Chomsky Normal Form (CNF)

1. A → BC: A non-terminal produces two non-terminals.
2. A → a: A non-terminal produces a single terminal.
3. S → ε: The start symbol can produce the empty string, but only if ε is part of the language.

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   Conversion Steps:
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• Eliminate ε-Productions: Remove all productions of the form A → ε, except when ε is in the language.
• Eliminate Unit Productions: Remove productions of the form A → B, where A and B are non-terminals.
• Convert to Binary Productions: Replace productions with more than two non-terminals (e.g., A → BCD) with binary productions (e.g., A → BX, X → CD).
• Replace Terminals in Mixed Productions: For rules like A → aB, introduce a new non-terminal (e.g., C → a) and rewrite the rule as A → CB.

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Greibach Normal Form (GNF)

• A → aα: A non-terminal produces a terminal followed by a string of non-terminals (α can be empty).

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  Conversion Steps:
• Eliminate Left Recursion: Remove productions where a non-terminal derives itself in the leftmost position.
• Ensure Proper Order: Rewrite the grammar so that the right-hand side of each rule starts with a terminal or empty string.
• Substitute Non-Terminals: Replace non-terminals on the right-hand side with their equivalent productions until all rules conform to GNF.

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Applications of Normal Forms

• Parsing Algorithms: CNF is particularly useful for algorithms like the Cocke-Younger-Kasami (CYK) algorithm, which is a bottom-up parsing method for context-free grammars.
• Theoretical Properties: Normal forms help in proving theoretical properties of languages and grammars, making them essential in the study of formal languages and automata theory.

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  In summary, normal forms like CNF and GNF standardize context-free grammars, facilitating easier parsing and analysis in computational models. Understanding these forms is crucial for anyone studying automata theory and formal languages.
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