0fe61lecture1

8/7/2019 0fe61lecture1

1/60

To build a system to solve a problem

Define the problem precisely. This definition mustinclude precise specification of what the initialsituation(s) will be as well as what final situation

constitute acceptable solution to the problem. Analyze the problem. A few very important featurescan have an immense impact on the appropriatenessof various possible techniques for solving the problem.

Isolate and represent the task knowledge that isnecessary to solve the problem

Choose the best problem-solving technique and applyit to the particular problem.


2/60

2

Water Jug Problem

3 liters 4 liters

x, y real numbers

x integers between 0 and 4

x, y between 0 and 4


3/60

3

Water Jugs cont.

Rules : x0 (pour d ltr water out of the 3 ltr jug)

( Dump 4 liters )

( Dump 3 liters )

| x+y >= 4

| x+y >= 3

| x+y


4/60

Defining the problem as a state space

search

Basic Search Problem

Given [S,s,O,G)

S- is the implicitly specify set of states s is the start state

O is the set of state transition operators

G is the set of goal states


5/60

Defining the problem as a state space

search

State Space - Define a state space that contains allthe possible configuration of the relevant objects.

Start Space - Specify one or more states within

that space describe possible situation from whichthe problem-solving process may start.

Goal State - Specify one or more states would beacceptable as solutions to the problem.

Operators Specify a set of rules that describethe actions available.


6/60

6

Examples of Problems

Toy Problems :

Water jug.

Missionaries and Cannibals.

8 Queens. 8 Puzzle. Real Problems :

Schedules. Traveling Salesman.

Robot navigation. Language Analysis (Parsers, Grammars). VLSI design


7/60

Production System

Since search forms the core of many

intelligent processes, it is useful to structure AI

programs in a way that facilitates describing

and performing the search process.

Production systems provide such structures


8/60

8

Production

1. DATA initial database.

2. Until DATA satisfies termination conditiondo:

3. begin:

Select some rule, R, in the set of rules that canbe applied to DATA.

DATA result of applying R to DATA.4. end.


9/60

A Production system

A set of rules, each consisting of a left side (a pattern) thatdetermines the applicability of the rule and a right side thatdescribes the operation to be performed.

One or more knowledge/database that contain whatever

information is appropriate for the particular task. Somepart of the database may be permanent, while other partsof it may pertain only to the solution of the currentproblem.

A control strategy that specifies the order in which the rules

will be compared to the database and a way of resolvingthe conflicts that arise when several rules match at once.

A rule applier.


10/60

10

Search Through State-Space

Initial state

Goal state


11/60

Control strategies

How to decide which rule to apply next duringthe process of searching for a solution to aproblem.

As it is going to affect that how quicklyproblem is solved or even whether problem isfinally solved.

Good control strategy The first requirement is that it cause motion

The second requirement is that it be systematic


12/60

Good Control strategy Cause Motion

Water jug problem. Suppose we implemented the simple

control strategy of starting each time at the top of the list

of rules and choosing the first applicable one. If we didthat we, we would never solve the problem. We would

continue indefinitely filling the 4- gallon jug with water.

Good strategy is that it be systematic.

The requirement that a control strategy be be systematiccorresponds to the need for global motion (over the

course of several steps) as well as local motion (over the

course of a single step)


13/60

copyright 2005 all rightsreserved

L. Manevitz Lecture 2 13

Control via Search Techniques

Uninformed :

Breadth First.

Depth First. Backtracking.

Informed Heuristic :

Hill Climbing. Best First, A*.


14/60

Breadth- First search

Create a variable called NODE-LIST and set it to theinitial state

Until a goal state is found or NODE-LIST is empty do

A) remove the first element from NODE_LIST and call it E,If NODE-LIST was empty, quit

B) For each way that each rule can match the statedescribed in E do

Apply the rule to generate a new state

If the new state is a goal state, quit and return this state Otherwise, as the new state to the end of NODE-LIST

Which data structure should it use for NODE-LIST?


15/60

15

Breadth First cont.


16/60

16

Breadth First

b = branching factor d=depth

1 + b + b + + b

Time Complexity O(b )

Space Complexity O(b )

2 d

d

d


17/60

Depth First Search

If the initial state is a goal state, quit and returnsuccess.

Otherwise, do the following until success or

failure is signaled Generate a successor, E, of the initial state. If there are

no more successors, signal failure.

Call Depth-First Search with as the initial state

If success is returned, signal success. Otherwisecontinue in this loop.

Which data structure should it use ?


18/60

18

Depth First cont.


19/60

Combinatorial Explosion

To solve a problem there can be too many

combination for a small set of possibilities,

Eg Travelling Sales Problem even for 10 cities!10 possibilities. 3628800.

Branch and bound strategies


20/60

For real world problems, it is often hard to measure

precisely the value of a particular solution.

Rarely do we actually need the optimum solution; a

good approximation usually serve very well.

In fact, there is some evidence that people, when

they solve problems, are not optimizers but rather

are satisfiers.

They seek solution that satisfies some set of

requirements and as soon as they find one they quit.

Eg. finding the parking space.


21/60

Heuristic Search

A heuristic is a technique that improves theefficiency of a search process, possibly by sacrificing

claims of completeness. A good general purpose heuristic that is useful for a

variety of combinatorial problem is the nearestneighbor heuristic.


22/60

Applying nearest neighbour heuristic

on travelling sales man problem

Applying it to Traveling salesman Probelm Arbitrarily select a starting city

To select the next city, look at all cities not yet visited and

select the one closest to the current city . Go to it. Repeat step 2 until all cities have been visited.

This procedure executes in time proportional to N2.a significant improvement over !N.


23/60

A heuristic function is a function that maps from problem

state descriptions to measures of desirability, usually

represented as numbers.

Which aspects of the problem state are considered, how those aspects are evaluated

and the weights given to individual aspects are chosen in such

a way that the value of the heuristic function at a given node

in the search process gives as good estimate as possible of

whether that node is on the desired path to a solution.

The purpose of a heuristic function is to guide the search

process in the most profitable direction by suggesting which

path to follow first when more than one is available.


24/60

Problem Characteristics


25/60

Is the problem decomposable into a set

independent smaller or easier subprogram.

At each step, it checks to see whether the problem itis working on is immediately solvable. If so then the

answer is returned directly.

Otherwise, the integrator checks to see whether it

can be decomposed into smaller problem.


26/60

Can solution step be ignored or undone?

Suppose we are trying to prove a mathematical

theorem, if we realize that the lemma is no help at

all, everything we need to know is still true, we can

just proceed as we should have in the first place. In 8 puzzle problem after sliding a number, can we

slide another number in place of first without

backtrack?

In a chess game, can a person after taking a move

backtrack to original move?


27/60

Problems are

Ignorable

Recoverable

irrecoverable


28/60

Is the universe predictable

Certain-outcome problem (8 puzzle)-

Every time we make a move, we know exactly what will

happen.

We can use planning to avoid having undo actualmoves, although it will be necessary to backtrack past

those moves one at a time during planning process.

Uncertain outcome problem (chess, bridge)


29/60

Is good solution Absolute or relative?

Consider the problem of answer questions based ona database of simple facts Marcus was a man.

Marcus was pompeian.

Marcus was born in 40 AD. All men are mortal.

All pompeians died when the volcano erupted in 79 AD

No mortal lives longer than 150 years.

It is now 2010 AD. Suppose the question is is marcus alive?

Consider the travelling sales man problem. There canbe any path problem or best path problem.


30/60

Is the solution a state or path?

Finding a consistent interpretation for the sentence

The bank president ate a dish of pasta salad with the

fork.

There are several components of this sentence,

which in isolation may have more than one

interpretation.

But the component must form a coherent whole andso they constrain each others interpretation.


31/60

Is the solution a state or path?

The word bank may refer either to a financial institution orto a side of a river.

The word dish is the object of the verb eat. It is possible thata dish was eaten. But more likely that the pasta salad in the

dish was eaten. Pasta salad is a salad containing pasta. But there are other

ways meaning can be formed from pairs of nouns.

The phrase with the fork could modify several parts of thesentence. In this case it modifies the verb eat. But if thephrase had been with vegetables then the modificationwould be different.

What in case of water jug problem.


32/60

What is the role of knowledge?

Does the chess require knowledge? If require

than how much?

Consider the problem of scanning daily newspaper to decide the reason for inflation and

how it can be tackled?


33/60

Does the task require interaction with

a person?

Consider the problem of proving mathematicaltheorems.

Solitary Problems the computer is given a problem

description and produces and answer with noimmediate communication and with no demand foran explanation?

Conversational problem there is intermediate

communication between a person and computer,either to provide additional assistance to thecomputer or to provide additional information theuser or both.


34/60

Heuristic Search Techniques

Generate & Test

Hill Climbing

Best First Search Problem Reduction

Constraint Satisfaction


35/60

Generate-and-Test

Algorithm

Generate a possible solution. For some problem, this means

generating a particular point in the problem state space.

Test to see if this is actually a solution by computing the chosen point

or the endpoint of chosen path to the set of acceptable goal states.

If a solution has been found quit. Otherwise return to step 1.

It is an exhaustive search of the problem space.

It is also known as British Museum Algorithm.


36/60

Hill Climbing

Simple Hill Climbing

Evaluate the initial state, If it is also a goal state, then return it and

quit. Otherwise continue with the initial state as the current state.

Loop until a solution is found or until there are no new operator left to

be applied in the current state Select an operator that has not yet been applied to the current state and

apply it to produce a new state.

Evaluate the new state

If it is a goal state, then return it and quit

If it is a goal state but it is better than the current state, then make it thecurrent state.

If it is not better than the current state, then continue in the loop.


37/60

The key difference between this algorithm and

the Generate-and-Test is the use of an

evaluation function as a way to inject task

specific knowledge into control process.


38/60

Hill Climbing

Steepest-Ascent Hill Climbing

Evaluate the initial state. If it is also a goal state, then return it and

quit. Otherwise, continue with the initial state as the current state.

Loop until a solution is found or until a complete iteration produces no

change to current state. Let succ be a state such that any possible successor of the current state

will be better than succ.

For each operator that applied to the current state do:

Apply the operator and generate a new state.

Evaluate the new state. If it is a goal state, then return it and quit. If notcompare it to succ. If it is better, then set succ to this state. If it is not better,

leave succ alone.

If succ is better than current state, set current state to succ.


39/60

To apply steepest-ascent hill climbing, we mustconsider all perturbation of the initial state andchoose the best.

Both basic and steepest-ascent hill climbing may failto find a solution.

Either algorithm may terminate not by finding a goalstate but by getting to a state from which no better

states can be generated. This will happen if the program has reached either alocal maximum, a plateau or a ridge.


40/60

A Local maximum is a state that is better than

all its neighbors but is not better than some

other states farther away.

A plateau is a flat area of the search space in

which a whole set of neighboring state have

the same value. On a plateau, it is not possible

to determine best direction in which to moveby making local comparisons.


41/60

A ridge is a special kind of local maximum. It is an area of the search spacethat is higher then surrounding area and that itself has a slope.

Each point on a ridge looks to the algorithm like a local maximum, eventhough the point is part of a curve leading to a better optimum

Ways of dealing with these problems

Backtrack Backtrack to some earlier node and try going in a different direction.

To implement this, maintain a list of paths almost taken and go back to one of themif the path that was taken leads to a dead end. Good for local maximum

Make a big jump in some direction to try to get to new section of the searchspace. Good for plateau.

Apply two or more rules before doing the test. This corresponds to moving in

several direction at once. Good for dealing with ridges


42/60

Best First Search

Or Graphs At each step best first search process (BFS), we select the

most promising of the nodes we have generated so far.

This is done by applying an appropriate heuristic function

each of them. expand the chosen node by using the rules to generate its

successors.

If one of them is a solution, we can quit.

If not, all those new nodes are added to the set of nodes

generated so far. Again the most promising node is selected and the process

continues.


43/60

To implement such a graph-search procedure, will

need to use two list of node

OPEN Nodes that have been generated and have had the

heuristic function applied to them but which have not yetbeen examined. OPEN is actually a priority queue in which

the elements with the highest priority are those with the

most promising value of the heuristic function.

CLOSED nodes that have been examined. We need to

keep these nodes in memory if want to search graph

rather than a tree, since whenever a new node is

generated, we need to check whether it has been

generated before.


44/60

Algorithm

Start with open containing the initial state.

Until a goal is found or there are no nodes left on open do:

Pick the best node on open Generate its successor

For each successor do

If it has not been generated before, evaluate it, add it to open and

record its parent.

If it has been generated before, change the parent if this new path isbetter than the previous one. In that case, update the cost of getting

to this node and to any successors that this node may already have.


45/60

A

B (3) C (5) D (1)

A

B (3) C (5) D

E (4) F (6)

A

B C D

E FG (6)H(5)

A

B C D

E FG H

I(2) J(1)

A


46/60

g a measure of the cost of getting from initial state to the

current node. G is not an estimate of anything but it is exact

sum of the costs of applying each of the rules that were

applied along the best path to the node.

The function h is an estimate of the additional cost of getting

from the current node to a goal state. This is the place where

knowledge of problem domain exploited

f, represent an estimate of the cost of getting from the initial

state to a goal state along the path that generated the currentnode.


47/60

A* Algorithm

Start with OPEN containing only the initial node. Set that

nodes g value to 0, its h value to whatever it is, and its f

value to h+ g. Set Closed to the empty list.

Until a goal node is found repeat

If there are no nodes on open, report failure

Otherwise, pick the node on OPEN with the lowest f value. Call it

BESTNODE. Remove it from OPEN. Place it on CLOSED.

Check if best node is goal state, if so then report and exit.

Otherwise generate the successor of BESTNODE (but do not setBESTNODE to point to them yet).


48/60

For Each successor

Set successor to point back to BESTNODE.

Computer g(Successor) = g(BESTNODE) + the cost of getting from

BESTNODE to successor.

Check if successor is the as any node on open (i.e. already generatedbut not processed) then call that node OLD.

Check whether it is cheaper to get to OLD via its current parent or to

SUCCESSOR via BESTNODE by comparing their g values.

If OLD is cheaper, then we need do nothing. If SUCCESSOR is cheaper,

then reset OLDs parent link to point to BESTNODE, record the newcheaper path in g(OLD), and update f(OLD).


49/60

If SUCCESSOR was not on OPEN, See If It is on

CLOSED. If so, call the node on CLOSED as OLD and

add OLD to the list of BESTNODE succesor.

Check to se if the new path or the old path is betteras previous and set the parent link and g & f values

appropriately.

If we have just found a better path to OLD, we must

propagate the improvement to OLDs Successors.


50/60

If successor was not already on either OPEN or

CLOSED, then put it on open and add it to the

list of BESTNODEs successors.

Compute

f(successor)=g(successor)+h(successor)


51/60

Function g lets us choose which node to expand next on thebasis not only of how good the node itself look, but also onthe basis of how good the path to the node was.

If we case about the path consideration of g is important.

If we only care about getting to a solution somehow, we candefine g always to be 0.

If we want to find a path involving the fewest number of steps, then we set the cost of going from a node to itssuccessor as a constant, usually 1.

If we want to find the cheapest path and some operator costmore than others, then we set the cost of going from a nodeto another to reflect those costs, then we set the cost of goingfrom one node to another to relect those costs.


52/60

If h is a perfect estimator of h, then A* will converge

immediately to the goal with no search.

If value of h is always 0, then search will be

controlled by g. If value of g is 0 then search is random

If the value of g is always 1, the search will Breadth first

search.

If we can maintain that h never overestimates h,then A* algorithm is guaranteed to find that h an

optimal path to a goal, if one exists.


53/60

This implementation is for graph.

If can be simplified to apply to trees by not

bothering to check whether a new node is

already on OPEN or CLOSEd. This makes it

faster to generate node.


54/60

Problem Reduction

AND-OR Graphs

It is useful for representing the solution of problems that can be solved by decomposing

them into a set of smaller problem, all of whichmust then be solved.

This decomposition, or reduction, generates arcthat we call AND arcs.

One AND arc may point to any number of successor nodes, all of which must be solved inorder for the arc to point to a solution.


55/60

Goal : Acquire TV Set

Goal: Steal TV Set Goal : Earn Some Money Goal : Buy TV Set


56/60


57/60

Change the f estimate of the newly expanded nodeto reflect the new information provided by itssuccessors.

Propagate this change backward through the graph.

If any node contains a successor arc whosedescendants are all solved, label the node itself asSOLVED.

At each node that is visited while going up the graph,

decide which of its successor arcs is the mostpromising and mark it as part of the current bestpath. This may cause the current best path tochange.


58/60


59/60

59

Points to Consider when Representing

Problems

Decomposable ?

Can partial steps be ignored or undone ?

Predictable ?

A

A

B

CB

C

Theorem proving vs. chess


60/60

Points to Consider when Representing

Problems cont.

Is good solution easily recognizable ?

Is Knowledge Base consistent ?

How much Knowledge is needed ?

Traveling Salesman

Big Data Base and limitations of logic

Chess, Newspaper-Understanding

Date post:	09-Apr-2018
Category:	Documents
Upload:	piyush-datir
View:	215 times
Download:	0 times

0fe61lecture1

Documents