Chomsky Normal Form || Converting CFG to CNF || TOC || FLAT || Theory of Computation

Summary

This video explains how to convert a Context-Free Grammar (CFG) into Chomsky Normal Form (CNF). It outlines the necessary steps, including simplifying the grammar by eliminating useless symbols, epsilon productions, and unit productions. The video also covers replacing terminal symbols with non-terminals and ensuring all productions follow the CNF structure—either a non-terminal producing two non-terminals or a non-terminal producing a single terminal. By following this process, viewers will learn to transform any CFG into a simplified and standardized CNF form, making it easier to analyze or parse in computational settings.

Takeaways

• 😀 Chomsky Normal Form (CNF) is a specific way to express Context-Free Grammar (CFG) where productions are in the form A → BC (two non-terminals) or A → a (a terminal).
• 😀 A CFG is in CNF if the left-hand side of every production has one non-terminal and the right-hand side contains either one terminal or two non-terminals.
• 😀 To convert a CFG to CNF, the grammar must be simplified by eliminating useless symbols, epsilon productions (empty strings), and unit productions (single non-terminal productions).
• 😀 In CNF, all terminals appearing in the right-hand side of a production must be replaced with new non-terminals representing those terminals.
• 😀 Example: For a production like S → ba, replace the terminal symbols 'b' and 'a' with non-terminals C and D (e.g., C → b, D → a), so S → ba becomes S → CD.
• 😀 If a production has more than two non-terminals on the right-hand side (e.g., A → XYZ), it must be split into smaller productions (e.g., A → XY and X → ZE).
• 😀 The process begins by simplifying the CFG, checking for useless symbols, epsilon productions, and unit productions, and removing them if present.
• 😀 After simplification, replace all terminal symbols in the productions with appropriate non-terminals to fit CNF format.
• 😀 Productions with three or more non-terminals are split into multiple productions, each containing only two non-terminals on the right-hand side.
• 😀 After applying the necessary changes, all productions should be in the CNF format, where the left-hand side contains one non-terminal, and the right-hand side is either one terminal or two non-terminals.

Q & A

What is Chomsky Normal Form (CNF)?

Chomsky Normal Form (CNF) is a standard way of representing context-free grammars (CFG). A CFG is in CNF if its production rules are of two types: 1) A production of the form A → BC, where both B and C are non-terminals, and 2) A production of the form A → a, where a is a terminal symbol.

What are the basic rules for a CFG to be in Chomsky Normal Form?

The basic rules for a CFG to be in Chomsky Normal Form are: 1) Each production should either be A → BC (two non-terminals on the right-hand side) or A → a (one terminal on the right-hand side). 2) The left-hand side must contain exactly one non-terminal.

What are the steps involved in converting a CFG into CNF?

The steps involved in converting a CFG into CNF are: 1) Simplify the grammar by eliminating useless symbols, epsilon productions, and unit productions. 2) Replace terminals on the right-hand side of the productions with new non-terminals. 3) Rewrite productions that do not conform to CNF (e.g., productions with more than two non-terminals on the right-hand side).

What are useless symbols in a context-free grammar?

Useless symbols in a context-free grammar are symbols that do not appear in any derivation starting from the start symbol. These can be removed from the grammar as they do not contribute to generating strings of the language.

What are epsilon productions, and how are they handled during CNF conversion?

Epsilon productions are productions of the form A → ε, where ε represents the empty string. During CNF conversion, epsilon productions must be eliminated because CNF does not allow empty strings. This is typically done by removing epsilon productions and adjusting other rules accordingly.

What are unit productions, and how are they eliminated in the CNF conversion process?

Unit productions are productions of the form A → B, where both A and B are non-terminals. To eliminate unit productions, the production A → B is replaced by the rules of B, thus eliminating the intermediate non-terminal.

Why is it necessary to replace terminals on the right-hand side of a production with non-terminals?

It is necessary to replace terminals on the right-hand side with non-terminals to ensure that the production conforms to CNF. CNF requires that the right-hand side of a production consists of either two non-terminals or a single terminal. Replacing terminals ensures this structure is maintained.

In the given example, how do you handle the production S → b A?

In the production S → b A, the terminal symbol 'b' is replaced with a new non-terminal, say C. This results in the production S → C A, where C represents the terminal b. This transformation ensures the production adheres to CNF.

How do you handle the production A → A B during CNF conversion?

The production A → A B has more than two symbols on the right-hand side, which violates CNF. To fix this, a new non-terminal (say F) is introduced to split the production into A → F A and F → B. This ensures the production follows CNF, where the right-hand side contains only two non-terminals.

What does the final CNF for the example grammar look like?

The final CNF for the example grammar consists of productions like: S → C A | E A, A → D | C | F A, F → B, B → D | C, E → r, C → b, and D → a. Each production follows the rules of CNF, with either two non-terminals or a single terminal on the right-hand side.