What is the difference between a recursively enumerable and recursive language?

What is the difference between a recursively enumerable and recursive language?

The difference between a recursively enumerable language and a recursive language lies in their acceptance properties.

A recursively enumerable language, also known as a partially decidable language, is a language for which there exists a Turing machine that can accept all the strings in the language, but may either reject or loop indefinitely on strings not in the language. In other words, there is a Turing machine that can enumerate all the strings in the language, but it may not halt on strings outside the language.

On the other hand, a recursive language, also known as a decidable language, is a language for which there exists a Turing machine that can accept all the strings in the language and reject all the strings not in the language. In this case, the Turing machine will always halt, either accepting or rejecting the input string.

In summary, the main difference is that a recursively enumerable language may have strings that cause the Turing machine to loop indefinitely, while a recursive language guarantees that the Turing machine will always halt.