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# Construct RE for given finite automata with Arden’s theorem

There are two methods for converting a Deterministic Finite Automata (DFA) to Regular expression (RE). These methods are as follows −

- Arden's Theorem Method.
- State Elimination Method.

Let us understand the Arden's Theorem method in detail.

## Arden's Theorem

Let P and Q be the two regular expressions.

If P does not contain null string, then the following equation in R, viz R = Q + RP,

Which has a unique solution by R = QP*

Here,

- The finite automata (FA) does not have epsilon moves.
- It must have only initial state q1.
- Its states are q1, q2, q3,....qn. The final state may be some qi where i<=n.
- qi is the regular expression representing a set of strings accepted by the finite automata even though qi is a final state.

Now let us consider a problem, that how to construct given finite automata with the help of Arden's theorem.

## Problem

Find out the regular expression for the given finite automata by applying Arden's theorem.

## Solution

Let us see the equations.

q_{0 }= q_{1} + q_{20}+ E

q_{1} =q_{0}0

q_{2} =q_{0}1

q_{3} = q_{1}0 + q_{2}1+ q_{3}(0+1)

Let us solve q_{0} first, as shown below−

q_{0} = q_{1} + q_{20}+ E

q_{0} = q_{1} + q_{0}10+ E

q_{0} = q_{0}(01 +10)+ E

q_{0} = E(01+10)*

q_{0} = (01+10)*

·: R = Q+ RP

=> QP* where

R =q_{0} ,Q =e, P= (01 + 10)

Thus, the regular expression will be as follows −

r = {01+10)*

Since q0 is a final state, we are interested in q0 only.

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