Describe the concept of recursively enumerable sets.

Recursively enumerable sets, also known as computably enumerable sets or simply r.e. sets, are a fundamental concept in the field of formal languages and computability theory. These sets play a crucial role in understanding the limits of computation and the properties of formal languages.

A set is said to be recursively enumerable if there exists a Turing machine that can enumerate its elements. In other words, a set is recursively enumerable if there is an algorithm that can generate a sequence of strings, where each string either belongs to the set or does not belong to the set. This algorithm may not halt for strings that do not belong to the set, but it will eventually produce all the strings that do belong to the set.

Formally, a set A is recursively enumerable if there exists a Turing machine M such that:
1. If x belongs to A, then M will eventually halt and accept x.
2. If x does not belong to A, then M may either halt and reject x, or it may run indefinitely without halting.

The key idea behind recursively enumerable sets is that they can be effectively generated or recognized by a Turing machine. This means that there is an algorithmic procedure that can be followed to generate or recognize the elements of the set. However, it does not guarantee that there is an algorithm that can decide whether a given string belongs to the set or not. In other words, recursively enumerable sets may contain undecidable or partially decidable problems.

Recursively enumerable sets have several important properties. Firstly, they are closed under union, intersection, and complementation. This means that if two sets are recursively enumerable, their union, intersection, and complement are also recursively enumerable. Secondly, recursively enumerable sets can be enumerated in any order, meaning that there are infinitely many possible ways to enumerate the elements of a recursively enumerable set. Finally, recursively enumerable sets are not necessarily decidable, meaning that there may not exist an algorithm that can decide whether a given string belongs to the set or not.

In summary, recursively enumerable sets are sets that can be effectively generated or recognized by a Turing machine. They are an important concept in formal languages and computability theory, providing insights into the limits of computation and the properties of formal languages.