Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
Bartleby Related Questions Icon

Related questions

bartleby

Concept explainers

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.
expand button
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.
Expert Solution
Check Mark
Knowledge Booster
Background pattern image
Computer Science
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Text book image
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Text book image
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
Text book image
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Text book image
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Text book image
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education