Prerequisite – Designing finite automata

**Reversal : ** Reversing a language means reversing the each string in the language.

**Steps to Reversal:**

- Draw the states as it is.
- Make final state as initial state and initial states as a final state
- Reverse the edges
- Loop will remain same
- Remove the inappropriate transition state.

**Note : **

1. Not every reversal of DFA lead to DFA

2. Reversal process goes like this:

Let’s understand the Reversal process in Deterministic Finite Automata (DFA) with the help of below example.

**Example-1:**

Designing a DFA for the set of string over {a, b} such that string of the language start symbol ‘a’. The desired language will be formed:

L1 = {a, aa, ab, aab, aaabb, aabab, .......}

In L1, each string have starting element a. State Transition Diagram for the language L1:

This DFA accepts all the string starting with a. Here, State A and C are normal state but state B is final state. Now, Reversing the L1 language which gives the final result of the language L2.

State Transition Diagram of L2 (reverse of L1):

Thus as we see that L2 have been obtained through reversal process and this final NFA accept all the language containing strings ending with a symbol ‘a’.

l = {a, ba, bba, ababa, ......}

**Explanation: **

- Same states (A, B, C) are draw as present in original state diagram
- Since A is the initial state make it as final state.
- Since B is the final state make it as initial state.
- Reversing the edges, since previously edges point A to B and A to C .Change the direction, now edges points toward B to A and C to A.
- Assign the values as same as original.
- Draw the loop as in original state diagram
- Since there is no incident edge on the state C so, we can reduce this state C.
- Since there is no transition for ‘a’ and ‘b’ in state A .Therefore resultant FA is NFA.

**Example-2:**

Designing a DFA for the set of string over {a, b} such that string of the language contain even number of’a’. The desired language will be formed:

L1 = {, aa, aab, aba, aab, aababa, .......}

In L1, each string having even number of element a.

State Transition Diagram for the language L1:

This DFA accepts all the string containing even number of element a.Here, State A is initial state and state A is final state. Now, Reversing the L1 language which gives the final result of the language L2.

State Transition Diagram of L2 (reverse of L1):

Thus as we see that L2 have been obtained through reversal process and this final DFA accepts all the string containing even number of element a which is same as the original language.

L2 = {, aa, aab, aba, aab, aababa, .......}

**Explanation:**

- Same states (A, B) are draw as present in original state diagram.
- Since A is the initial state make it as final state.
- Since A is the final state make it as initial state.
- Reversing the edges, since previously edges point A to B and so on.Change the direction, now edges points toward B to A and so on.
- Assign the values as same as original.
- Since there is transition for ‘a’ and ‘b’ in state A and B.Therefore resultant FA is DFA.
- Since State diagram after and before reversal are same.Therefore L1 = L2

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