Automata Theory Questions Long
Recursively enumerable languages, also known as recursively enumerable sets or simply RE languages, are a class of languages in automata theory that have certain properties and limitations. Let's discuss these properties and limitations in detail:
1. Definition: A language L is recursively enumerable if there exists a Turing machine that halts and accepts any string in L, but may either halt and reject or loop indefinitely for strings not in L.
2. Semi-decidability: Recursively enumerable languages are also referred to as semi-decidable languages because there exists a Turing machine that can decide whether a given string is in the language, but it may not halt for strings not in the language. In other words, if a string is in the language, the Turing machine will eventually accept it, but if a string is not in the language, the Turing machine may either reject it or loop indefinitely.
3. Universal Turing machine: There exists a universal Turing machine that can simulate any other Turing machine on any input. This property allows us to construct a Turing machine that can enumerate all strings in a recursively enumerable language. However, this enumeration may not terminate for strings not in the language.
4. Closure properties: Recursively enumerable languages are closed under union, concatenation, and Kleene star operations. This means that if L1 and L2 are recursively enumerable languages, then L1 ∪ L2, L1 · L2, and L1* are also recursively enumerable. However, recursively enumerable languages are not closed under complementation. The complement of a recursively enumerable language may or may not be recursively enumerable.
5. Limitations: Recursively enumerable languages have certain limitations compared to decidable languages. Unlike decidable languages, there is no algorithm that can determine whether a given Turing machine accepts any string or not. This means that there is no algorithm to decide whether a given recursively enumerable language is empty or not. Additionally, there is no algorithm to determine whether two recursively enumerable languages are equal or not.
6. Undecidability: The complement of a recursively enumerable language, known as a co-recursively enumerable language, is not recursively enumerable. This implies that there are languages that are not recursively enumerable, and hence, not semi-decidable.
In conclusion, recursively enumerable languages possess the property of semi-decidability, allowing a Turing machine to accept any string in the language but potentially loop indefinitely for strings not in the language. They have closure properties under union, concatenation, and Kleene star operations, but not under complementation. Recursively enumerable languages have limitations such as the inability to determine emptiness or equality of languages, and the existence of languages that are not recursively enumerable.