**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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Answered: 7. Given that L and L' (L complement)…

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

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Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

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Concept explainers

Computational Systems

The computation systems can be defined as the systems that are capable of solving a problem that includes calculations either mathematical or logical, and are able to produce the result as an output. For example, a simple calculator that is given a set of numbers and operators as input and produces the result as an output can be defined as a computational system. The above example is a simple illustration only, the computational systems range from simple devices to complex devices such as the systems that control the SpaceX companies, Starlink satellites which are being quite recently seen in the skies at many places globally.

Question

**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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