The family of recursively enumerable languages is a subset of the family of recursive languages.
The Correct Answer and Explanation is:
Yes, the family of recursively enumerable languages is indeed a subset of the family of recursive languages. To understand this, let’s break down the definitions of these two types of languages.
- Recursively Enumerable Languages (RE):
A language is called recursively enumerable if there exists a Turing machine that can accept all strings in the language. This means the machine will halt for all strings that belong to the language, but for strings that do not belong, the machine may either halt or run forever. In other words, the machine may not halt on inputs outside the language, so it doesn’t guarantee termination for all inputs. - Recursive Languages (R):
A language is recursive if there exists a Turing machine that always halts and correctly decides membership for every string in the language. This means for every string, the machine will eventually stop and tell you whether it belongs to the language or not.
Why RE is a subset of R:
- Every recursive language is recursively enumerable because if we can always decide membership (i.e., the Turing machine halts for every input), we can certainly enumerate the strings in the language by accepting them. Since the machine halts on every input, the enumeration will work correctly.
- However, the converse is not true: not every recursively enumerable language is recursive. For a recursively enumerable language, there may be some strings for which the Turing machine doesn’t halt, which means we can’t always decide membership in finite time. Therefore, recursively enumerable languages are not guaranteed to be recursive.
Conclusion:
The family of recursive languages is strictly larger than the family of recursively enumerable languages. Hence, RE is a subset of R.
