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Designing Deterministic Finite Automata (Set 10)
  • Last Updated : 01 Jun, 2021

Prerequisite: Designing finite automata 
In this article, we will see some designing of Deterministic Finite Automata (DFA).
Problem-1: Construction of a minimal DFA accepting set of strings over {a} in which {an | n≥0, n≠2 i.e, ‘n’ should be greater than 0 and not equal to 2}. 
Explanation: The desired language will be like: 
 

L1 = {ε, a, aaa, aaaa, aaaaa, ..................}

Here ε is taken as string because value of ‘n’ is greater than or equal to zero and rest of the strings are having ‘a’ to the power of any positive natural number but not 2. 
Below language is not accepted by this DFA because some of the string containing ‘a’ to the power 2. 
 

L2 = {aa, aaaaa, ..........}

This language L2 is not accepted by this required DFA because of its string containing ‘a’ to the power of 2. 
The state transition diagram of the desired language will be like below: 
 

In the above DFA, the initial and final state’W’ on getting ‘a’ as the input it transits to a final state ‘X’. The final state ‘X’ on getting ‘a’ as the input it transits to a state ‘Y’. The state ‘Y’ on getting ‘a’ as the input it transits to a final state ‘Z’ which on getting any number of ‘a’ it remains in the state of itself. 
 



Python Implementation:

 

Python3




             
def stateW(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateX function   
        if (n[0]=='a'):
            stateX(n[1:])
        
             
         
def stateX(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else
        #if at zero index
        #'a' found call
        #stateY function   
        if (n[0]=='a'):
            stateY(n[1:])
         
def stateY(n):
    #if length of n become 0
    #then print not accepted
    if(len(n)==0):
        print("string not accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateZ function   
        if (n[0]=='a'):
            stateZ(n[1:])
         
def stateZ(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateZ function   
        if (n[0]=='a'):
            stateZ(n[1:])
             
#take input
n=input()
 
#call stateW function
#to check the input
stateW(n)

Problem-2: Construction of a minimal DFA accepting set of strings over {a} in which {an | n≥0, n≠2, n≠4 i.e, ‘n’ should be greater than 0 and not equal to 2 and 4}. 
Explanation: The desired language will be like: 
 

L1 = {ε, a, aa, aaaaa, aaaaaa, .................. }

Here ε is taken as string because value of ‘n’ is greater than or equal to zero and rest of the strings are having ‘a’ to the power of any positive natural number but not 2 and 4. 
Below language is not accepted by this DFA because some of the string containing ‘a’ to the power of 2 and 4. 
 

L2 = {aa, aaaaa, aaaaaaaaaa, ............. }

The state transition diagram of the desired language will be like below: 
 

In the above DFA, the initial and final state ‘A’ on getting ‘a’ as the input it transits to a final state ‘B’. The final state ‘B’ on getting ‘a’ as the input it transits to a state ‘C’. The state ‘C’ on getting ‘a’ as the input it transits to a final state ‘D’. The final state ‘D’ on getting ‘a’ as the input it transits to a state ‘E’. The state ‘E’ on getting ‘a’ as the input it transits to a final state ‘F’. The final state ‘F’ on getting ‘a’ as the input it remains in the state of itself. 
 

Python Implementation:

 

Python3




             
def stateA(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateB function   
        if (n[0]=='a'):
            stateB(n[1:])
        
             
         
def stateB(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else
        #if at zero index
        #'a' found call
        #stateC function   
        if (n[0]=='a'):
            stateC(n[1:])
         
def stateC(n):
    #if length of n become 0
    #then print not accepted
    if(len(n)==0):
        print("string not accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateD function   
        if (n[0]=='a'):
            stateD(n[1:])
         
def stateD(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateE function   
        if (n[0]=='a'):
            stateE(n[1:])
 
def stateE(n):
    #if length of n become 0
    #then print not accepted
    if(len(n)==0):
        print("string not accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateF function   
        if (n[0]=='a'):
            stateF(n[1:])
             
def stateF(n):
    #if length of n become 0
    #then print accepted
    if(len(n)==0):
        print("string accepted")
         
    else:
        #if at zero index
        #'a' found call
        #stateF function   
        if (n[0]=='a'):
            stateF(n[1:])           
             
#take input
n=input()
 
#call stateA function
#to check the input
stateA(n)

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