7. Given that L and L' (L complement) are both r.e. languages, prove that L is recursive.

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**Problem 7: Proving Recursiveness of a Language**

Given that \( L \) and \( L' \) (the complement of \( L \)) are both recursively enumerable (r.e.) languages, prove that \( L \) is recursive.

**Explanation:**

- **Recursive Languages:** A language is recursive if there exists a Turing machine that will halt and decide for every input whether that input belongs to the language. In other words, a language is recursive if it is both r.e. and co-r.e.

- **Recursively Enumerable Languages:** A language is recursively enumerable if there exists a Turing machine which will enumerate all the strings in the language; however, it may not halt for strings not in the language.

- **Complement Languages:** The complement of a language \( L \), denoted \( L' \), consists of all strings that are not in \( L \).

**Proof Overview:**

To prove that \( L \) is recursive, given that both \( L \) and \( L' \) are r.e., we can construct a Turing machine that simultaneously enumerates the strings of \( L \) and \( L' \). For any input string, the machine will run the enumeration process for both \( L \) and \( L' \) in parallel:

1. If the string is found in the enumeration of \( L \), the machine halts and accepts.
2. If the string is found in the enumeration of \( L' \), the machine halts and rejects.

Since the machine will definitely halt on every input (as it will find the string either in \( L \) or \( L' \)), this implies \( L \) is recursive.
Transcribed Image Text:**Problem 7: Proving Recursiveness of a Language** Given that \( L \) and \( L' \) (the complement of \( L \)) are both recursively enumerable (r.e.) languages, prove that \( L \) is recursive. **Explanation:** - **Recursive Languages:** A language is recursive if there exists a Turing machine that will halt and decide for every input whether that input belongs to the language. In other words, a language is recursive if it is both r.e. and co-r.e. - **Recursively Enumerable Languages:** A language is recursively enumerable if there exists a Turing machine which will enumerate all the strings in the language; however, it may not halt for strings not in the language. - **Complement Languages:** The complement of a language \( L \), denoted \( L' \), consists of all strings that are not in \( L \). **Proof Overview:** To prove that \( L \) is recursive, given that both \( L \) and \( L' \) are r.e., we can construct a Turing machine that simultaneously enumerates the strings of \( L \) and \( L' \). For any input string, the machine will run the enumeration process for both \( L \) and \( L' \) in parallel: 1. If the string is found in the enumeration of \( L \), the machine halts and accepts. 2. If the string is found in the enumeration of \( L' \), the machine halts and rejects. Since the machine will definitely halt on every input (as it will find the string either in \( L \) or \( L' \)), this implies \( L \) is recursive.
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