Chomsky Normal Form (CNF)

Chomsky Normal Form (CNF) is a way to represent a context-free grammar in a simplified form. In CNF, all production rules have the following forms:

1. A -> BC (Binary Rule)
2. A -> a (Terminal Rule)

Where A, B, and C are non-terminal symbols, and a is a terminal symbol.

Finding the Minimum Productions for Length 4

To derive a string of length 4 in Chomsky Normal Form, we need to consider all possible combinations of binary and terminal rules. Let's break down the process step by step:

Step 1: Start by introducing the initial non-terminal symbol S.

Step 2: Generate all possible combinations of binary and terminal rules to derive a string of length 4.

We can represent the length 4 string as follows:

S -> AB
A -> BC
B -> CD
C -> DE

Here, each non-terminal symbol represents a length 1 string. By chaining the non-terminal symbols together, we can derive a string of length 4.

Step 3: Finally, we convert the grammar to Chomsky Normal Form by eliminating unit rules and long rules.

In our case, there are no unit rules or long rules, so we don't need to perform any further conversions.

Calculating the Minimum Productions

To calculate the minimum number of productions, we count the number of binary and terminal rules used in the derivation.

In our case, we have used 4 binary rules and 3 terminal rules, resulting in a total of 7 productions.

Therefore, the correct answer is '7'.

Summary

To derive a string of length 4 in Chomsky Normal Form, we need a minimum of 7 productions. This involves introducing the initial non-terminal symbol, generating all possible combinations of binary and terminal rules, and converting the grammar to Chomsky Normal Form if necessary. By following these steps, we can obtain the desired string length while adhering to the rules of CNF.