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 ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$, language ${\displaystyle L_{1}\cup L_{2}}$ is regular.

Proof

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

Let

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

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

Construct

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

where

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

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

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

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

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

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

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

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

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

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

${\displaystyle \vdots }$

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

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

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

Furthermore,

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

and

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

The above path can be rewritten as

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

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

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

References

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