Construct a DFA which accept the language L = {anbm | n > =1, (m) mod 3 = 1}

Problem: Construct a DFA which accept the language L = {aⁿb^m | n > =1, (m) mod 3 = 1}. Explanation: For constructing the DFA, the following things to be remember:

 which means any no of elements, and
 =  which means any no of elements greater than 1.

Examples:

Input: a a b b b           
Output: NOT ACCEPTED
// n = 2 (>=1), m=3 ((3) mod3 != 1)

Input: a a a b        
Output: ACCEPTED
 // n = 3 (>=1), m = 1 ((1) mod 3= 1)

Input: b b b b        
Output: NOT ACCEPTED
// n = 0(must be >=1), m = 4 ((4) mod 3 = 1) 

Approaches:
It’s construction should contain the following steps:

• Step-1: Construct FA for means having any number of a’s greater than one.
• Step-2: Construct FA for means having exactly
• Step-3: Construct FA for means having b’s equal to multiple of 3 .
• Step-4: Concatenate the three FA and make single DFA. Always maintain the order of a, b and c.

Given DFA has following states. State 2 leads to the acceptance of the string.Whereas states 0, 1, 3, 4 and 5 leads to the rejection of the string. DFA State Transition Diagram:

Let’s see code for the demonstration:
C/C++
// C program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n >=1, (m)mod 3=1}
#include <stdio.h>
#include <string.h>

// dfa tells the number associated
// string end in which state.
int dfa = 0;

// This function is for
// the starting state (Q0)of DFA
void start(char c)
{
    if (c == 'a') {
        dfa = 1;
    }
    else if (c == 'b') {
        dfa = 5;
    }

    // -1 is used to check for any invalid symbol
    else {
        dfa = -1;
    }
}

// This function is for the first state (Q1) of DFA
void state1(char c)
{
    if (c == 'a') {
        dfa = 1;
    }
    else if (c == 'b') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}

// This function is for the second state (Q2) of DFA
void state2(char c)
{
    if (c == 'b') {
        dfa = 3;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the third state (Q3)of DFA
void state3(char c)
{
    if (c == 'b') {
        dfa = 4;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the forth state (Q4)of DFA
void state4(char c)
{
    if (c == 'b') {
        dfa = 2;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the fifth state (Q5) of DFA
void state5(char c)
{
    dfa = -1;
}

int isAccepted(char str[])
{
    // store length of string
    int i, len = strlen(str);

    for (i = 0; i < len; i++) {
        if (dfa == 0)
            start(str[i]);

        else if (dfa == 1)
            state1(str[i]);

        else if (dfa == 2)
            state2(str[i]);

        else if (dfa == 3)
            state3(str[i]);

        else if (dfa == 4)
            state4(str[i]);

        else if (dfa == 5)
            state5(str[i]);

        else
            return 0;
    }
    if (dfa == 2)
        return 1;
    else
        return 0;
}

// driver code
int main()
{
    char str[] = "aaabbbb";
    if (isAccepted(str))
        printf("ACCEPTED");
    else
        printf("NOT ACCEPTED");
    return 0;
}

// This code is contributed by SHUBHAMSINGH10.
Java
// Java program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n >=1, (m)mod 3=1}
import java.util.*;
class GFG {

// dfa tells the number associated
// string end in which state.
static int dfa = 0;

// This function is for
// the starting state (Q0)of DFA
static void start(char c)
{
    if (c == 'a') {
        dfa = 1;
    }
    else if (c == 'b') {
        dfa = 5;
    }

    // -1 is used to check for any invalid symbol
    else {
        dfa = -1;
    }
}

// This function is for the first state (Q1) of DFA
static void state1(char c)
{
    if (c == 'a') {
        dfa = 1;
    }
    else if (c == 'b') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}

// This function is for the second state (Q2) of DFA
static void state2(char c)
{
    if (c == 'b') {
        dfa = 3;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the third state (Q3)of DFA
static void state3(char c)
{
    if (c == 'b') {
        dfa = 4;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the forth state (Q4)of DFA
static void state4(char c)
{
    if (c == 'b') {
        dfa = 2;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the fifth state (Q5) of DFA
static void state5(char c)
{
    dfa = -1;
}

static int isAccepted(char str[])
{
    // store length of string
    int i, len = str.length;

    for (i = 0; i < len; i++) {
        if (dfa == 0)
            start(str[i]);

        else if (dfa == 1)
            state1(str[i]);

        else if (dfa == 2)
            state2(str[i]);

        else if (dfa == 3)
            state3(str[i]);

        else if (dfa == 4)
            state4(str[i]);

        else if (dfa == 5)
            state5(str[i]);

        else
            return 0;
    }
    if (dfa == 2)
        return 1;
    else
        return 0;
}

// Driver code
public static void main(String[] args) {
   char str[] = "aaabbbb".toCharArray();
    if (isAccepted(str)==1)
        System.out.println("ACCEPTED");
    else
        System.out.println("NOT ACCEPTED");
    }
}

// This code is contributed by Rajput-Ji
Python 3
# Python3 program to implement DFS that accepts
# all Stringing which follow the language
# L = {a ^ n b ^ m | n >= 1, (m)mod 3 = 1 }

# This function is for the dfa = starting
# dfa = state (zeroth) of DFA
def start(c):
    if (c == 'a'):
        dfa = 1
    elif (c == 'b'):
        dfa = 5

    # -1 is used to check for any
    # invalid symbol
    else:
        dfa = -1
    return dfa

# This function is for the first
# dfa = state of DFA
def state1(c):
    if (c == 'a'):
        dfa = 1
    elif (c == 'b'):
        dfa = 2
    else:
        dfa = -1
    return dfa

# This function is for the second
# dfa = state of DFA
def state2(c):
    if (c == 'b'):
        dfa = 3
    elif (c == 'a'):
        dfa = 5
    else:
        dfa = -1
    return dfa

# This function is for the third
# dfa = state of DFA
def state3(c):
    if (c == 'b'):
        dfa = 4
    elif (c == 'a'):
        dfa = 5
    else:
        dfa = -1
    return dfa

# This function is for the fouth
# dfa = state of DFA
def state4(c):
    if (c == 'b'):
        dfa = 2
    elif (c == 'a'):
        dfa = 5
    else:
        dfa = -1
    return dfa


# This function is for the fifth
# dfa = state of DFA
def state5(c):
    dfa = -1
    return dfa

def isAccepted(String):

    # store length of Stringing
    l = len(String)

    # dfa tells the number associated
    # with the present dfa = state
    dfa = 0
    for i in range(l):
        if (dfa == 0):
            dfa = start(String[i])

        elif (dfa == 1):
            dfa = state1(String[i])

        elif (dfa == 2) :
            dfa = state2(String[i])

        elif (dfa == 3) :
            dfa = state3(String[i])

        elif (dfa == 4) :
            dfa = state4(String[i])

        elif (dfa == 5) :
            dfa = state5(String[i])
        else:
            return 0
    if(dfa == 2) :
        return 1
    else:
        return 0

# Driver code
if __name__ == "__main__" :
    String = "aaabbbb"
    if (isAccepted(String)) :
        print("ACCEPTED")
    else:
        print("NOT ACCEPTED")

# This code is contributed by SHUBHAMSINGH10.
C#
// C# program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n >=1, (m)mod 3=1}
using System;
public class GFG {

// dfa tells the number associated
// string end in which state.
static int dfa = 0;

// This function is for
// the starting state (Q0)of DFA
static void start(char c)
{
    if (c == 'a') {
        dfa = 1;
    }
    else if (c == 'b') {
        dfa = 5;
    }

    // -1 is used to check for any invalid symbol
    else {
        dfa = -1;
    }
}

// This function is for the first state (Q1) of DFA
static void state1(char c)
{
    if (c == 'a') {
        dfa = 1;
    }
    else if (c == 'b') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}

// This function is for the second state (Q2) of DFA
static void state2(char c)
{
    if (c == 'b') {
        dfa = 3;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the third state (Q3)of DFA
static void state3(char c)
{
    if (c == 'b') {
        dfa = 4;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the forth state (Q4)of DFA
static void state4(char c)
{
    if (c == 'b') {
        dfa = 2;
    }
    else if (c == 'a') {
        dfa = 5;
    }
    else {
        dfa = -1;
    }
}

// This function is for the fifth state (Q5) of DFA
static void state5(char c)
{
    dfa = -1;
}

static int isAccepted(char []str)
{
    // store length of string
    int i, len = str.Length;

    for (i = 0; i < len; i++) {
        if (dfa == 0)
            start(str[i]);

        else if (dfa == 1)
            state1(str[i]);

        else if (dfa == 2)
            state2(str[i]);

        else if (dfa == 3)
            state3(str[i]);

        else if (dfa == 4)
            state4(str[i]);

        else if (dfa == 5)
            state5(str[i]);

        else
            return 0;
    }
    if (dfa == 2)
        return 1;
    else
        return 0;
}

// Driver code
public static void Main(String[] args) {
   char []str = "aaabbbb".ToCharArray();
    if (isAccepted(str)==1)
        Console.WriteLine("ACCEPTED");
    else
        Console.WriteLine("NOT ACCEPTED");
    }
}
// This code has been contributed by 29AjayKumar
Output:

ACCEPTED