Prerequisite: Basic Knowledge of Finite Automata
Problem 1
Construction of a minimal NFA accepting a set of strings over {a, b} in which each string of the language starts with 'ab'.
Explanation
The desired language will be like:
L1 = {ab, abba, abaa, abbb ...........}
Here as we can see that each string of the above language starts with 'ab' and end with any alphabet either 'a' or 'b'. But the below language is not accepted by this NFA because none of the string of below language starts with 'ab'.
L2 = {ba, ba, babaaa..............}
The state transition diagram of the desired language will be like below:
In the above NFA, the initial state 'X' on getting 'a' as the input it transits to a state 'Y'. The state 'Y' on getting 'b' as the input it transits to a final state 'Z'. The final state 'Z' on getting either 'a' or 'b' as the input it remains in the state of itself.
Python Implementation
def stateX(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
#stateY function
if (n[0]=='a'):
stateY(n[1:])
#if at zero index
#'b' then print
#not accepted
elif (n[0]=='b'):
print("string not accepted")
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' then print
#not accepted
if (n[0]=='a'):
print("string not accepted")
#if at zero index
#'b' found call
#stateZ function
elif (n[0]=='b'):
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
#'b' found call
#stateZ function
if (n[0]=='a'):
stateZ(n[1:])
#if at zero index
#'b' found call
#stateZ function
elif (n[0]=='b'):
stateZ(n[1:])
#take input
#n=input()
n="a"
print(n," ")
#call stateX function
#to check the input
stateX(n)
n="ba"
print(n," ")
#call stateX function
#to check the input
stateX(n)
n="abab"
print(n," ")
#call stateX function
#to check the input
stateX(n)
n="ab"
print(n," ")
#call stateX function
#to check the input
stateX(n)
Output:
a
string not accepted
ba
string not accepted
abab
string accepted
ab
string accepted
Problem 2
Construction of a minimal NFA accepting a set of strings over {a, b} in which each string of the language is not starting with 'ab'.
Explanation
The desired language will be like:
L1 = { 𝝐, a, b, aa, bb, ba, bba, bbaa, ...........}
Here as we can see that each string of the above language is not starting with 'ab' but can end with either 'a' or 'b'.
But the below language is not accepted by this NFA because some of the string of below language starts with 'ab'.
L2 = {ab, aba, ababaab..............}
The state transition diagram of the desired language will be like below:
In the above NFA, the initial state 'X' on getting 'b' as the input it transits to a state 'Y' , and on getting 'a' transit to state 'Z' .The state 'Y' on getting either 'a' or 'b' as the input it transits to a final state 'Y' . The final state 'Z' on getting 'a' moves to the state 'Y' and on 'b' moves to the dead state 'S'.
Python Implementation
def stateX(n):
#if length of n is 0
#then print accepted
if(len(n)==0):
print("string accepted")
else:
#if at zero index
#'a' found then
#stateS function
if (n[0]=='a'):
stateS(n[1:])
#if at zero index
#'b' then call
#stateY function
elif (n[0]=='b'):
stateY(n[1:])
def stateY(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:])
#if at zero index
#'b' found call
#stateZ function
elif (n[0]=='b'):
stateZ(n[1:])
def stateS(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:])
#if at zero index
#'b' found call
#"string not accepted"
elif (n[0]=='b'):
print("string not accepted")
def stateZ(n):
#if length of n become 0
#then print accepted
if(len(n)==0):
print("string accepted")
else:
#if at zero index
#'b' found call
#stateZ function
if (n[0]=='a'):
stateZ(n[1:])
#if at zero index
#'b' found call
#stateZ function
elif (n[0]=='b'):
stateZ(n[1:])
#take input
#n=input()
n=""
print("epsilon ")
#call stateX function
#to check the input
stateX(n)
n="a"
print(n," ")
#call stateX function
#to check the input
stateX(n)
n="aa"
print(n," ")
#call stateX function
#to check the input
stateX(n)
n="ab"
print(n," ")
#call stateX function
#to check the input
stateX(n)
n="ba"
print(n," ")
#call stateX function
#to check the input
stateX(n)
Output:
epsilon
string accepted
a
string accepted
aa
string accepted
ab
string not accepted
ba
string accepted
