# Automata | Regular Expression

Automata | Regular Expression with automata tutorial, finite automata, dfa, nfa, regexp, transition diagram in automata, transition table, theory of automata, examples of dfa, minimization of dfa, non deterministic finite automata, etc.

# Regular Expression

• The language accepted by finite automata can be easily described by simple expressions called Regular Expressions. It is the most effective way to represent any language.
• The languages accepted by some regular expression are referred to as Regular languages.
• A regular expression can also be described as a sequence of pattern that defines a string.
• Regular expressions are used to match character combinations in strings. String searching algorithm used this pattern to find the operations on a string.

For instance:

In a regular expression, x* means zero or more occurrence of x. It can generate {e, x, xx, xxx, xxxx, .....}

In a regular expression, x+ means one or more occurrence of x. It can generate {x, xx, xxx, xxxx, .....}

## Operations on Regular Language  The various operations on regular language are:

Union: If L and M are two regular languages then their union L U M is also a union.

```  1. L U M = {s | s is in L or s is in M}
```

Intersection: If L and M are two regular languages then their intersection is also an intersection.

```  1. L ⋂ M = {st | s is in L and t is in M}
```

Kleen closure: If L is a regular language then its Kleen closure L1* will also be a regular language.

```  1. L* = Zero or more occurrence of language L.
```

### Example 1:

Write the regular expression for the language accepting all combinations of a's, over the set ∑ = {a}

Solution:

All combinations of a's means a may be zero, single, double and so on. If a is appearing zero times, that means a null string. That is we expect the set of {ε, a, aa, aaa, ....}. So we give a regular expression for this as:

```	R = a*
```

That is Kleen closure of a.

### Example 2:

Write the regular expression for the language accepting all combinations of a's except the null string, over the set ∑ = {a}

Solution:

The regular expression has to be built for the language

```L = {a, aa, aaa, ....}
```

This set indicates that there is no null string. So we can denote regular expression as:

```R = a+
```

### Example 3:

Write the regular expression for the language accepting all the string containing any number of a's and b's.

Solution:

The regular expression will be:

```r.e. = (a + b)*
```

This will give the set as L = {ε, a, aa, b, bb, ab, ba, aba, bab, .....}, any combination of a and b.

The (a + b)* shows any combination with a and b even a null string.