Recursively enumerable set

Definition

A recursively enumerable set is a collection of elements for which there exists a Turing machine that will list all the members of the set, possibly without ever halting if the element is not in the set. This concept connects closely to the limitations of formal systems, illustrating the boundaries of what can be computed or decided within those systems. It highlights that while you can enumerate members of such a set, determining membership for arbitrary elements may be undecidable.

5 Must Know Facts For Your Next Test

1. Every finite set is recursively enumerable, as you can list all its elements in a finite amount of time.
2. All recursive sets are recursively enumerable, but not all recursively enumerable sets are recursive; some may not have an algorithm to decide membership.
3. The existence of a Turing machine that lists the elements of a recursively enumerable set does not guarantee it will stop for every input.
4. The relationship between recursively enumerable sets and the Halting Problem illustrates fundamental limits on computation and decidability.
5. Recursively enumerable sets play a crucial role in the theory of computation and help in understanding the boundaries between what can be computed and what cannot.

Review Questions

• How does the concept of recursively enumerable sets illustrate the limitations of formal systems?
  • Recursively enumerable sets demonstrate the limitations of formal systems by showing that while you can generate members of certain sets using algorithms, you may not be able to determine if arbitrary elements belong to those sets. This distinction highlights that not every property can be computed or decided within formal systems, especially when faced with undecidable problems like the Halting Problem.
• Compare recursively enumerable sets and decidable sets in terms of their computational properties.
  • Recursively enumerable sets can be generated by a Turing machine that lists their members, but this process may never halt for non-members. In contrast, decidable sets allow for an algorithm to determine membership for any element in finite time. Therefore, while all decidable sets are recursively enumerable, not all recursively enumerable sets are decidable due to their potentially infinite listing nature.
• Evaluate the implications of recursively enumerable sets on understanding computational limits and algorithms.
  • The existence of recursively enumerable sets reveals significant insights into computational limits by demonstrating that certain problems cannot be solved by algorithms. This understanding is critical when considering real-world computing applications, as it emphasizes that some questions—like whether a Turing machine halts—are fundamentally unresolvable. This distinction informs how researchers approach problem-solving in computer science and mathematics, fostering an appreciation for what is computably achievable versus what remains elusive.

Related terms

Turing Machine: An abstract computational model that defines an algorithm's operation, consisting of a tape, a head for reading and writing symbols, and a set of states.

Decidable Set: A set for which there exists an algorithm that can determine whether any given element is a member of the set in a finite amount of time.

Halting Problem: A decision problem that asks whether a given Turing machine will halt or run forever on a specific input; it is proven to be undecidable.