Chapter 02-artificialPROBLEM SOLVING.ppt

Chapter 2
Problems, Problem Spaces
and Search

Contents
• Defining the problem as a State Space Search
• Production Systems
• Control Strategies
• Breadth First Search
• Depth First Search
• Heuristic Search
• Problem Characteristics
• Is the Problem Decomposable?
• Can Solution Steps be ignored or undone?
• Production system characteristics
• Issues in the design of search programs

To build a system to solve a
problem
1. Define the problem precisely
2. Analyze the problem
3. Isolate and represent the task
knowledge that is necessary to solve
the problem
4. Choose the best problem-solving
techniques and apply it to the
particular problem.

Defining the problem as State
Space Search
• The state space representation forms the basis of
most of the AI methods.
• Its structure corresponds to the structure of
problem solving in two important ways:
– It allows for a formal definition of a problem as the need to
convert some given situation into some desired situation
using a set of permissible operations.
– It permits us to define the process of solving a particular
problem as a combination of known techniques (each
represented as a rule defining a single step in the space) and
search, the general technique of exploring the space to try
to find some path from current state to a goal state.
– Search is a very important process in the solution of hard
problems for which no more direct techniques are available.

Example: Playing Chess
• To build a program that could “play chess”, we
could first have to specify the starting
position of the chess board, the rules that
define the legal moves, and the board positions
that represent a win for one side or the other.
• In addition, we must make explicit the
previously implicit goal of not only playing the
legal game of chess but also winning the game,
if possible,

Playing chess
• The starting position can be described as an 8by 8 array where each
position contains a symbol for appropriate piece.
• We can define as our goal the check mate position.
• The legal moves provide the way of getting from initial state to a goal
state.
• They can be described easily as a set of rules consisting of two parts:
– A left side that serves as a pattern to be matched against the current
board position.
– And a right side that describes the change to be made to reflect the move
• However, this approach leads to large number of rules 10120
board
positions !!
• Using so many rules poses problems such as:
– No person could ever supply a complete set of such rules.
– No program could easily handle all those rules. Just storing so many rules
poses serious difficulties.

Defining chess problem as State
Space search
• We need to write the rules describing the
legal moves in as general a way as possible.
• For example:
– White pawn at Square( file e, rank 2) AND Square(
File e, rank 3) is empty AND Square(file e, rank 4)
is empty, then move the pawn from Square( file e,
rank 2) to Square( file e, rank 4).
• In general, the more succintly we can
describe the rules we need, the less work we
will have to do to provide them and more
efficient the program.

Water Jug Problem
• The state space for this problem can be
described as the set of ordered pairs of
integers (x,y) such that x = 0, 1,2, 3 or 4 and y
= 0,1,2 or 3; x represents the number of gallons
of water in the 4-gallon jug and y represents
the quantity of water in 3-gallon jug
• The start state is (0,0)
• The goal state is (2,n)

Production rules for Water Jug
Problem
• The operators to be used to solve the
problem can be described as follows:
Sl No Current state Next State Descritpion
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

Production rules
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
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

To solve the water jug
problem
• Required a control structure
that loops through a simple cycle
in which some rule whose left
side matches the current state
is chosen, the appropriate
change to the state is made as
described in the corresponding
right side, and the resulting
state is checked to see if it
corresponds to goal state.
• One solution to the water jug
problem
• Shortest such sequence will
have a impact on the choice of
appropriate mechanism to guide
the search for solution.
Gallons in the
4-gallon jug
Gallons in the
3-gallon jug
Rule applied
0 0 2
0 3 9
3 0 2
3 3 7
4 2 5 or 12
0 2 9 0r 11
2 0

Formal Description of the
problem
1. Define a state space that contains all the
possible configurations of the relevant
objects.
2. Specify one or more states within that
space that describe possible situations from
which the problem solving process may start
( initial state)
3. Specify one or more states that would be
acceptable as solutions to the problem.
( goal states)
4. Specify a set of rules that describe the
actions ( operations) available.

Production Systems
A production system consists of:
• A set of rules, each consisting of a left side that determines the
applicability of the rule and a right side that describes the operation
to be performed if that rule is applied.
• One or more knowledge/databases that contain whatever information
is appropriate for the particular task. Some parts of the database
may be permanent, while other parts of it may pertain only to the
solution of the current problem.
• A control strategy that specifies the order in which the rules will be
compared to the database and a way of resolving the conflicts that
arise when several rules match at once.
• A rule applier

Production system
In order to solve a problem:
• We must first reduce it to one for which a precise
statement can be given. This can be done by defining
the problem’s state space ( start and goal states) and
a set of operators for moving that space.
• The problem can then be solved by searching for a
path through the space from an initial state to a goal
state.
• The process of solving the problem can usefully be
modeled as a production system.

Control Strategies
• How to decide which rule to apply next
during the process of searching for a
solution to a problem?
• The two requirements of good control
strategy are that
– it should cause motion.
– It should be systematic

Breadth First Search
• Algorithm:
1. Create a variable called NODE-LIST and set it
to initial state
2. 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 state
described in E do:
i. Apply the rule to generate a new state
ii. If the new state is a goal state, quit and return this state
iii. Otherwise, add the new state to the end of NODE-LIST

BFS Tree for Water Jug problem
(0,0)
(4,0) (0,3)
(4,3) (0,0) (1,3) (4,3) (0,0) (3,0)

Algorithm: Depth First Search
1. If the initial state is a goal state, quit and
return success
2. Otherwise, do the following until success or
failure is signaled:
a. Generate a successor, E, of initial state. If there
are no more successors, signal failure.
b. Call Depth-First Search, with E as the initial
state
c. If success is returned, signal success. Otherwise
continue in this loop.

Backtracking
• In this search, we pursue a singal branch of the tree until it
yields a solution or until a decision to terminate the path is made.
• It makes sense to terminate a path if it reaches dead-end,
produces a previous state. In such a state backtracking occurs
• Chronological Backtracking: Order in which steps are undone
depends only on the temporal sequence in which steps were
initially made.
• Specifically most recent step is always the first to be undone.
• This is also simple backtracking.

Advantages of Depth-First
Search
• DFS requires less memory since only the
nodes on the current path are stored.
• By chance, DFS may find a solution
without examining much of the search
space at all.

Advantages of BFS
• BFS will not get trapped exploring a
blind alley.
• If there is a solution, BFS is guaranteed
to find it.
• If there are multiple solutions, then a
minimal solution will be found.

TSP
• A simple motion causing and systematic control
structure could solve this problem.
• Simply explore all possible paths in the tree and
return the shortest path.
• If there are N cities, then number of different paths
among them is 1.2….(N-1) or (N-1)!
• The time to examine single path is proportional to N
• So the total time required to perform this search is
proportional to N!
• For 10 cities, 10! = 3,628,800
• This phenomenon is called Combinatorial explosion.

Branch and Bound
• Begin generating complete paths, keeping
track of the shortest path found so far.
• Give up exploring any path as soon as its
partial length becomes greater than the
shortest path found so far.
• Using this algorithm, we are guaranteed to
find the shortest path.
• It still requires exponential time.
• The time it saves depends on the order in
which paths are explored.

Heuristic Search
• A Heuristic is a technique that improves the efficiency of a
search process, possibly by sacrificing claims of completeness.
• Heuristics are like tour guides
• They are good to the extent that they point in generally
interesting directions;
• They are bad to the extent that they may miss points of
interest to particular individuals.
• On the average they improve the quality of the paths that are
explored.
• Using Heuristics, we can hope to get good ( though possibly
nonoptimal ) solutions to hard problems such asa TSP in non
exponential time.
• There are good general purpose heuristics that are useful in a
wide variety of problem domains.
• Special purpose heuristics exploit domain specific knowledge

Nearest Neighbor Heuristic
• It works by selecting locally superior alternative at
each step.
• Applying to TSP:
1. Arbitrarily select a starting city
2. To select the next city, look at all cities not yet visited
and select the one closest to the current city. Go to next
step.
3. Repeat step 2 until all cities have been visited.
– This procedure executes in time proportional to N2
– It
is possible to prove an upper bound on the error it
incurs. This provides reassurance that one is not
paying too high a price in accuracy for speed.

Heuristic Function
• This is a function that maps from problem state
descriptions to measures of desirsability, 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 an estimate as possible of
whether that node is on the desired path to a solution.
• Well designed heuristic functions can play an important
part in efficiently guiding a search process toward a
solution.

Example Simple Heuristic
functions
• Chess : The material advantage of our
side over opponent.
• TSP: the sum of distances so far
• Tic-Tac-Toe: 1 for each row in which we
could win and in we already have one
piece plus 2 for each such row in we
have two pieces

Problem Characteristics
• In order to choose the most appropriate method for a particular
problem, it is necessary to analyze the problem along several key
dimensions:
– Is the problem decomposable into a set of independent smaller or
easier sub problems?
– Can solution steps be ignored or at least undone if they prove
unwise?
– Is the problem’s universe predictable?
– Is a good solution to the problem obvious without comparison to all
other possible solutions?
– Is the desired solution a state of the world or a path to a state?
– Is a large amount of knowledge absolutely required to solve the
problem or is knowledge important only to constrain the search?
– Can a computer that is simply given the problem return the solution
or will the solution of the problem require interaction between the
computer and a person?

Is the problem Decomposable?
• Whether the problem can be
decomposed into smaller problems?
• Using the technique of problem
decomposition, we can often solve very
large problems easily.

Blocks World Problem
• Following operators
are available:
• CLEAR(x) [ block x
has nothing on it]->
ON(x, Table)
• CLEAR(x) and
CLEAR(y) -> ON(x,y)
[ put x on y]
C
A B
A
B
C
Start: ON(C,A)
Goal:
ON(B,C) and
ON(A,B)
ON(B,C)
ON(B,C) and ON(A,B)
ON(B,C)
ON(A,B)
CLEAR(A) ON(A,B)
CLEAR(A) ON(A,B)

Can Solution Steps be ignored or
undone?
• Suppose we are trying to prove a math theorem.We can prove a lemma.
If we find the lemma is not of any help, we can still continue.
• 8-puzzle problem
• Chess: A move cannot be taken back.
• Important classes of problems:
– Ignorable ( theorem proving)
– Recoverable ( 8-puzzle)
– Irrecoverable ( Chess)
• The recoverability of a problem plays an important role in determining
the complexity of the control structure necessary for the problem’s
solution.
– Ignorable problems can be solved using a simple control structure that never
backtracks
– Recoverable problems can be solved by a slightly more complicated control
strategy that does sometimes make mistakes
– Irrecoverable problems will need to be solved by systems that expends a
great deal of effort making each decision since decision must be final.

Is the universe Predictable?
• Certain Outcome ( ex: 8-puzzle)
• Uncertain Outcome ( ex: Bridge, controlling a
robot arm)
• For solving certain outcome problems, open
loop approach ( without feedback) will work
fine.
• For uncertain-outcome problems, planning can
at best generate a sequence of operators that
has a good probability of leading to a solution.
We need to allow for a process of plan revision
to take place.

Is a good solution absolute or
relative?
• Any path problem
• Best path problem
• Any path problems can often be solved in
a reasonable amount of time by using
heuristics that suggest good paths to
explore.
• Best path problems are computationally
harder.

Is the solution a state or a
path?
• Examples:
– Finding a consistent interpretation for the
sentence “The bank president ate a dish of pasta
salad with the fork”. We need to find the
interpretation but not the record of the
processing.
– Water jug : Here it is not sufficient to report that
we have solved , but the path that we found to the
state (2,0). Thus the a statement of a solution to
this problem must be a sequence of operations
( Plan) that produces the final state.

What is the role of
knowledge?
• Two examples:
– Chess: Knowledge is required to constrain the search for a solution
– Newspaper story understanding: Lot of knowledge is required even
to be able to recognize a solution.
• Consider a problem of scanning daily newspapers to decide which
are supporting the democrats and which are supporting the
republicans in some election. We need lots of knowledge to
answer such questions as:
– The names of the candidates in each party
– The facts that if the major thing you want to see done is have
taxes lowered, you are probably supporting the republicans
– The fact that if the major thing you want to see done is improved
education for minority students, you are probably supporting the
democrats.
– etc

Does the task require Interaction
with a person?
• The programs require intermediate
interaction with people for additional inputs
and to provided reassurance to the user.
• There are two types of programs:
– Solitary
– Conversational:
• Decision on using one of these approaches will
be important in the choice of problem solving
method.

Problem Classification
• There are several broad classes into
which the problems fall. These classes
can each be associated with generic
control strategy that is appropriate for
solving the problems:
– Classification : ex: medical diagnostics,
diagnosis of faults in mechanical devices
– Propose and Refine: ex: design and planning

Production System
Characteristics
1. Can production systems, like problems, be described by a set
of characteristics that shed some light on how they can easily
be implemented?
2. If so, what relationships are there between problem types and
the types of production systems best suited to solving the
problems?
• Classes of Production systems:
– Monotonic Production System: the application of a rule never
prevents the later application of another rule that could also have
been applied at the time the first rule was selected.
– Non-Monotonic Production system
– Partially commutative Production system: property that if
application of a particular sequence of rules transforms state x
to state y, then permutation of those rules allowable, also
transforms state x into state y.
– Commutative Production system

Monotonic Production Systems
• Production system in which the
application of a rule never prevents the
later application of another rule that
could also have been applied at the time
the first rule was applied.
• i.e., rules are independent.

Commutative Production system
• A partially Commutative production system
has a property that if the application of a
particular sequence of rules transform state
x into state y, then any permutation of those
rules that is allowable, also transforms state
x into state y.
• A Commutative production system is a
production system that is both monotonic and
partially commutative.

Partially Commutative, Monotonic
• These production systems are useful for solving
ignorable problems.
• Example: Theorem Proving
• They can be implemented without the ability to
backtrack to previous states when it is discovered
that an incorrect path has been followed.
• This often results in a considerable increase in
efficiency, particularly because since the database
will never have to be restored, It is not necessary to
keep track of where in the search process every
change was made.
• They are good for problems where things do not
change; new things get created.

Non Monotonic, Partially
Commutative
• Useful for problems in which changes occur
but can be reversed and in which order of
operations is not critical.
• Example: Robot Navigation, 8-puzzle, blocks
world
• Suppose the robot has the following ops: go
North (N), go East (E), go South (S), go West
(W). To reach its goal, it does not matter
whether the robot executes the N-N-E or N-
E-N.

Not partially Commutative
• Problems in which irreversible change occurs
• Example: chemical synthesis
• The ops can be :Add chemical x to the pot, Change the
temperature to t degrees.
• These ops may cause irreversible changes to the potion being
brewed.
• The order in which they are performed can be very important in
determining the final output.
• (X+y) +z is not the same as (z+y) +x
• Non partially commutative production systems are less likely to
produce the same node many times in search process.
• When dealing with ones that describe irreversible processes, it
is partially important to make correct decisions the first time,
although if the universe is predictable, planning can be used to
make that less important.

Four Categories of Production
System
Monotonic NonMonotonic
Partially
Commutative
Theorem
proving
Robot
Navigation
Not Partially
Commutative
Chemical
Synthesis
Bridge

Issues in the design of search
programs
• The direction in which to conduct the
search (forward versus backward
reasoning).
• How to select applicable rules
( Matching)
• How to represent each node of the
search process (knowledge
representation problem)

Summary
• Four steps for designing a program to
solve a problem:
1. Define the problem precisely
2. Analyse the problem
3. Identify and represent the knowledge
required by the task
4. Choose one or more techniques for
problem solving and apply those
techniques to the problem.

Chapter 02-artificialPROBLEM SOLVING.ppt

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