Designing Non-Deterministic Finite Automata (Set 1)
Prerequisite:
In this article, we will see some designing of Non-Deterministic Finite Automata (NFA).
Problem-1:
Construction of a minimal NFA accepting a set of strings over {a, b} in which each string of the language starts with ‘a’.
Explanation:
The desired language will be like:
L1 = {ab, abba, abaa, ...........}
Here as we can see that each string of the above language starts with ‘a’ 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 ‘a’.
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 final state ‘Y’. The final state ‘Y’ on getting either ‘a’ or ‘b’ as the input it remains in the state of itself.
Python Implementation:
C++
#include <iostream> #include <string> void stateX( const std::string& n); void stateY( const std::string& n); void stateX( const std::string& n) { // if length of n becomes 0 // then print not accepted if (n.empty()) { std::cout << "String not accepted" << std::endl; } else { // if at zero index // 'a' found call // stateY function if (n[0] == 'a' ) { stateY(n.substr(1)); } // if at zero index // 'b' then print // not accepted else if (n[0] == 'b' ) { std::cout << "String not accepted" << std::endl; } } } void stateY( const std::string& n) { // if length of n becomes 0 // then print accepted if (n.empty()) { std::cout << "String accepted" << std::endl; } else { // if at zero index // 'a' found call // stateY function if (n[0] == 'a' ) { stateY(n.substr(1)); } // if at zero index // 'b' found call // stateY function else if (n[0] == 'b' ) { stateY(n.substr(1)); } } } int main() { // take input std::string inputString= "ababa" ; // call stateX function // to check the input stateX(inputString); return 0; } //This code is contributed by utkarsh |
Java
public class StateMachine { public static void stateX(String n) { // if length of n becomes 0 // then print not accepted if (n.isEmpty()) { System.out.println( "String not accepted" ); } else { // if at zero index // 'a' found call // stateY function if (n.charAt( 0 ) == 'a' ) { stateY(n.substring( 1 )); } // if at zero index // 'b' then print // not accepted else if (n.charAt( 0 ) == 'b' ) { System.out.println( "String not accepted" ); } } } public static void stateY(String n) { // if length of n becomes 0 // then print accepted if (n.isEmpty()) { System.out.println( "String accepted" ); } else { // if at zero index // 'a' found call // stateY function if (n.charAt( 0 ) == 'a' ) { stateY(n.substring( 1 )); } // if at zero index // 'b' found call // stateY function else if (n.charAt( 0 ) == 'b' ) { stateY(n.substring( 1 )); } } } public static void main(String[] args) { // take input String inputString = "ababa" ; // call stateX function // to check the input stateX(inputString); } } |
Python3
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 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 :]) #if at zero index #'b' found call #stateY function elif (n[ 0 ] = = 'b' ): stateY(n[ 1 :]) #take input n = input () #call stateA function #to check the input stateX(n) |
C#
using System; class StateMachine { static void StateX( string n) { // if length of n becomes 0 // then print not accepted if ( string .IsNullOrEmpty(n)) { Console.WriteLine( "String not accepted" ); } else { // if at zero index // 'a' found call // StateY function if (n[0] == 'a' ) { StateY(n.Substring(1)); } // if at zero index // 'b' then print // not accepted else if (n[0] == 'b' ) { Console.WriteLine( "String not accepted" ); } } } static void StateY( string n) { // if length of n becomes 0 // then print accepted if ( string .IsNullOrEmpty(n)) { Console.WriteLine( "String accepted" ); } else { // if at zero index // 'a' found call // StateY function if (n[0] == 'a' ) { StateY(n.Substring(1)); } // if at zero index // 'b' found call // StateY function else if (n[0] == 'b' ) { StateY(n.Substring(1)); } } } static void Main() { // take input string inputString = "ababa" ; // call StateX function // to check the input StateX(inputString); } } |
output:
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 ‘a’.
Explanation:
The desired language will be like:
L1 = {ba, bba, bbaa, ...........}
Here as we can see that each string of the above language is not starting with ‘a’ 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 ‘a’.
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 final state ‘Y’. The final state ‘Y’ on getting either ‘a’ or ‘b’ as the input it remains in the state of itself.
Python Implementation:
Python3
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 #'b' found call #stateY function if (n[ 0 ] = = 'b' ): stateY(n[ 1 :]) #if at zero index #'a' then print #not accepted elif (n[ 0 ] = = 'a' ): print ("string not accepted") 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 #stateY function if (n[ 0 ] = = 'a' ): stateY(n[ 1 :]) #if at zero index #'b' found call #stateY function elif (n[ 0 ] = = 'b' ): stateY(n[ 1 :]) #take input n = input () #call stateA function #to check the input stateX(n) |