Understanding Context-Free Languages: L1, L2, and L3

Let's analyze each of the given languages to determine if they are context-free. A language is context-free if it can be represented by a Context-Free Grammar (CFG) or recognized by a Pushdown Automaton (PDA). Languages involving comparisons like $m=n$, $m>n$, or $m<n$ for the counts of symbols in a specific order (like $a^m b^n$) are typically context-free. Languages involving inequalities like $m \ne n$ or $m \ne 2n$ can often be expressed as the union of two context-free languages (the 'greater than' case and the 'less than' case), and since the union of two context-free languages is context-free, these languages can also be context-free.

Analyzing Language L1: $L_1 = \{a^m b^n \mid m \neq n\}$

Language $L_1$ consists of strings of the form $a^m b^n$ where the number of 'a's ($m$) is not equal to the number of 'b's ($n$). This can be broken down into two cases:

• Case 1: $m > n$. This corresponds to the language $\{a^m b^n \mid m > n\} = \{a^{n+k} b^n \mid k > 0 \}$.
• Case 2: $m < n$. This corresponds to the language $\{a^m b^n \mid m < n\} = \{a^m b^{m+k} \mid k > 0 \}$.

Both languages $\{a^m b^n \mid m > n\}$ and $\{a^m b^n \mid m < n\}$ are context-free. For example, a PDA can recognize $\{a^m b^n \mid m > n\}$ by pushing 'a' for each 'a', then popping 'a' for each 'b'. If the stack is not empty after processing all 'b's, the string is accepted. Similarly, a PDA can recognize $\{a^m b^n \mid m < n\}$. Since $L_1$ is the union of these two context-free languages, $L_1$ is also context-free.

Analyzing Language L2: $L_2 = \{a^m b^n \mid m = 2n + 1\}$

Language $L_2$ consists of strings where the number of 'a's ($m$) is exactly one more than twice the number of 'b's ($n$). This can be written as $\{a^{2n+1} b^n \mid n \ge 0\}$. This is a standard pattern for context-free languages. A PDA can recognize this language. For instance, for each 'a', push one symbol onto the stack. Then, for each 'b', pop two symbols from the stack. After processing the entire string, if the stack contains exactly one symbol, the string is accepted. Alternatively, push two symbols for each 'a' and one for each 'b', matching appropriately. This structure is clearly recognizable by a PDA, proving $L_2$ is context-free.

Analyzing Language L3: $L_3 = \{a^m b^n \mid m \neq 2n\}$

Language $L_3$ consists of strings of the form $a^m b^n$ where the number of 'a's ($m$) is not equal to twice the number of 'b's ($n$). Similar to $L_1$, this can be broken down into two cases:

• Case 1: $m > 2n$. This corresponds to the language $\{a^m b^n \mid m > 2n\} = \{a^{2n+k} b^n \mid k > 0 \}$.
• Case 2: $m < 2n$. This corresponds to the language $\{a^m b^n \mid m < 2n\} = \{a^m b^n \mid 2n > m \}$.

Both languages $\{a^m b^n \mid m > 2n\}$ and $\{a^m b^n \mid m < 2n\}$ are context-free. A PDA can recognize $\{a^m b^n \mid m > 2n\}$ by pushing symbols for 'a' and popping two symbols for each 'b'. If symbols remain on the stack after 'b's, accept. Similarly, a PDA can recognize $\{a^m b^n \mid m < 2n\}$. Since $L_3$ is the union of these two context-free languages, $L_3$ is also context-free.

Conclusion on Context-Free Nature

Based on the analysis, languages $L_1$, $L_2$, and $L_3$ can all be recognized by a Pushdown Automaton or generated by a Context-Free Grammar. Therefore, all three languages are context-free.

The statement that $L_1$, $L_2$, and $L_3$ are context free languages is correct.

Revision Table: Language Properties

Additional Information: Context-Free Languages and Automata

Context-Free Languages (CFLs) are a class of formal languages that are widely studied in computer science, particularly in the fields of automata theory and formal language theory. They are defined by Context-Free Grammars (CFGs). A key property of CFLs is their equivalence to languages accepted by Pushdown Automata (PDAs).

• Context-Free Grammar (CFG): A CFG consists of a set of production rules that describe how symbols can be combined to form strings in the language. Rules have the form $A \rightarrow \beta$, where $A$ is a single non-terminal symbol and $\beta$ is a string of terminals and/or non-terminals.
• Pushdown Automaton (PDA): A PDA is a type of automaton that is similar to a finite automaton but with an additional memory component called a stack. The stack allows the PDA to remember arbitrarily deep information about the input seen so far, which is crucial for recognizing languages that require counting or matching, like $a^n b^n$.

Languages like $\{a^n b^n\}$, $\{a^n b^{2n}\}$, $\{a^n b^n c^n\}$ are classic examples used to illustrate the boundary between context-free and non-context-free languages. While $\{a^n b^n\}$ and $\{a^n b^{2n}\}$ are CFLs (as a PDA can match the counts using the stack), $\{a^n b^n c^n\}$ is not context-free because a single stack is insufficient to simultaneously match the counts of three symbols like this.

The languages in this question, $L_1$, $L_2$, and $L_3$, involve counts of only two distinct symbols ($a$ and $b$) appearing in a specific order ($a^*$ followed by $b^*$). This structure is well-suited for recognition by a PDA.