3 Steps To Convert To Chomsky Normal Form

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The Chomsky Normal Form (CNF) is a fundamental concept in formal language theory, particularly in the context of context-free grammars. It is a standardized way of representing context-free grammars, making it easier to parse and analyze the structure of sentences. In this article, we will delve into the world of Chomsky Normal Form and provide a step-by-step guide on how to convert a given context-free grammar to CNF.

What is Chomsky Normal Form?

The Chomsky Normal Form is a specific format for context-free grammars, where all production rules are in one of the following two forms:

• A → BC, where A, B, and C are non-terminal symbols
• A → a, where A is a non-terminal symbol and 'a' is a terminal symbol

In other words, all production rules in a CNF grammar are either of the form "non-terminal → non-terminal non-terminal" or "non-terminal → terminal". This standardized format simplifies the process of parsing and analyzing 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 find all non-terminal symbols that can be derived from the empty string. We can do this by identifying all production rules of the form A → ε and then finding all non-terminal symbols that can be derived from A.

For example, consider the following context-free grammar:

S → AB A → a B → ε

To remove the null production B → ε, we need to find all non-terminal symbols that can be derived from B. In this case, the only non-terminal symbol that can be derived from B is S. Therefore, we can remove the null production B → ε and add a new production rule S → A.

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 find all production rules of the form A → B and then replace B with its corresponding production rules.

For example, consider the following context-free grammar:

S → A A → a

To remove the unit production S → A, we need to replace A with its corresponding production rule. In this case, the only production rule for A is A → a. Therefore, we can remove the unit production S → A and add a new production rule S → a.

Step 3: Convert to Chomsky Normal Form

The final step in converting a context-free grammar to Chomsky Normal Form is to convert all production rules to the standard form A → BC or A → a.

To do this, we need to find all production rules that are not in the standard form and then convert them to the standard form.

For example, consider the following context-free grammar:

S → aA A → bB B → c

To convert this grammar to Chomsky Normal Form, we need to convert all production rules to the standard form. In this case, we can convert the production rules as follows:

S → aA becomes S → aA' and A' → A A → bB becomes A → bB' B → c becomes B' → c

The resulting grammar in Chomsky Normal Form is:

S → aA' A' → A A → bB' B' → c

In conclusion, converting a context-free grammar to Chomsky Normal Form is a three-step process that involves removing null productions, removing unit productions, and converting all production rules to the standard form. By following these steps, we can simplify the process of parsing and analyzing context-free languages.

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