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```									Problem Solving
Topics

• Problem Solving
• Searching Methods
• Game Playing
Introduction
• Problem solving is mostly based on searching.

• Every search process can be viewed as a traversal of a
directed graph in which each node represents a problem
state and each arc represents a relationship between the
states represented by the nodes it connects.

• The search process must find a path through the graph,
starting at an initial state and ending in one or more final
states. The graph is constructed from the rules that define
the allowable moves in the search space. Most search
programs represent the graph implicitly in the rules to
avoid combinatorial explosion and generate explicitly only
those parts that they decide to explore.
Introduction

• Goal : a description of a desired solution (may be a state
(8-puzzle) or a path (traveling salesman)).
• Search space: set of possible steps leading from initial
conditions to a goal.
• State: a snapshot of the problem at one stage of the
solution. The idea is to find a sequence of operators that
can be applied to a starting state until a goal state is
reached.
• A state space: the directed graph whose nodes are states
and whose arcs are the operators that lead from one state to
another.
• Problem solving is carried out by searching through the
space of possible solutions for ones that satisfy a goal.
Example: Water jug problem

• Given two jugs one 4 gallons and the other 3 gallons. The
goal is to get 2 gallons in 4 gallon jug.

Assumptions:
• You can fill the jug from the pump
• You can pour water out of the jug onto the ground
• You can pour water from one jug to another
Example: Water jug problem

State space representation
Search

Five important issues that arise in search
techniques are:
• The direction of the search
• The topology of the search process
• Representation of the nodes
• Selecting applicable rules
• Using heuristic function to guide the search
The Direction of the Search
• Forward : Data directed search. Start search from the
initial state.

To reason forward, the left sides (the preconditions) are
matched against the current state and the right sides (the
results) are used to generate new nodes until the goal is
reached

• Backward: Goal directed search. Start search from the
goal state.

To reason backward the right sides are matched against the
current node and the left sides are used to generate new
nodes representing new goal states to be achieved.
The Direction of the Search
Factors influencing the choice between forward vs. backward
chaining are:
• relative number of goal states to start states – move from
the smaller set of states to the larger

• branching factor – move in the direction with the lower
branching factor

• explanation of reasoning – proceed in the direction that
corresponds more closely with the way the user thinks.
The Direction of the Search
Examples
Branching factor
In theorem proving goal state is the theorem to be proved and the
initial state is the set of axioms.
• From small set of axioms large number of theorems can be
proved. This large set of theorems must go back to the small set
of axioms. Branching factor is greater going forward from
axioms to theorems. Backward reasoning is more appropriate
• If the branching factor is same in both directions then relative
number of start states to goal states determine the direction of
search.
• Bi-directional search, start from both ends and meet somewhere
in between. The disadvantage of this technique is search may
bypass each other.
Explanation of reasoning
MYCIN program that diagnoses infectious diseases uses
backward reasoning to determine the cause of patient's
illness.
A doctor may reason as follows:
If an organism has a set of properties (lab results) then it is
likely that the organism is X.
Even though the evidence is most likely documented in the
reverse direction
(IF (ORGANISM X) (PROPERTIES Y)) CF
• The rules justify why certain tests should be performed.
The Topology of the Search
• Trees
The Topology of the Search

• Graphs
The topology of the search

1. Check if the generated node already exists
2. If not, add the node
3. If exists, then do:
a) Set the node that is being expanded to point to the already existing
node corresponding to its successor, rather than to the new one.
The new one can be thrown away.
b) If looking for the best path, check if the new path is better. If
worse do nothing. If better record the new path as the correct path
to use to get to the node, and propagate the corresponding change
in cost down through successor nodes as necessary.
Disadvantage of this topology is that cycles may occur and
there is no guarantee for termination
Representation of the nodes

• Arrays
• Ordered pairs
• Predicates
Representation of the nodes

State: location of 8 number tiles
Operators: blank moves left, right, left or down
Goal test: state matches the configuration on the right
Path cost: each step cost 1, i.e. path length for search tree depth.
Representation of the nodes

Possible state representations in LISP (0 is the blank)
•(0 2 3 1 8 4 7 6 5 )
•((0 2 3) (1 8 4) (7 6 5))
•((0 1 7) (2 8 6) (3 4 5))

The representation depends on: how easy to compare,operate on,
and store (size).
Goal Test

>(defvar *goal-state* ‘(1 2 3 8 0 4 7 6 5 ))

>(equal *goal-state* ‘(1 2 3 8 0 4 7 6 5 ))
t
Operators

Functions from state to subset of states
• drive to neighboring city
• place piece on chess board
• add person to meeting schedule
• slide a tile in 8-puzzle

Matching Conflict resolution:
• order (priority)
• recency
• Indexing
Using heuristic function to guide the
search
• It is frequently possible to find rules which will increase
the chance of success. Such rules are termed heuristics and
a search involving them is termed a heuristic search.
• A heuristic function is a function that maps from problem
state description to measure of desirability

Heuristics for the 8-puzzle problem could
be:
• the number of displaced tiles
• distance of displaced tiles
Implementing heuristic evaluation
example: 8-puzzle  functions
2   8   3                                 fails to
1   6   4           5         6      0
distinguish
7   5

2   8   3
2    8   3                                                        1   2   3
3           4     0
1       4
1    6   4                                                        8       4

7        5              7   6   5
7   6   5

Start
2   8   3                                 Goal
5          6     0
1   6   4

7   5
more accurate
(1)        (2)   (3)
(1) tiles out of place
(2) sum of distance out of place
(3) 2*number of direct tile reversals
Evaluation of Search Strategies

• Time complexity: how many nodes expanded so
far?
• Space complexity: how many nodes must be
stored in node-list at any given time?
• Completeness: if solution exists, guaranteed to be
found?
• Optimality: guaranteed to find the best solution?
Components of Implicit State-Space
Graphs
There are three basic components to an implicit representation
of a state-space graph.
1. A description with which to label the start node. This
description is some data structure modeling the initial
state of the environment.
2. Functions that transform a state description representing
one state of the environment into one that represents the
state resulting after an action.
These functions are usually called operators. When an
operator is applied to a node, it generates one of that
node’s successor’s.
3. A goal condition, which can be either a True-False valued
function on state descriptions or a list of actual instances
of state descriptions that correspond to goal states.
Types of Search

There are three broad classes of search processes:
1) Uninformed- Blind Search-
– There is no specific reason to prefer one part of the
search space to any other, in finding a path from initial
state to goal state.
– systematic, exhaustive search

• depth-first-search
Types of Search

2) Informed – Heuristic search - there is specific information
to focus the search.
–   Hill climbing
–   Branch and bound
–   Best first
–   A*
3) Game playing – there are at least two partners opposing to
each other.
– Minimax (a, b pruning)
– Means ends analysis
Search Algorithms

–   find solution path thro’ problem space
–   keep track of paths from start to goal nodes
–   define optimal path if > 1 solution (circumstances)
–   avoid loops (prevent reaching goal)
Depth-first search
Uses generate and test strategy. Nodes are generated by
applying the applicable rules. Then, each generated node
is tested if it is the goal. Nodes are generated in a
systematic form. It is an exhaustive search of the problem
space.
1. Form a one element queue consisting of the root node
2. Until the queue is empty or the goal has been reached,
determine if the first element in the queue is the goal
node.
a) If the first element is the goal do nothing.
b) If the first element is not the goal node remove the first element
from the queue and add the first element's children if any to the
front of the queue.
3. If the goal node has been found announce success,
otherwise announce failure.
Depth-first search
- lists: keep track of progress through state space
- open states generated but children not examined
begin
open := [Start]
/initialise
close := [];
while open <> [] do
begin
remove leftmost state from open, call it X;
if X is a goal then return (success)
else begin
generate children of X;
put X on close;
eliminate children of X on open or close;
/loop check
put remaining children on the left end of open
/queue
end
end;
return (failure)
/no states left
end.
Depth-first search

Node visit order: 1 2 4 8 9 5 10 11 3 6 12 13 7 14 15
Queuing function: enqueue at left
Depth-first search
Evolution of the closed
and open lists
1. [1] – []
2. [2 3] – [1]
3. [4 5 3] – [1 2 ]
4. [8 9 5 3 ] – [1 2 4]
……………….
Depth-first Evaluation

Branching factor b, depth of solutions d, max depth m:
• Incomplete: may wonder down the wrong path. Bad
for deep and infinite depth state space
• Time: bm nodes expanded (worst case)
• Space: bm (just along the current path)
• Does not guarantee the shortest path. Good when
there are many shallow goals.

• It will first explore all paths of length one, then
two and if a solution exists it will find it at the
exploration of the paths of length N. There is a
guarantee of finding a solution if one exists.

• It will find the shortest path from the solution, it
may not be the best one.

The algorithm:
1. Form one element queue consisting of the root node
2. Until the queue is empty or the goal has been reached,
determine if the first element in the queue is the goal node.
a) If the first element is the goal do nothing.
b) If the first element is not the goal node remove the first
element from the queue and add the first element's children
if any to the back of the queue.
3. If the goal node has been found announce success,
otherwise announce failure.
begin
open := [Start]
/initialise
close := [];
while open <> [] do
begin
remove leftmost state from open, call it X;
if X is a goal then return (success)
else begin
generate children of X;
put X on close;
eliminate children of X on open or close;
/loop check
put remaining children on the right end of open
/queue
end
end;
return (failure)
/no states left
end.

• Node visit order (goal test): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
• Queuing function: enqueue at end ( add expanded node at the end of
the list)

• Evolution of the open and
closed lists
1. [1] – [ ]
2. [2 3 ] – [ 1 ]
3. [3 4 5 ] – [1 2 ]
4. [4 5 6 7 ] – [1 2 3 ]
…………..
First Search
The lisp implementation of breadth first search maintains the
open list as a first in first out (FIFO) structure.

(cond ((null *open*) nil)
(t (let ((state (car *open*)))
(cond ((equal state *goal*) ‘success)
(t (setq *closed* (cons state *closed*))
(setq *open* (append (cdr *open*) (generate-
descendants state *moves*))) ;*moves*:list of the funcs that generate
the moves.
First Search
(setq *open* (list start))
(setq *closed* nil)
(setq *qoal* goal)
First Search
generate-descendants takes a state and returns a list of its children.
(defun generate-descendants (state moves)
(cond ((null moves) nil)
(t (let (child (funcall (car moves) state))
(rest (generate-descendants state (cdr *moves*))))
(cond ((null child) rest)
((member child rest :test #‘equal) rest)
((member child *open* :test #‘equal) rest)
((member child *closed* :test #‘equal) rest)
(t (cons child rest)))))))

By binding the global variable *move* to an appropriate list of move functions
this search algorithm may be used to search any state space in breadth first
search fashion.

•   Branching factor b, depth of solution d:
•   Complete: it will find the solution if it exists
•   Time. 1 + b + b2 + …+ bd
•   Space: bk where k is the current depth
•   Space is more problem than time in most cases
•   Time is also a major problem nonetheless
Heuristic Search
Reasons for heuristics
- impossible for exact solution,
- no exact solution but an acceptable one
- fallible due to limited information

Intelligence for a system with limited processing resources
consists in making wise choices of what to do next

Heuristics = Search Algorithm + Measure
Hill climbing
Hill climbing is depth first search with a heuristic
measurement that orders choices as nodes are
expanded.The algorithm is the same only 2b differs
slightly.
1. Form a one element queue consisting of the root node
2. Until the queue is empty or the goal has been reached,
determine if the first element in the queue is the goal node.
a) If the first element is the goal do nothing.
b) If the first element is not the goal node remove the first
element from the queue, sort the first elements children, if
any by estimated remaining distance, and add the first
element's children if any to the front of the queue.
3. If the goal node has been found announce success,
otherwise announce failure.
Hill climbing
Hill climbing
Problems that may arise:
• A local maximum, is a state that is better than all its
neighbors, but is not better than some other states farther
away. At a local maximum, all moves appear to make
things worse.

• A plateau, A whole set of neighboring states have the
same value.It is not possible to determine the best
direction.

• A ridge,. Higher than surrounding area but can not be
traversed by single move in any one direction.
Hill climbing
Some ways of dealing with these:
• Backtrack: local maximum
• Make a big jump in one direction to try to get to
new section of search space (plateau)
• Apply two or more rules before doing the test.
This corresponds to moving in several directions
at once (ridges).
Best-first Search
Best-first search is a combination of depth-first and breadth-
first search algorithms. Forward motion is from the best
open node (most promising) so far, no matter where it is on
the partially developed tree.

The second step of the algorithm changes as:
2. Until the queue is empty or the goal has been reached,
determine if the first element in the queue is the goal node.
a) If the first element is the goal do nothing.
b) If the first element is not the goal node remove the first
element from the queue and add the first element's
children if any to the queue and sort the entire queue by
estimated remaining distance..
procedure best_first_search;         Best-First Search
begin
open := [Start]; closed = [];
while open <> [] do
remove leftmost state from open, call it X;
if X = goal then return path from Start to X
else begin
generate children of X; for each child of X do
case
the child is not on open or closed:
begin
assign child heuristic value;
end;
the child is already on open:
if the child reached by shorter path
then give state on open shorter path
the child is already on closed:
if child reached shorter path then
begin
remove state from closed;
end;
end case;
put X on closed;
re-order states on open by heuristic method
end;
return failure
end.
Example of best-first search
A-5
1. open = [A5]; closed = []
2. eval A5; open = [B4, C4, D6];
B-4       C-4      D-6
closed = [A5]
3. eval B4; open = [C4, E5, F5, D6];
E-5 F-5 G-4    H-3 I         J
closed = [B4, A5]
4. eval C4; open = [H3, G4, E5, F5, D6]
K   L    M    N O-2 P-3     Q        R     closed = [C4, B4, A5]
5. eval H3; open = [O2, P3, G4, E5, F5,
D6]; closed = [H3, C4, B4, A5]
S   T
6. eval O2; open = [P3, G4, E5, F5, D6]
closed = [O2, H3, C4, B4, A5]
7. eval P3; the solution is found!
Branch and Bound Search

• Shortest path is always chosen for expansion. The
path first reaching the destination is optimal

• In order to be certain that supposed solution is not
longer than one or more incomplete paths, instead
of terminating when a path is found, terminate
when the shortest incomplete path is longer than
the shortest complete path.
Branch and Bound Search
To conduct a branch and bound search:
1. Form a queue of partial paths. Let the initial queue consist of the zero
length, zero step path from the root node to no where.
2. Until the queue is empty or the goal has been reached, determine if the
first path in the queue reaches the goal node.
a) If the first path reaches the goal do nothing.
b) If the first path does not reach the goal node,
i) remove the first path from the queue
ii) form new paths from the removed path by extending one step
iii) add the new paths to the queue
iv) sort the queue by cost accumulated so far, with least cost paths in
front
3. If the goal node has been found announce success, otherwise announce
failure.
Branch and Bound Search
Branch and Bound Search
Branch and Bound Search

c(total length) = d(already traveled) + e(distance remaining)
• If the guesses are not perfect, and a bad overestimate
somewhere along the true optimal path may cause us to
wonder off that optimal path permanently.
• But underestimates cannot cause the right path to be
overlooked. An underestimate of the distance remaining
yields an underestimate of the total path, u(total path length).
u(total path length) = d(already traveled) + u(distance
remaining)
Branch and Bound Search

• If a total path is found by extending the path with the
smallest underestimate repeatedly, no further work need be
done once all incomplete path distance estimates are longer
than some complete path distance. This is true, because a
real distance along a completed path can not be shorter
than an underestimate of the distance.
• To conduct a branch and bound search with
underestimates:
• 2b4) sort the queue by the sum of cost accumulated so far
and a lower bound estimate of the cost remaining, with the
least cost paths in front.
A* Search
• Dynamic-programming principal holds that when looking
for the best path from S to G, all paths from S to any
intermediate node, I, other than the minimum length path
from S to I, can be ignored.
• The A* procedure is branch and bound search in a graph
space with an estimate of remaining distance, combined
with dynamic programming principle.
• If one can show that h(n) never overestimates the cost to
reach the goal, then it can be shown that the A* algorithm
is both complete and optimal.
A* Search
To do A* search with lower bound estimates:

2b4) sort the queue by the sum of the cost
accumulated so far and a lower bound
estimate of the cost remaining, with least
cost paths in front.
2b5) If two or more paths reach a common
node, delete all those paths except for one
that reaches the common node with the
minimum cost.
Recursive Search in Prolog
3X3 knight’s tour problem

move(1, 6)    move(3, 4)   move(6, 7)
move(1, 8)    move(3, 8)   move(6, 1)
move(2,7)     move(4, 3)   move(7, 6)
move(2, 9)    move(4, 9)   move(7, 2)
move(8,3)     move(9,4)
move(8,1)     move(9,2)
Recursive Search in Prolog
predicates
path(integer, integer, integer*)
clauses
path(Z, Z, L).
path(X, Y, L):- move(X, Z), not(member(Z, L),
path(Z, Y, [Z|L])
/*x is the member of the list if X is the head of the list or x is a member of
the tail */
member(X, [X|T]).
member(X, [Y|T]):- member(X, T).
goal
path(1, 3, [1]).
Farmer-Wolf-Cabbage Problem

• A Farmer with his wolf, goat and cabbage come to the
edge of a river they wish to cross. There is a boat at the
river’s edge, but of course only the farmer can raw it. The
boat also can carry only two things. If the wolf is ever left
alone with the goat, the wolf will eat the goat. If the goat is
ever left alone with the cabbage, the goat will eat the
cabbage. Devise a sequence of crossings of the river so
that all four characters arrive safely on the other side of the
river.
• The problem implementation.
Search Algorithms in LISP
Example: Farmer, wolf, goat and cabbage problem.

uses depth first search

states are represented as list of four elements. eg: (w e w e)
represents the farmer and the goat on the west bank, and
wolf and the cabbage on the east bank.

make-state takes as arguments the locations of the farmer,
wolf, goat and cabbage and returns a state and four access
functions, farmer-side, wolf-side, goat-side, and cabbage-
side, which take a state and return the location of an
individual.
(defun make-state (f w g c)(list f w g c))

(defun farmer-side (state)
(nth 0 state))

(defun wolf-side (state)
(nth 1 state))

(defun goat-side (state)
(nth 2 state))

(defun cabbage-side (state)
(nth 3 state))
(defun farmer-takes-self(state)
(make-state(opposite (farmer-side state))
(wolf-side state)
(goat-side state)
(cabbage-side state)))

In the above procedure a new state is returned regardless of its
being safe or not.
A safe function should be defined so that it returns nil if a state is
not safe.
>(safe ‘(w w w w))          ;safe state, return unchanged
>(safe ‘(e w w e))           ; wolf eats goat, return nil.

(defun safe(sate)
(cond((and (equal(goat-side state) (wolf-side state))
(not(equal(farmer-side state) (wolf-side state)))
nil)     ; wolf eats goat
((and(equal(goat-side state) (cabbage-side state))
(not(equal(farmer-side state) (goat-side state)))
nil)     ; goat eats cabba
(t state)))
;return nil for unsafe states
;filter the unsafe states
(defun farmer-takes-self(state)
(safe (make-state(opposite (farmer-side state))
(wolf-side state)
(goat-side state)
(cabbage-side state))))

(defun opposite( side)
(cond ((equal side ‘e) ‘w)
((equal side ‘w) ‘e)))
(defun farmer-takes-wolf(state)
(cond((equal (farmer-side state) (wolf-side state))
(safe (make-state (opposite (farmer-side state))
(oppsite (wolf-side state))
(goat-side state)
(cabbage-side state))))
(t nil)))
(defun farmer-takes-goat(state)
(cond((equal (farmer-side state) (goat-side state))       ;
farmer and on the same side
(safe (make-state (opposite (farmer-side state))
(wolf-side state)
(oppsite (goat-side state))
(cabbage-side state))))
(t nil)))
(defun farmer-takes-cabbage(state)
(cond((equal (farmer-side state) (cabbage-side state))
(safe (make-state (opposite (farmer-side state))
(wolf-side state)
(goat-side state)
(oppsite (cabbage-side state)))))
(t nil)))
(defun path(state goal)
(cond((equal state goal) ‘success)
(t (or (path (farmer-takes-self state) goal)
(path (farmer-takes-wolf state) goal)
(path (farmer-takes-goat state) goal)
(path (farmer-takes-cabbage state) goal)))))

To prevent path from attempting to generate the children of a nil state, it
must first check whether the created state is nil. If it is, the path should
return nil.

In this definition there is the probability of going into a loop, repeating the
same states over and over again. Third parameter, been-list, which keeps
track of these visited states, is passed to path. member predicate is used to
make sure that the current state is not the member of the been-list.
(defun path(state goal been-list)
(cond ((null state) nil)
((equal state goal)(reverse (cons state been-list)))
((not (member state been-list :test ‘equal))
(or (path (farmer-takes-self state) goal (cons state been-list))
(path (farmer-takes-wolf state) goal (cons state been-list))
(path (farmer-takes-goat state) goal (cons state been-list))
(path (farmer-takes-cabbage state) goal (cons state been-
list))))))
*moves* is a list of functions that generate the moves.
In the farmer, goat and cabbage problem *moves* would be
defined by
(setq *moves* ‘(farmer-takes-self farmer-takes-wolf farmer-takes-goat
farmer-takes-cabbage))

(setq *open* (list start))
(setq *closed* nil)
(setq *qoal* goal)
generate-descendants takes a state and returns a list of its children.
It also disallows duplicates in the list of children and eliminates any
children that are already in the open or closed list.

(defun generate-descendants (state moves)
(cond ((null moves) nil)
(t (let (child (funcall (car moves) state))
(rest (generate-descendat state (cdr *moves*))))
(cond ((null child) rest)
((member child rest :test ‘equal) rest)
((member child *open* :test ‘equal) rest)
((member child *closed* :test ‘equal) rest)
(t (cons child rest)))))))
By binding the global variable *move* to an appropriate list of move functions
this search algorithm may be used to search any state space in breadth first
search fashion.
The lisp implementation of breadth first search maintains the
open list as a first in first out (FIFO) structure. Open, closed and
goal are defined as global variables.

(cond ((null *open*) nil)
(t (let ((state (car *open*)))
(cond ((equal state *goal*) ‘success)
(t (setq *closed* (cons state *closed*))
(setq *open* (append (cdr *open*)
(generate-descendants state *moves*)))