APPLICATION requirement. Here we discuss the application of

APPLICATION
OF FUZZY GRAPH THEORY IN FINITE STATE AUTOMATA

V. Mythili

We Will Write a Custom Essay Specifically
For You For Only $13.90/page!


order now

Assistant
Professor, Department of Mathematics,

Prince Shri
Venkateshwara Padmavathi Engineering College, Ponmar, Chennai-600127.

[email protected]

 

Abstract
– Multistage
decision making is a kind of dynamic process. A required goal is not achieved
by solving a single decision problem but by solving a sequence of decision
making problems. These decision making problems which represent stages in
overall multistage decision making are dependent on one another in dynamic
sense. Automata theory deals with the definitions and properties of
mathematical models of computation. Finite state automata are used in text processing,
compilers and hardware design. In this paper we give a procedure to obtain an
optimal sequence of input states of finite state automata.

Key words: Transition graph, Transition table,
Finite state automata, Directed graph, Strings, Languages.       

I
INTRODUCTION

The
process of computation involves solving problems by communicating them to a
computational model by means of a suitable language. Also we need some model to
recognize these languages. A simple kind of machines called finite automata
will fulfil this requirement. Here we discuss the application of fuzzy graph
theory in fuzzy finite state automaton through multistage decision making.

 Fuzzy sets originated in a seminal paper by
Lotfi A.Zadeh in 1965. Since then it has grown by leaps and bounds and
innumerable number of papers has appeared in various journals. Applications of
fuzzy sets and fuzzy logic were ushered in by Mamdani through a paper in 1975.
The development in applications was so drastic that within 15 years both
consumer products like cameras, washing machines, TV and industrial products
based on fuzzy logic controllers were rolled out in the market.

The
origin of graph theory started with the Konisberg bridge problem in 1735. Euler
studied the Konisberg bridge problem and constructed a structure that solves
the problem which is referred to as an Eulerian graph. Graph theory is a
delightful playground for the exploration of proof techniques in discrete
mathematics and its results have applications in many areas of computing,
social and natural sciences. The initial model for any article on graph theory
was the elegant text by J.A.Bondy and U.S.R Murty, Graph Theory with
applications (Macmillan/North-Holland 1976). Graph theory is still young and
no consensus has emerged on how the introductory material should be presented.

Currently
concepts of graph theory are highly utilized by computer science applications
especially in areas of data mining, image segmentation, clustering and
networking. Fuzzy sets paved the way for a new method of philosophical thinking
“Fuzzy Logic” which is now an essential concept in artificial intelligence. The
most important property of fuzzy set is that it consists of a class of objects
that satisfy a certain property or several properties.

In
1957 John Myhill invented Transition graphs. A Transition graph is a directed
labelled graph whose nodes correspond to the states of the finite automaton and
edges correspond to the transition. Each edge of the graph corresponds to the
transition from one state to another state. Therefore each edge is labelled
with the corresponding input symbol, which is written on the edge. If a path is
followed from the start state to the final state and the symbols on the edges
contained in the path are combined together, a string will be generated, which
is accepted by the finite automaton.

II FUZZY NUMBER

 

A Triangular
Fuzzy number

                  It is a Fuzzy number
represented by 3 points (a1, a2, a3) as A (a1, a2,
a3) with membership function

 

                   mA(x) =    0                      if   
x a3

 

 

B Trapezoidal Fuzzy number

It is a  
Fuzzy number represented by 4 points 
(a1,a2,a3,a4)  as A(a1,a2,a3,a4)
with membership function

 

mA(x)  =        
0                          if   x
a4

 

Deterministic finite state automaton

A Deterministic finite state automaton M
is defined as a 5-tuple ( Q, S,
d,
q0, F )

where

(i)                
Q represents a
finite non empty set of states

(ii)              
S denotes a finite non empty set of input symbols
from which input string is made.

(iii)            
d is a transition function which may be defined as d
: Q´S®Q

Here
q0Î
Q is called as the initial state, F Í
Q is the non empty finite set of final states or accepting states. For finite
automata, a finite set of final states can be specified. However, for a given
input, there will always be only one final state. Changes in the input string
may provide some other final states.

The
transition function of finite state automata is generally represented by a
transition table. The number of rows in a transition table is equal to the
number of states in Q , while the number of columns is equal to the number of
input symbols in S.
The entry for each row is generated by using d
: Q ´S®Q.
The starting state in a transition table is denoted by an incoming arrow
whereas the final states are enclosed in double circles.

 

 

 

III DIRECTED GRAPH

A
directed graph is a set of points with arrows connecting some of the points.
The points are called nodes or vertices, the arrows are called directed edges.
The number of arrows pointing from a particular node is the out degree of that
node and the number of arrows pointing to a particular node is the in degree.

EXAMPLE

 

 

 

 

 

 

 

 

 

                                                             Fig1.
Directed Graph

 

            In a directed graph we represent an
edge from i to j as a pair (i,j). The formal description of a directed graph G
is (V,E) where V is the set of nodes and E is the set of edges. The formal
description of figure 1 is  G =
{{1,2,3,4,5}, {(1,2),(1,5),(2,1),(2,4),(5,4),(5,6),(6,1),(6,3)}}

Alphabet

            An
alphabet is a non empty finite set denoted byS.
The members of the alphabet are the symbols of the alphabet. A string over an
alphabet is a finite sequence of symbols from that alphabet, written next to
one another and not separated by commas.

EXAMPLE

           Let  S
= {0,1} be an alphabet. Then the strings are {0011,0000011111, 111100011100,}

DEFINITION

       Transition function is slightly
cumbersome when it is used for every input symbol. Hence extended transition
function is used which is denoted by d*
and is defined by d*:
Q´S*®
Q as

d*(q0,
aw)
= d*(
d(q0,
a),w)
and d*(q,e)
= q where e
is the empty string.

EXAMPLE

Let
us design a DFA to check for numbers that are divisible by 4.

            We know that if any number is
divided by 4, it may give remainder 0, 1, 2 or 3 where remainder 0 implies that
the number is divisible by 4. Hence there are 4 possible states for the DFA.
Let us represent these states as q0, q1, q2, q3. The next step is to
generate the transition table in which the number of rows would be equal to 4
corresponding to q0, q1, q2, q3 and
number of columns would be equal to 10 corresponding to all possible inputs (
0,1,2,3,4,5,6,7,8,9)                                  

States/Input

0

1

2

3

4

5

6

7

8

9

q0

q0

q1

q2

q3

q0

q1

q2

q3

q0

q1

q1

q2

q3

q0

q1

q2

q3

q0

q1

q2

q3

q2

q0

q1

q2

q3

q0

q1

q2

q3

q0

q1

q3

q2

q3

q0

q1

q2

q3

q0

q1

q2

q3

 

Table
1. Transition Table

The DFA
is defined as Q = { q0,q1,q2,q3 }

S
=   ( 0,1,2,3,4,5,6,7,8,9)

q0
is the initial state and F = {q0}
is the final state because a number is divisible by 4 only if it produces a
remainder 0.

Theorem                                                                             

            Let A= (X,Z,f) be a fuzzy automaton
where X & Z are respectively the set of input and output states and f:ZxX®Z is the state – transition function of A. The
optimal sequence of decisions x0, x1,x2…..xN-1  of decision can be obtained by successively
maximizing values xN-K in  CN-K(zN-K)=max
minAN-K(xN-K)CN-K+1(zN-K+1)

                                    XN-K          

for
K= 1,2, ……. N where zN-K+1 = f ( zN-K,
xN-K)

 

Proof
     

             Let A0, A1, A2…..AN-1  be the fuzzy input states which could be
considered as constraints and fuzzy internal state CN as fuzzy goal
in a fuzzy decision making, We may conceive of fuzzy decision as a fuzzy set on
xn defined by D = Ã0 Ç Ã1Ç……CN where Ãt is a
cylindric extension of At from X to XN  for each t = 0,1,…. N-1 and CN is
the fuzzy set on xN that induces CN on Z. That is for any
sequence x0, x1,x2…..xN-1   viewed as a sequence of decision, the membership
grade of D is defined by

D(x0,
x1,x2…..xN-1 
) = minA0(x0), A1(x1)……..
AN-1 (xN-1), CN(zn )
——————— (1)

Where
zN is a uniquely determined by 
x0, x1,x2…..

xN-1  and   z0
through zt+1 = f (zt, xt) we have to
find a sequence of input states.

Example

             Let A = (x,z,f) be an automaton where input
states X = {x1,x2} output states Z ={ z1, z2,
z3 } and the state transition function. Let their be two
states and the fuzzy goal at t =2 is C2 = .3/ z1
+ 1/z2 ­+ .8/z3 . 
At time t= 0, t=1. Let the fuzzy constraints be A0 = .7/x1
+ 1/x2 and A1 = 1/x1 + .6/x2
obtain an optimal sequence of input states.

 

Solution

The
best decision X1  for each
state z1 Î Z at time t=1
is

Z1        z1      z2         z3

X1            x2      x1         x2

          The best decision X0  for each state z0 Î Z at time t=0 is

Z0        z1                z2               z3

X0           x2      x1 or x2         x1 or x2        

           The goal is satisfied to the degree
.6 when the intial state is Z2  regardless of which of the two maximising
decisions is used.

IV CONCLUSION

A
decision problem conceived in terms of fuzzy dynamic programming is viewed as a
decision problem regarding a fuzzy finite state automaton.  A fuzzification of dynamic programming
extends its practical utility since it allows decision makers to express their
goals, constrains, decisions and so on. In approximate fuzzy terms whenever
desirable.

REFERENCES

 

1                                                             Zadeh
L.A.Fuzzy sets ,Information and control,8(1965).

2                                                             Animesh Kumar
Sharma,Padamwar.B.V,Dewangar.C.L.-Trends in Fuzzy Graphs- IJIRSET,2 (2013)
4636-4639.

3                                                             Bhattacharya.p,
Some Remarks of fuzzy graphs, pattern Recognition Left, 6 (1987) 297-302.

4
          NagoorGani.A, Chandrasekaran.V.T., A first look at Fuzzy Graph
Theory,Allied publishers Pvt. Ltd (2010).

5
          George J Klir and Bo Yuan, Fuzzy
sets and Fuzzy Logic, PHI Learning pvt ltd.

6                                                             J.
Kavikumar, Member, IAENG, Azme Bin Khamis and Rozaini Bin Roslan Bipolar-valued
Fuzzy Finite    Switchboard State Machines
Proceedings of the World Congress on Engineering and Computer Science 2012 Vol
I WCECS 2012, October 24-26, 2012, San Francisco, USA.

7     Frank  Harary, Graph Theory , Narosa Publishing
House Pvt Ltd.

8     Douglas B.West,
Introduction to Graph Theory, Second Edition, Pearson India Education

          Services Pvt Ltd.

M. GATTO AND G. GUARDABASSM. GATTO AND G.
GUARDABASS

                                                                                                                                                                                                      C

BA

x

Hi!
I'm Mack!

Would you like to get a custom essay? How about receiving a customized one?

Check it out