Formal Languages Questions Long
Recursively enumerable sets, also known as recursively enumerable languages, are sets of strings that can be recognized by a Turing machine. These sets can be manipulated and operated upon using various operations. The operations that can be performed on recursively enumerable sets are as follows:
1. Union: The union of two recursively enumerable sets A and B is the set that contains all the strings that belong to either A or B. In other words, if a string is in A or B, it is in the union of A and B.
2. Intersection: The intersection of two recursively enumerable sets A and B is the set that contains all the strings that belong to both A and B. In other words, if a string is in both A and B, it is in the intersection of A and B.
3. Complement: The complement of a recursively enumerable set A is the set that contains all the strings that do not belong to A. In other words, if a string is not in A, it is in the complement of A.
4. Concatenation: The concatenation of two recursively enumerable sets A and B is the set that contains all the strings that can be formed by concatenating a string from A with a string from B. For example, if A = {a, b} and B = {c, d}, then the concatenation of A and B is {ac, ad, bc, bd}.
5. Kleene Star: The Kleene star operation on a recursively enumerable set A is the set that contains all possible concatenations of zero or more strings from A. For example, if A = {a, b}, then the Kleene star of A is {ε, a, b, aa, ab, ba, bb, aaa, ...} where ε represents the empty string.
6. Difference: The difference between two recursively enumerable sets A and B is the set that contains all the strings that belong to A but do not belong to B. In other words, if a string is in A but not in B, it is in the difference of A and B.
These operations allow us to manipulate and combine recursively enumerable sets in various ways, enabling us to perform computations and solve problems in formal languages and automata theory.