Post's problem

Definition

Post's problem is a fundamental question in mathematical logic and computability theory that asks whether there exists a non-trivial, recursively enumerable set that is not Turing computable. This problem connects deeply with various concepts in recursive functions, particularly how sets relate to each other within the framework of Turing degrees and the hierarchies of hyperarithmetical sets.

5 Must Know Facts For Your Next Test

1. Post's problem specifically highlights the gap between recursively enumerable sets and Turing computable sets, illustrating that there are complexities in defining computability.
2. The priority method was developed as a way to construct examples related to Post's problem by ensuring that various properties can coexist without contradiction.
3. An important aspect of Post's problem is its connection to the concept of degrees of unsolvability, as it helps classify sets based on their computational complexity.
4. Post's problem remains unresolved, which means researchers continue to explore potential non-computable sets and their implications within recursion theory.
5. The investigation into Post's problem has led to significant developments in understanding the structure of Turing degrees and their relationships with hyperarithmetical sets.

Review Questions

• How does Post's problem illustrate the relationship between recursively enumerable sets and Turing computable sets?
  • Post's problem shows that while some recursively enumerable sets can be computed by algorithms, there exist non-trivial examples that cannot be computed at all, revealing a gap between these two categories. This distinction emphasizes that not all problems that can be listed algorithmically are solvable, prompting deeper investigations into the nature of computability.
• Discuss the role of the priority method in addressing Post's problem and constructing relevant examples.
  • The priority method serves as a crucial tool in tackling Post's problem by allowing mathematicians to create specific recursively enumerable sets that fulfill desired properties without leading to contradictions. This method prioritizes certain conditions over others during construction, enabling researchers to demonstrate the existence of non-computable sets and thereby shedding light on the broader implications for computability and recursion theory.
• Evaluate the significance of Post's problem in the context of understanding Turing degrees and hyperarithmetical sets.
  • Post's problem is significant as it directly influences how we comprehend the hierarchy of Turing degrees and their relation to hyperarithmetical sets. By probing into the nuances of what it means for a set to be recursively enumerable but not computable, researchers gain insight into the structure and classification of problems based on their computational difficulty. This exploration not only deepens our understanding of theoretical computer science but also paves the way for future advancements in mathematical logic.

Related terms

Turing degree: A Turing degree is a measure of the level of unsolvability of a set of natural numbers, determining how 'complex' a problem is in terms of Turing computability.

Recursively enumerable set: A recursively enumerable set is a type of set whose members can be listed by an algorithm, although it may not be possible to decide membership for all elements.

Priority method: The priority method is a technique used in recursion theory to construct specific sets or functions that meet certain criteria by prioritizing different requirements during their construction.