Let $G = (V, \Sigma, S, P)$ be a context-free grammar in Chomsky Normal Form with $\Sigma = \{a, b, c\}$ and $V$ containing 10 variable symbols including the start symbol $S$. The string $w = a^{30} b^{30} c^{30}$ is derivable from $S$. The number of steps (application of rules) in the derivation $S \rightarrow^* w$ is ________

In Chomsky Normal Form (CNF), all production rules are strictly of the following two forms:

1. $A \rightarrow BC$ (where $A, B, C \in V$)

2. $A \rightarrow a$ (where $A \in V, a \in \Sigma$)

For any context-free grammar in CNF:

• To derive a terminal string $w$ of length $n = |w|$, exactly $n - 1$ rule applications of type $A \rightarrow BC$ are required to produce a sentential form of $n$ non-terminals.

• Exactly $n$ rule applications of type $A \rightarrow a$ are required to replace the $n$ non-terminals with terminal symbols.

Thus, the total number of derivation steps for any string of length $n$ is:

Here, the given string is $w = a^{30}b^{30}c^{30}$.