Classification of Context Free Grammars for GATE CSE Exam

Deciphering the Classification of Context Free Grammars

The classification of CFGs hinges on two distinct properties:

1) Classification Based on the Number of Generated Strings

• A CFG is deemed non-recursive if it yields a finite set of strings, thereby becoming a non-recursive grammar.
• A CFG is considered complete if it can generate an infinite array of strings, hence the term repeating grammar.

During the compilation process, the parser forms a derivation tree or a parse tree from the source code, utilizing the language’s grammar. The grammar must be devoid of ambiguity to ensure accurate parsing.

2) Classification Based on the Number of Derivation Trees

• A CFG is termed clear/unambiguous if there exists only one derivation tree.
• A CFG is labeled unclear/ambiguous if it can generate multiple derivation trees.

Illustrative Examples

Recursive Grammars

1) A->AxA

A->z

The language generated by the above grammar would be: {z, zxz, zxzxz,…}, which is infinite.

2) A-> Yy

Y->Bz|w

The language generated by the above grammar would be: {wy, wzy, wzzzy …}, which is infinite.

Note: A recursive CFG that is devoid of any useless rules will invariably produce an infinite language.

Non-Recursive Grammars

A->Yy

Y->z|w

The language generated by the above grammar would be: {zy, wy}, which is finite.

Different Types of Recursive Grammars

Recursive CFGs can be further divided based on the nature of recursion in the grammar:

• Left Recursive Grammar (possessing left recursion)
• Right Recursive Grammar (bearing right recursion)
• General Recursive Grammar (containing general recursion)

Note: A linear grammar is a special type of CFG that generates sentences with no more than one non-terminal on the right side.

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Further Reading:

• Context Free Grammar
• Context Sensitive Grammar (CSG)
• Context Sensitive Language (CSL)
• Undecidable Problem about Turing Machine