Understanding Context-Free Languages and Language Operations

The question asks us to consider a specific language, L, defined as the set of strings with $n$ 'a's followed by $n$ 'b's, where $n$ is any non-negative integer. We need to determine which statements about languages derived from L are correct, specifically regarding their context-free nature.

The language is given by $L = \{a^n b^n | n \ge 0\}$. This language includes strings like the empty string ($\varepsilon$, when $n=0$), $ab$ (when $n=1$), $aabb$ (when $n=2$), $aaabbb$ (when $n=3$), and so on. It is a well-known example of a context-free language (CFL). We can generate this language using a context-free grammar, for example: $S \to aSb \mid \varepsilon$.

Analyzing Statement S1: L squared ($L^2$)

Statement S1 says that $L^2$ is a context-free language. The language $L^2$ is defined as the concatenation of L with itself, i.e., $L \cdot L$. A string is in $L^2$ if it can be formed by taking a string from L and appending another string from L.

Let $w_1 \in L$ and $w_2 \in L$. Then $w_1 = a^{n_1} b^{n_1}$ for some $n_1 \ge 0$, and $w_2 = a^{n_2} b^{n_2}$ for some $n_2 \ge 0$. The concatenation is $w_1 w_2 = a^{n_1} b^{n_1} a^{n_2} b^{n_2}$. So, $L^2 = \{a^{n_1} b^{n_1} a^{n_2} b^{n_2} | n_1 \ge 0, n_2 \ge 0\}$.

A fundamental property of context-free languages is closure under concatenation. If $L_1$ and $L_2$ are CFLs, then their concatenation $L_1 \cdot L_2$ is also a CFL.

Since L is a CFL, and $L^2 = L \cdot L$, the language $L^2$ must also be a context-free language.

Therefore, Statement S1 is true.

Analyzing Statement S2: L to the power k ($L^k$)

Statement S2 says that $L^k$ is a context-free language for any given $k \ge 1$. The language $L^k$ is the concatenation of L with itself, k times: $L^k = L \cdot L \cdot \dots \cdot L$ (k times).

A string in $L^k$ has the form $w_1 w_2 \dots w_k$, where each $w_i \in L$. $w_i = a^{n_i} b^{n_i}$ for some $n_i \ge 0$. So, a string in $L^k$ looks like $a^{n_1} b^{n_1} a^{n_2} b^{n_2} \dots a^{n_k} b^{n_k}$ for $n_1, n_2, \dots, n_k \ge 0$.

As established in the analysis of S1, the concatenation of two CFLs is a CFL. By extending this property, the concatenation of any finite number of CFLs is also a CFL.

Since L is a CFL, and $L^k$ is the concatenation of a finite number (k) of copies of L, $L^k$ must also be a context-free language for any $k \ge 1$.

Therefore, Statement S2 is true.

Analyzing Statement S3: Complement of L ($\bar{L}$) and Kleene Star of L ($L^*$)

Statement S3 says that $\bar{L}$ and $L^*$ are context-free languages. We need to examine both parts separately.

Kleene Star ($L^*$)

The language $L^*$ is the set of all strings formed by concatenating zero or more strings from L. This includes the empty string, all strings in L, all strings in $L^2$, all strings in $L^3$, and so on. $L^* = \bigcup_{i=0}^{\infty} L^i = \{\varepsilon\} \cup L \cup L^2 \cup L^3 \cup \dots$

A fundamental property of context-free languages is closure under the Kleene star operation. If L is a CFL, then $L^*$ is also a CFL.

Since L is a CFL, its Kleene star $L^*$ must also be a context-free language.

Complement ($\bar{L}$)

The language $\bar{L}$ is the complement of L, meaning it contains all strings over the alphabet $\Sigma = \{a, b\}$ that are not in L. $\bar{L} = \Sigma^* \setminus L$.

The class of context-free languages is generally NOT closed under complementation. However, a special subclass of CFLs, called Deterministic Context-Free Languages (DCFLs), IS closed under complementation.

The language $L = \{a^n b^n | n \ge 0\}$ is a Deterministic Context-Free Language. A deterministic pushdown automaton (DPDA) can recognize this language by pushing 'a's onto the stack and then matching 'b's with the stacked 'a's.

Since L is a DCFL, its complement $\bar{L}$ is also a DCFL. All DCFLs are also CFLs.

Alternatively, we can describe $\bar{L}$ as the set of strings that are not of the form $a^n b^n$. These are strings containing 'ba' OR strings of the form $a^i b^j$ where $i \ne j$. $\bar{L} = \{w \in \{a,b\}^* \mid w \text{ contains } ba \} \cup \{ a^i b^j \mid i \ne j \}$. The language $\{w \in \{a,b\}^* \mid w \text{ contains } ba \}$ is regular (and thus CFL). The language $\{ a^i b^j \mid i \ne j \}$ is also CFL. It can be written as $\{ a^i b^j \mid i > j \} \cup \{ a^i b^j \mid i < j \}$. Both $\{ a^i b^j \mid i > j \}$ and $\{ a^i b^j \mid i < j \}$ are CFLs, and the union of two CFLs is a CFL. Since $\bar{L}$ is the union of two CFLs, it is a CFL.

Both $\bar{L}$ and $L^*$ are context-free languages.

Therefore, Statement S3 is true.

Conclusion on Context-Free Language Properties

Based on the analysis of each statement:

• Statement S1: $L^2$ is context-free. (True)
• Statement S2: $L^k$ is context-free for any $k \ge 1$. (True)
• Statement S3: $\bar{L}$ and $L^*$ are context-free. (True)

All three statements are correct regarding the language $L = \{a^n b^n | n \ge 0\}$.

Revision Table: Language Properties

Additional Information: Context-Free Language Closure Properties

Understanding the closure properties of language classes is crucial in formal language theory. Closure under an operation means that if you apply the operation to languages within that class, the resulting language is also within the same class.

• Union: If $L_1$ and $L_2$ are CFLs, $L_1 \cup L_2$ is a CFL.
• Concatenation: If $L_1$ and $L_2$ are CFLs, $L_1 \cdot L_2$ is a CFL.
• Kleene Star: If L is a CFL, $L^*$ is a CFL.
• Homomorphism: If L is a CFL and $h$ is a homomorphism, $h(L)$ is a CFL.
• Inverse Homomorphism: If L is a CFL and $h$ is a homomorphism, $h^{-1}(L)$ is a CFL.
• Intersection with Regular Language: If L is a CFL and R is a regular language, $L \cap R$ is a CFL.

CFLs are NOT closed under:

• Intersection: The intersection of two CFLs is not necessarily a CFL (e.g., $\{a^n b^n c^m\}$ and $\{a^m b^n c^n\}$ are CFLs, but their intersection $\{a^n b^n c^n\}$ is not).
• Complement: The complement of a CFL is not necessarily a CFL. However, DCFLs are closed under complementation. Since $L = \{a^n b^n | n \ge 0\}$ is a DCFL, its complement is a CFL.