5 Steps To Convert To Chomsky Normal Form

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Converting a context-free grammar to Chomsky Normal Form (CNF) is a crucial step in many applications of formal language theory, such as parsing and compiler design. In this article, we will delve into the process of converting a context-free grammar to Chomsky Normal Form, exploring the key steps involved and providing practical examples to illustrate each step.

Chomsky Normal Form is a standardized way of representing context-free grammars, where all production rules are in one of the following forms: A → BC or A → a, where A, B, and C are non-terminal symbols, and a is a terminal symbol. This normalization process is essential for efficient parsing and analysis of context-free languages.

Step 1: Remove Null Productions

The first step in converting a context-free grammar to Chomsky Normal Form is to remove all null productions. A null production is a production rule of the form A → ε, where A is a non-terminal symbol and ε represents the empty string.

To remove null productions, we need to identify all non-terminal symbols that can produce the empty string. We can do this by creating a set of nullable non-terminals, which we can then use to eliminate null productions from the grammar.

For example, consider the following context-free grammar:

S → AB A → aA | ε B → bB | ε

In this grammar, both A and B are nullable non-terminals, as they can produce the empty string. To remove null productions, we can add new production rules that allow A and B to produce the empty string explicitly:

S → AB | A | B A → aA | a B → bB | b

By adding these new production rules, we have effectively removed the null productions from the grammar.

Step 2: Remove Unit Productions

The second step in converting a context-free grammar to Chomsky Normal Form is to remove all unit productions. A unit production is a production rule of the form A → B, where A and B are non-terminal symbols.

To remove unit productions, we need to identify all unit productions in the grammar and replace them with new production rules that avoid unit productions. We can do this by creating a set of unit production rules, which we can then use to eliminate unit productions from the grammar.

For example, consider the following context-free grammar:

S → AB A → B B → bB | ε

In this grammar, the production rule A → B is a unit production. To remove this unit production, we can replace it with a new production rule that avoids unit productions:

S → AB A → bB | ε B → bB | ε

By replacing the unit production with a new production rule, we have effectively removed the unit production from the grammar.

Step 3: Replace Mixed Strings

The third step in converting a context-free grammar to Chomsky Normal Form is to replace all mixed strings with new non-terminal symbols. A mixed string is a production rule of the form A → αβ, where α is a non-empty string of terminal symbols and β is a non-empty string of non-terminal symbols.

To replace mixed strings, we need to identify all mixed strings in the grammar and create new non-terminal symbols to replace them. We can then use these new non-terminal symbols to rewrite the production rules in a standardized form.

For example, consider the following context-free grammar:

S → aBc B → bB | ε

In this grammar, the production rule S → aBc is a mixed string. To replace this mixed string, we can create a new non-terminal symbol, C, and use it to rewrite the production rule:

S → aC C → Bc B → bB | ε

By replacing the mixed string with a new non-terminal symbol, we have effectively standardized the production rule.

Step 4: Shorten Right-Hand Sides

The fourth step in converting a context-free grammar to Chomsky Normal Form is to shorten all right-hand sides to length 2. A right-hand side is the string of symbols on the right-hand side of a production rule.

To shorten right-hand sides, we need to identify all production rules with right-hand sides of length greater than 2 and create new non-terminal symbols to shorten them. We can then use these new non-terminal symbols to rewrite the production rules in a standardized form.

For example, consider the following context-free grammar:

S → aBcD B → bB | ε C → cC | ε D → dD | ε

In this grammar, the production rule S → aBcD has a right-hand side of length 4. To shorten this right-hand side, we can create new non-terminal symbols, E and F, and use them to rewrite the production rule:

S → aE E → BF F → cD B → bB | ε C → cC | ε D → dD | ε

By shortening the right-hand side, we have effectively standardized the production rule.

Step 5: Finalize Chomsky Normal Form

The final step in converting a context-free grammar to Chomsky Normal Form is to ensure that all production rules are in one of the two standardized forms: A → BC or A → a, where A, B, and C are non-terminal symbols, and a is a terminal symbol.

To finalize Chomsky Normal Form, we need to review all production rules and ensure that they are in one of the two standardized forms. We can do this by checking that all production rules have right-hand sides of length 2 and that all non-terminal symbols produce either two non-terminal symbols or a single terminal symbol.

By following these five steps, we can convert a context-free grammar to Chomsky Normal Form, ensuring that all production rules are in a standardized form that is efficient for parsing and analysis.

In conclusion, converting a context-free grammar to Chomsky Normal Form is an essential step in many applications of formal language theory. By following the five steps outlined in this article, we can ensure that our grammars are in a standardized form that is efficient for parsing and analysis. Whether you are a student of formal language theory or a practitioner in the field, understanding Chomsky Normal Form is crucial for working with context-free grammars.

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