Build a DFA to accept a binary string containing “01” i times and “1” 2j times

Given a binary string str, the task is to build a DFA that accepts given binary string if it contains “01” i times and “1” 2j times, i.e.,
L={(01)^i (1)^{2j} \text{ where }i\geq1\text{ and }j\geq1}

Examples:

Input: str = “011111”
Output: Accepted
Explanation:
The string follows the language as: (01)1(1)2*2

Input: str = “01111”
Output: Not Accepted

DFA or Deterministic Finite Automata is a finite state machine which accepts a string(under some specific condition) if it reaches a final state, otherwise rejects it.



In DFA, there is no concept of memory, therefore we have to check the string character by character, beginning with the 0th character. The input set of characters for the problem is {0, 1}. For a DFA to be valid, there must a transition rule defined for each symbol of the input set at every state to a valid state. Therefore, the following steps are followed to design the DFA:

  1. Create initial stage and make the transition of 0 and 1 to next possible state.
  2. Transition of 0 is always followed by transition of 1.
  3. Make an initial state and transit its input alphabets, i.e, 0 and 1 to two different states.
  4. Check for acceptance of string after each transition to ignore errors.
  5. First, make DfA for minimum length string then go ahead step by step.
  6. Define Final State(s) according to the acceptance of string.

Step by Step Approach to design a DFA:

  • Step 1: Minimum possible acceptable string is 0111, i.e, (01)1 (11)1. So, create an initial state “A” that make transition of 0 to state “B” and then transition of 1 from “B” to state “C” then transition of 1 from “C” to “D”, then transition of 1 from “D” to “E” as shown in diagram make this stage “E” is final state.

  • Step 2: Now, think about the string having consecutive (01) and then followed by consecutive (11) to end the string. Hence, when i>1, make a transition of “0” from state “C” to state “B” and make a transition of “1” from the state “E” to state “D”. Hence, strings like 010111, 011111, 0101111111, etc. are acceptable now.

  • Step 3: We have done with all kind of strings possible to accept. But, there are few input alphabets which are not transited to any of the states. In this case, all these kind of input will be sent to some dead state to block their further transitions that are not acceptable. Input alphabets of the dead state will be sent to the dead state itself. Therefore, the final design of the DFA is:

Below is the implementation of the above approach:

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# Python program for the given
# language
  
# Function for the state A
def checkstatea(n):
    if(len(n)%2!=0 or len(n)<4):
        print("string not accepted")
    else:    
        i=0
  
        # State transition to B
        # if the character is 0
        if(n[i]=='0'):
            stateb(n[1:])
        else:
            print("string not accepted")
  
# Function for the state B
def stateb(n):
    i=0
    if(n[i]=='0'):
        print("string not accepted")
  
    # State transition to C
    # if the character is 1
    else:
        statec(n[1:])
  
# Function for the state C
def statec(n):
    i=0
  
    # State transition to D
    # if the character is 1
    if(n[i]=='1'):
        stated(n[1:])
  
    # State transition to B
    # if the character is 0
    else:
        stateb(n[1:])
  
# Function for the state D
def stated(n):
    i=0
    if(len(n)==1):
        if(n[i]=='1'):
            print("string accepted")
        else:
            print("string not accepted")
    else:
  
        # State transition to E
        # if the character is 1
        if(n[i]=='1'):
            statee(n[1:])
        else:
            print("string not accepted")   
  
# Function for the state E     
def statee(n):
    i=0
    if(len(n)==1):
        if(n[i]=='0'):
            print("string not accepted")
        else:
            print("string accepted")
           
    else:
        if(n[i]=='0'):
            print("string not accepted")
        stated(n[1:])
       
       
# Driver code
if __name__ == "__main__":
  
    n = "011111"
    checkstatea(n)
      
  

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Output:

string accepted

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