Regular Expressions in Automata Theory

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A regular expression is basically a shorthand way of showing how a regular language is built from the base set of regular languages.

The symbols are identical which are used to construct the languages, and any given expression that has a language closely associated with it.

A Regular Expression can be recursively defined as follows −

• ε is a Regular Expression indicates the language containing an empty string. (L (ε) = {ε})
• φ is a Regular Expression denoting an empty language. (L (φ) = { })
• x is a Regular Expression where L = {x}
• If X is a Regular Expression denoting the language L(X) and Y is a Regular Expression denoting the language L(Y), then
  • X + Y is a Regular Expression corresponding to the language L(X) ∪ L(Y) where L(X+Y) = L(X) ∪ L(Y).
  • X . Y is a Regular Expression corresponding to the language L(X) . L(Y) where L(X.Y) = L(X) . L(Y)
  • R* is a Regular Expression corresponding to the language L(R*)where L(R*) = (L(R))*
• If we apply any of the rules several times from 1 to 5, they are Regular Expressions.

Examples of Regular Expressions

Example 1

If the regular expression is as follows −

$$\mathrm{a \:+\: b \: \cdot \: a*}$$

It can be written in fully parenthesized form as follows −

$$\mathrm{(a \:+\: (b \: \cdot \: (a*)))}$$

Regular Expressions Vs Languages

The symbols of the regular expressions are distinct from those of the languages. These symbols are given below −

Operators in Regular Expression

There are two binary operations on regular expressions (+ and ·) and one unary operator (*)

These are closely associated with the union, product and closure operations on the corresponding languages.

Example 2

The regular expression a + bc* is basically shorthand for the regular language {a} ∪ ({b} · ({c}*)).

Find the language of the given regular expression. It is explained below −

$$\mathrm{L(a \:+\: bc*) \\ \:\:\:\:\:=\: L(a)\: \cup\: L(bc*)\: \\ \:\:\:\:\:=\: L(a)\: \cup\: (L(b)\: \cdot \:L(c*)) \\ \:\:\:\:\:=\: L(a)\: \cup\: (L(b)\: \cdot \:L(c)*) \\ \:\:\:\:\:=\: \{a\}\: \cup\: (\{b\}\: \cdot \:\{c\}*) \\ \:\:\:\:\:=\: \{a\}\: \cup\: (\{b\}\: \cdot \:\{\wedge,\: c,\: c2,\: \dotso ,\: cn,\: \dotso , \}) \\ \:\:\:\:\:=\: \{a\}\: \cup\: \{b,\: bc,\: bc2,\: . . . ,\: bcn,\: . . . \} \\ \:\:\:\:\:=\: \{a,\: b,\: bc,\: bc2,\: \dotso ,\: bcn,\: \dotso \}}$$

Properties of Regular Expressions in TOC

All the properties held for any regular expressions R, E, F and can be verified by using properties of languages and sets.

Additive (+) Properties

The additive properties of regular expressions are as follows −

R + E = E + R

R + ∅ = ∅ + R = R

R + R = R

(R + E) + F = R + (E + F)

Product (.) Properties

The product properties of regular expressions are as follows −

R∅ = ∅R = ∅

R∧ = ∧R = R

(RE)F = R(EF)

Distributive Properties

The distributive properties of regular expressions are as follows −

R(E + F) = RE + RF

(R + E)F = RF + EF

Closure Properties

The closure properties of regular expressions are as follows −

∅* = ∧ * = ∧

R* = R*R* = (R*)* = R + R*

R* = ∧ + RR* = (∧ + R)R*

RR* = R*R

R(ER)* = (RE)*R

(R + E)* = (R*E*)* = (R* + E*)* = R*(ER*)*

All the properties can be verified by using the properties of languages and sets.

Example 1

Show that,

(∅ + a + b)* = a*(ba*)*

Using the properties above −

(∅ + a + b)* = (a + b)* (+ property)

= a*(ba*)* (closure property)

Example 2

Show that,

∧ + ab + abab(ab)* = (ab)*

Using the properties above −

∧ + ab + abab(ab)* = ∧ + ab(∧ + ab(ab)*)

= ∧ + ab((ab)*) (using R* = ∧ + RR*)

= ∧ + ab(ab)*= (ab)* (using R* = ∧ + RR* again)

Regular Expressions and Regular Set