## Kleene's Theorem --- Part 1

### Subjects to be Learned

• Union of FAs
• Concatenation of FAs
• Kleene Star of FAs
• Acceptance of regular languages by FAs

### Contents

In this unit we are going to learn Kleene's theorem. It states that any regular language is accepted by an FA and conversely that any language accepted by an FA is regular.

Theorem 1 (Part 1 of Kleene's theorem): Any regular language is accepted by a finite automaton.

Proof: This is going to be proven by (general) induction following the recursive definition of regular language.
Basis Step: As shown below the languages , {} and { a } for any symbol a in are accepted by an FA.

Inductive Step: We are going to show that for any languages L1 and L2 if they are accepted by FAs, then L1 L2 , L1L2 and L1* are accepted by FAs. Since any regular language is obtained from {} and { a } for any symbol a in by using union, concatenation and Kleene star operations, that together with the Basis Step would prove the theorem.

Suppose that L1 and L2 are accepted by FAs M1 = < Q1 , , q1,0 , 1 , A1 > and M2 = < Q2 , , q2,0 , 2 , A2 > , respectively. We assume that Q1 Q2 = without loss of generality since states can be renamed if necessary.
Then L1 L2 , L1L2 and L1* are accepted by the FAs Mu = < Qu , , qu,0 , u , Au > , Mc = < Qc , , qc,0 , c , Ac > and Mk = < Q2 , , qk,0 , k , Ak > , respectively, which are given below.

Mu = < Qu , , qu,0 , u , Au > :

Qu = Q1 Q2 { qu,0 } , where qu,0 is a state which is neither in Q1 nor in Q2 .
u = 1 2 { (qu,0, , { q1,0 , q2,0 } ) } , that is u(qu,0, ) = { q1,0 , q2,0 } . Note that u(qu,0, a ) = for all a in .
Au = A1 A2

Mc = < Qc , , qc,0 , c , Ac > :

Qc = Q1 Q2
qc,0 = q1,0
c = 1 2 { (q, , { q2,0 } ) | q A1 }
Ac = A2

Mk = < Qk , , qk,0 , k , Ak > :

Qk = Q1 { qk,0 } , where qk,0 is a state which is not in Q1 .
k = 1 { (qk,0, , { q1,0 } ) } { (q, , { qk,0 } ) | q A1 }
Ak = { qk,0 }

These NFA-s are illustrated below.

It can be proven, though we omit proofs, that these NFA-s , Mu, Mc and Mk , in fact accept L1 L2 , L1L2 and L1*, respectively.

End of Proof

Examples of Mu , Mc and Mk:

Example 1: An NFA- that accepts the language represented by the regular expression (aa + b)* can be constructed as follows using the operations given above.

Example 2: An NFA- that accepts the language represented by the regular expression ((a + b)a*)* can be constructed as follows using the operations given above.

### Test Your Understanding of Kleene's Theorem Part 1

Indicate which of the following statements are correct and which are not.
Click True or Fals , then Submit.

Next -- Kleene's Theorem --- Part 2

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