimal? No

Iterative deepening search

Iterative deepening search l =0

Iterative deepening search l =1

Iterative deepening search l =2

Iterative deepening search l =3

Iterative deepening search

� Number of nodes generated in a depth-limited search to depth d with branching factor b:

NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd

� Number of nodes generated in an iterative deepening search to depth d with branching factor b: search to depth d with branching factor b:

NIDS = (d+1)b0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd

� For b = 10, d = 5,� NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111� NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456

� Overhead = (123,456 - 111,111)/111,111 = 11%

Properties of iterative deepening search

� Complete? Yes

� Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd)

� Space? O(bd)

�Optimal? Yes, if step cost = 1�Optimal? Yes, if step cost = 1

Depth-limited search

= depth-first search with depth limit l,

i.e., nodes at depth l have no successors

� Recursive implementation:

Uniform-cost search

� Expand least-cost unexpanded node

� Implementation:� fringe = queue ordered by path cost

� Equivalent to breadth-first if step costs all equal

� Complete? Yes, if step cost ≥ ε� Complete? Yes, if step cost ≥ ε

� Time? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε)) where C* is the cost of the optimal solution

� Space? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε))

� Optimal? Yes – nodes expanded in increasing order of g(n)

Bi-directional SearchIntuition: Start searching from both the initial state and the goal state, meet in the middle.

Notes

• Not always possible to search

backwards

• How do we know when the

trees meet?

• Complete? Yes

• Optimal? Yes

• Time Complexity: O(bd/2), where d is the depth of the solution.

• Space Complexity: O(bd/2), where d is the depth of the solution.

trees meet?

• At least one search tree must

be retained in memory.

Summary of algorithms

Repeated states

�Failure to detect repeated states can turn a linear problem into an exponential one!

Graph search

Searching with partial information

� Sensorless problem (conformant)� Agent may have no idea where it is; solution is a sequence

� Contingency problem� Occurs for partially observable environment

� percepts provide new information about current state

often interleave search, execution� often interleave search, execution

� Exploration problem

� unknown states

� the agent must discover what state it is in and the actions to take

Example: vacuum world

� Single-state, start in #5. Solution?

Example: vacuum world

� Single-state, start in #5. Solution? [Right, Suck]

� Sensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Right goes to {2,4,6,8} Solution?

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, Suck, Left, Suck]

� Contingency

� Nondeterministic: Suck may dirty a clean carpet

� Partially observable: location, dirt at current location.

� Percept: [L, Clean], i.e., start in #5 or #7

Solution?

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, Suck, Left, Suck]

� Contingency

� Nondeterministic: Suck may dirty a clean carpet

� Partially observable: location, dirt at current location.

� Percept: [L, Clean], i.e., start in #5 or #7

Solution? [Right, if dirt then Suck]

Summary

� Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored

� Variety of uninformed search strategies

� Iterative deepening search uses only linear space and not much more time than other uninformed algorithms