Union of two regular languages

In formal language theory, and in particular the theory of nondeterministic finite automata, it is known that the union of two regular languages is a regular language. This article provides a proof of that statement.

Theorem

For any regular languages $L_{1}$ and $L_{2}$ , language $L_{1}\cup L_{2}$ is regular.

Proof

Since $L_{1}$ and $L_{2}$ are regular, there exist NFAs $N_{1},\ N_{2}$ that recognize $L_{1}$ and $L_{2}$ .

Let

N_{1} = (Q_{1},\ \Sigma ,\ T_{1},\ q_{1},\ A_{1})

N_{2} = (Q_{2},\ \Sigma ,\ T_{2},\ q_{2},\ A_{2})

Construct

N = (Q,\ \Sigma ,\ T,\ q_{0},\ A_{1}\cup A_{2})

where

Q = Q_{1}\cup Q_{2}\cup\{q_{0}\}

T(q,x) = \left\{\begin{array}{lll} T_{1}(q,x) & \mbox{if} & q\in Q_{1} \\ T_{2}(q,x) & \mbox{if} & q\in Q_{2} \\ \{q_{1}, q_{2}\} & \mbox{if} & q = q_{0}\ and\ x =\epsilon\\ \emptyset & \mbox{if} & q = q_{0}\ and\ x\neq\epsilon \end{array}\right.

In the following, we shall use $p\stackrel{x,T}{\rightarrow}q$ to denote $q\in E(T(p,x))$

Let $w$ be a string from $L_{1}\cup L_{2}$ . Without loss of generality assume $w\in L_{1}$ .

Let $w = x_{1}x_{2}\cdots x_{m}$ where $m\geq 0, x_{i}\in\Sigma$

Since $N_{1}$ accepts $x_{1}x_{2}\cdots x_{m}$ , there exist $r_{0}, r_{1},\cdots r_{m}\in Q_{1}$ such that

q_{1}\stackrel{\epsilon , T_{1}}{\rightarrow}r_{0}\stackrel{x_{1} , T_{1}}{\rightarrow}r_{1}\stackrel{x_{2} , T_{1}}{\rightarrow}r_{2}\cdots r_{m-1}\stackrel{x_{m} , T_{1}}{\rightarrow}r_{m}, r_{m}\in A_{1}

Since $T_{1}(q,x) = T(q,x)\ \forall q\in Q_{1}\forall x\in\Sigma$

r_{0}\in E(T_{1}(q_{1},\epsilon ))\Rightarrow r_{0}\in E(T(q_{1},\epsilon ))

r_{1}\in E(T_{1}(r_{0},x_{1} ))\Rightarrow r_{1}\in E(T(r_{0},x_{1} ))

\vdots

r_{m}\in E(T_{1}(r_{m-1},x_{m} ))\Rightarrow r_{m}\in E(T(r_{m-1},x_{m} ))

We can therefore substitute $T$ for $T_{1}$ and rewrite the above path as

$q_{1}\stackrel{\epsilon , T}{\rightarrow}r_{0}\stackrel{x_{1} , T}{\rightarrow}r_{1}\stackrel{x_{2} , T}{\rightarrow}r_{2}\cdots r_{m-1}\stackrel{x_{m} , T}{\rightarrow}r_{m}, r_{m}\in A_{1}\cup A_{2}, r_{0}, r_{1},\cdots r_{m}\in Q$

Furthermore,

\begin{array}{lcl} T(q_{0}, \epsilon) = \{q_{1}, q_{2}\} & \Rightarrow & q_{1}\in T(q_{0}, \epsilon)\\ \\ & \Rightarrow & q_{1}\in E(T(q_{0}, \epsilon))\\ \\ & \Rightarrow & q_{0}\stackrel{\epsilon , T}{\rightarrow}q_{1} \end{array}

and

q_{0}\stackrel{\epsilon , T}{\rightarrow}q_{1}\stackrel{\epsilon , T}{\rightarrow}r_{0}\Rightarrow q_{0}\stackrel{\epsilon , T}{\rightarrow}r_{0}

The above path can be rewritten as

q_{0}\stackrel{\epsilon , T}{\rightarrow}r_{0}\stackrel{x_{1} , T}{\rightarrow}r_{1}\stackrel{x_{2} , T}{\rightarrow}r_{2}\cdots r_{m-1}\stackrel{x_{m} , T}{\rightarrow}r_{m}, r_{m}\in A_{1}\cup A_{2}, r_{0}, r_{1},\cdots r_{m}\in Q

Therefore, $N$ accepts $x_{1}x_{2}\cdots x_{m}$ and the proof is complete.

Note: The idea drawn from this mathematical proof for constructing a machine to recognize $L_{1}\cup L_{2}$ is to create an initial state and connect it to the initial states of $L_{1}$ and $L_{2}$ using $\epsilon$ arrows.

References

Michael Sipser, Introduction to the Theory of Computation ISBN 0-534-94728-X. (See . Theorem 1.22, section 1.2, pg. 59.)

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