L = {wcw^R : w ϵ {a, b}^+}, where V = {a,b,c} (note: w^R is the reverse of string w).



The language \( L = \{wcw^R \mid w \in \{a, b\}^+\} \) is defined as a set of strings of the form:

- \( w \) is any non-empty string made up of the alphabet \( \{a, b\} \),
- \( w^R \) is the reverse of the string \( w \),
- There is exactly one occurrence of the character \( c \) in the middle, separating \( w \) and \( w^R \).

### Properties:
1. The string \( w \) must belong to \( \{a, b\}^+ \), meaning it's a non-empty string of \( a \)'s and \( b \)'s. For example, \( w = ab, ba, aab \), etc.
2. The reverse of \( w \), denoted \( w^R \), is obtained by reversing the order of the characters in \( w \). If \( w = ab \), then \( w^R = ba \).
3. The language contains strings where \( w \) and \( w^R \) are separated by the character \( c \), which acts as a delimiter.

### Example Strings:
- If \( w = a \), then \( w^R = a \), so the string is \( aca \).
- If \( w = ab \), then \( w^R = ba \), so the string is \( abcba \).
- If \( w = baa \), then \( w^R = aab \), so the string is \( baacaab \).

### Characteristics of the Language:
- The length of the strings in the language \( L \) is always odd, because \( w \) and \( w^R \) are symmetric around the character \( c \).
- This language is context-free but not regular because it requires memory to "match" the first part of the string \( w \) with its reverse \( w^R \), which is a common property of non-regular languages like palindromes.

### Grammar or Automaton:
- A context-free grammar (CFG) can be constructed to generate this language. The CFG would need to produce the same string before and after the central \( c \), which corresponds to the reverse of the string.
- A pushdown automaton (PDA) could recognize this language by pushing characters of \( w \) onto a stack and then popping them off when reading \( w^R \), using the \( c \) as a trigger to reverse the operation.

Would you like further details on constructing a context-free grammar or a pushdown automaton for this language?

Here are five related questions for deeper understanding:
1. How do we prove that this language is context-free but not regular?
2. What is the formal construction of a context-free grammar for this language?
3. How can we design a pushdown automaton to recognize the language \( L \)?
4. What are the closure properties of context-free languages, and how does this apply to \( L \)?
5. How would the presence of multiple \( c \)'s affect the language?

**Tip**: Regular languages cannot handle matching symbols across unbounded lengths, which is why \( L \) requires a context-free grammar or PDA.

L = {wcw^R : w ϵ {a, b}^+}, where V = {a,b,c} (note: w^R is the reverse of string w).

Explore the formal language L = {wcw^R : w ϵ {a, b}^+}, where w is a non-empty string over {a, b}, and w^R is its reverse. Learn how this language is context-free, but not regular, with examples and explanations on context-free grammars and PDAs. Suitable for advanced computer science students.

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