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.

q₀ = q₁ + q₂₀+ E

q₁ =q₀0

q₂ =q₀1

q₃ = q₁0 + q₂1+ q₃(0+1)

Let us solve q₀ first, as shown below−

q₀ = q₁ + q₂₀+ E

q₀ = q₁ + q₀10+ E

q₀ = q₀(01 +10)+ E

q₀ = E(01+10)*

q₀ = (01+10)*

·: R = Q+ RP

=> QP* where

R =q₀ ,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.