Design a DFA of a string with at least two 0’s and at least two 1’s

Data Structure Algorithms Computer Science Computers

A Deterministic Finite automaton (DFA) is a 5-tuples

M=(Q, Σ, δ,q0,F)

Where,

Q : Finite set called states.
Σ : Finite set called alphabets.
δ : Q × Σ → Q is the transition function.
q0 ϵ Q is the start or initial state.
F : Final or accept state.

Problem

Construct DFA of a string with at least two 0’s and at least two 1’s.

Solution

The language generated based on the given condition over the alphabet Σ ={0,1) is −

L={0011,001011,0001010,0011001,010101,……}

The given language accepts at least two zero’s means it can accept two or more than two zero’s and at least two one’s means it accepts two or more than two one’s.

Suppose,

Input − 00010
Output − string rejected

Because, the given input does not have at least two’s.

Even though it has at least zeros, it won’t accept the string.

To accept the string both conditions are satisfied, if one fails the string will not be accepted by the machine.

Input − 001001001
Output − string accepted

Now construct the DFA for given inputs −

State	Number of zero’s	Number of one's
→ q0	0	0
q1	0	1
q2	0	> =2
q3	1	0
q4	1	1
q5	1	> =2
q6	> =2	0
q7	> =2	1
*q8	> =2	> =2

The DFA will be as follows −

Explanation

If input is 1 then the number of 1 increases to 1. Move to state q1. If input is 0 then the number of 0 increases to 1. Move to state q3.

If input is 1 then the number of 1 increases to 2. Move to state q2. If input is 0 then the number of 0 increases to 1. Move to state q4.

If input is 1 then the number of 1 always increases by 1. And then remain in the same state. If input is 0 then the number of 0 increases to 1. Move to state q5.

If input is 1 then the number of 1 increases to 1. Move to state q4. If input is 0 then the number of 0 increases to 2. Move to state q6.

If input is 1 then the number of 1 increases to 2. Move to state q5. If input is 0 then number of 0 increases to 2. Move to state q7.

If input is 1 then the number of 1 always increases by 1. And then remain in the same state. If input is 0 then the number of 0 increases to 2. Move to state q8

If input is 1 then the number of 1 increase to 1. Move to state q7. If input is 0 then the number of 0 keeps increasing by 1. And then remain in the same state.

If input is 1 then the number of 1 increase to 2. Move to state q8. If input is 0 then the number of 0 keeps increasing by 1. And then remain in the same state.

If input is 1 then the number of 1 always increases by 1. And then remain in the same state. If input is 0 then the number of 0 keeps increasing by 1. And then remain in the same state.

Finally, if the string is finished, then it is ACCEPTED.

Bhanu Priya

Published on 15-Jun-2021 06:39:59

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