Chomsky Classification of Grammar || GATECSE || TOC

Summary

This video provides an in-depth explanation of Type 2 and Type 3 grammars in formal language theory. It explores the structure and rules of Context-Free Grammar (Type 2), where a single variable appears on the left-hand side, and Regular Grammar (Type 3), which has stricter constraints where the variable must appear only in the leftmost or rightmost positions on the right-hand side. The video also covers Left Linear and Right Linear Grammars and sets the stage for understanding how regular languages can be represented through these grammars.

Takeaways

• 😀 **Type 2 Grammar** allows any number of variables and terminals on the right-hand side, but the left-hand side must contain only one variable.
• 😀 **Type 3 Grammar** restricts the right-hand side to contain only one variable, which must appear either in the leftmost or rightmost position, but never in between.
• 😀 In **Type 3 Grammar**, if a variable appears, it must be positioned at the leftmost or rightmost place, not in the middle of the production.
• 😀 **Left Linear Grammar** (LLG) refers to a grammar where the variable appears at the **leftmost** position on the right-hand side of the production.
• 😀 **Right Linear Grammar** occurs when the variable appears at the **rightmost** position of the right-hand side.
• 😀 The combination of both a leftmost and rightmost variable in the same production is **not allowed** in Type 3 Grammar.
• 😀 In Type 3 Grammar, when no variable appears on the right-hand side, any combination of terminals (like 'a', 'b', 'aa') is allowed.
• 😀 The position of the variable in Type 3 Grammar must be either leftmost or rightmost, but **not both simultaneously** in any production.
• 😀 Type 3 grammar can be represented using **regular languages**, and its production rules can be written as left linear or right linear grammar.
• 😀 The next video will cover how to write left linear and right linear grammars for any given regular language and how to map them accordingly.

Q & A

What is the primary distinction between Type 2 and Type 3 grammars?

The primary distinction between Type 2 and Type 3 grammars lies in the structure of their production rules. Type 2 grammar (context-free grammar) allows the left-hand side of a production to contain a single variable, and the right-hand side can contain any combination of variables and terminals. Type 3 grammar (regular grammar), on the other hand, restricts the right-hand side to either a single terminal or a single variable placed at either the leftmost or rightmost position, but not both.

What are the constraints on the variable in Type 3 grammar?

In Type 3 grammar, the variable can only appear in one of two positions: the leftmost or the rightmost position of the right-hand side of a production. The variable cannot be located anywhere else on the right-hand side, nor can there be variables at both ends of the production.

Can Type 3 grammar have both terminals and variables in the right-hand side of a production?

Yes, Type 3 grammar can have both terminals and variables in the right-hand side of a production. However, if a variable is present, it must appear at either the leftmost or rightmost position, and the rest of the symbols can be any combination of terminals.

What is the difference between Left Linear Grammar (LLG) and Right Linear Grammar (RLG)?

Left Linear Grammar (LLG) refers to a Type 3 grammar where the variable appears as the leftmost symbol in the production rule. Right Linear Grammar (RLG), conversely, refers to a Type 3 grammar where the variable appears as the rightmost symbol. Both LLG and RLG are subsets of regular grammar.

Is it allowed to have a variable in both the leftmost and rightmost positions of a production in Type 3 grammar?

No, it is not allowed to have a variable in both the leftmost and rightmost positions of a production in Type 3 grammar. The grammar must have the variable either at the leftmost or rightmost position, but not both simultaneously.

How does the structure of Type 3 grammar relate to regular languages?

Type 3 grammar is directly related to regular languages because the right-hand side of production rules can consist of only terminals and variables placed at the ends (leftmost or rightmost). This structure enables the representation of regular languages, which can be described by finite automata and regular expressions.

Can a Type 3 grammar have multiple variables in a production?

No, a Type 3 grammar can only have a single variable on the right-hand side of a production. The variable must appear at either the leftmost or rightmost position. Multiple variables on the right-hand side are not allowed in Type 3 grammar.

What happens if a Type 3 production has no variables on the right-hand side?

If a Type 3 production has no variables on the right-hand side, it will consist solely of terminals. This is allowed and represents a basic terminal string in the language, which is typical for regular languages.

Why are productions like 'S → aSb' not allowed in Type 3 grammar?

'S → aSb' is not allowed in Type 3 grammar because it places the variable 'S' in the middle of the production, violating the restriction that the variable must only appear at the leftmost or rightmost position. This structure is not permissible in Type 3 grammar, which adheres to the rules of regular languages.

What is the significance of 't star' in Type 3 grammar?

't star' refers to a sequence of zero or more terminals (denoted as t), and it means that a production can contain any number of terminal symbols. This flexibility allows Type 3 grammar to generate regular languages, where any number of terminals can appear in a sequence, as long as the variable appears at one of the ends.