Properties of R.E Sets Video Lecture | Question Bank for GATE Computer Science Engineering - Computer Science Engineering (CSE)

Ans. Recursively enumerable (R.E) sets have the following properties: - They can be recognized by Turing machines, meaning there exists a Turing machine that halts and accepts if the input is in the set, but may run indefinitely if the input is not in the set. - They are also known as semidecidable sets because there is no guarantee that a Turing machine will halt and reject if the input is not in the set. - R.E sets are closed under union, meaning if two R.E sets exist, their union will also be an R.E set. - They are not closed under complement, i.e., the complement of an R.E set may not be an R.E set. - R.E sets are recursively enumerable because there exists an algorithm (Turing machine) that can list all the elements of the set.

Ans. Recursively enumerable (R.E) sets are recognized by Turing machines through a two-phase process. In the first phase, the Turing machine scans the input and verifies if it belongs to the set. If it does, the machine halts and accepts the input. In the second phase, if the input does not belong to the set, the machine may run indefinitely or enter an infinite loop. This is because R.E sets are not guaranteed to be decidable, meaning there is no guarantee that the machine will halt and reject if the input is not in the set. Therefore, the Turing machine recognizes R.E sets by either halting and accepting or running indefinitely.

Ans. No, recursively enumerable (R.E) sets are not the same as decidable sets. R.E sets are sets for which there exists a Turing machine that halts and accepts if the input is in the set, but may run indefinitely if the input is not in the set. In other words, R.E sets are semidecidable. On the other hand, decidable sets are sets for which there exists a Turing machine that halts and either accepts or rejects every input. Decidable sets are also known as computable sets. Therefore, while all decidable sets are recursively enumerable, not all recursively enumerable sets are decidable.

Ans. No, the complement of a recursively enumerable (R.E) set may not be an R.E set. R.E sets are not closed under complement, meaning that given an R.E set, there is no guarantee that its complement will also be an R.E set. This is because if there exists a Turing machine that recognizes the original R.E set, there is no guarantee that another Turing machine can recognize its complement. Therefore, the complement of an R.E set can be undecidable or not recursively enumerable.

Ans. The term "recursively enumerable" in the context of R.E sets is related to the concept of recursion in computer science. Recursion refers to the process of defining a function or algorithm in terms of itself. Similarly, R.E sets are sets that can be enumerated or listed using an algorithm (Turing machine). The concept of recursion is utilized in the design of Turing machines that recognize R.E sets. The Turing machine can repeatedly apply a set of rules or instructions to generate or enumerate the elements of the R.E set. Therefore, while the concept of recursion is not limited to R.E sets, it is closely related as it enables the enumeration of elements in an R.E set.