# Right and Left linear Regular Grammars

Regular grammar is a type of grammar that describes a regular language. A regular grammar is a mathematical object, G, which consists of four components, G = (N, E*,* P, S), where

**N:**non-empty, finite set of non-terminal symbols,**E:**a finite set of terminal symbols, or alphabet, symbols,**P:**a set of grammar rules, each of one having one of the forms- A ⇢ aB
- A⇢ a
- A ⇢∈, Here ∈=empty string, A, B ∈ N, a ∈ ∑

**S**∈ N is the start symbol.

This grammar can be of two forms:

- Right Linear Regular Grammar
- Left Linear Regular Grammar

#### Right Linear Regular Grammar

In this type of regular grammar, all the non-terminals on the right-hand side exist at the rightmost place, i.e; right ends.

**Examples :**

A ⇢ a, A ⇢ aB, A ⇢ ∈ where, A and B are non-terminals, a is terminal, and ∈ is empty string

S ⇢ 00B| 11SB ⇢ 0B| 1B| 0 | 1 where, S and B are non-terminals, and 0 and 1 are terminals

#### Left Linear Regular Grammar

In this type of regular grammar, all the non-terminals on the right-hand side exist at the leftmost place, i.e; left ends.

#### Examples :

A ⇢ a, A ⇢Ba, A ⇢ ∈ where, A and B are non-terminals, a is terminal, and ∈ is empty string

S ⇢B00 |S11 B ⇢B0 |B1 | 0 | 1 where S and B are non-terminals, and 0 and 1 are terminals

#### Left linear to Right Linear Regular Grammar

In this type of conversion, we have to shift all the left-handed non-terminals to right as shown in example given below:

Left linear Right linear A ->Ba A -> abaBB -> ab B -> epsilon OR A -> abBB -> a

So, this can be done to give multiple answers. Example explained above have multiple answers other than the given once.

#### Right linear to Left Linear Regular Grammar

In this type of conversion, we have to shift all the right-handed non-terminals to left as shown in example given below:

Right linear Left linear A -> aBA ->Baba B -> ab B -> epsilon OR A ->Bab B -> a

So, this can be done to give multiple answers. Example explained above have multiple answers other than the given once.