Post's theorem

Definition

Post's theorem is a fundamental result in recursion theory that establishes a clear relationship between recursively enumerable sets and the existence of certain degrees of unsolvability. It essentially shows that there are recursively enumerable sets that cannot be effectively enumerated or decided, providing insight into the limitations of computation and decision problems in mathematics.

5 Must Know Facts For Your Next Test

1. Post's theorem shows that there are infinitely many degrees of unsolvability, meaning that not all problems can be solved by Turing machines, reflecting the limits of computation.
2. The theorem highlights the existence of complete sets for recursively enumerable degrees, illustrating how certain problems can be as hard as any problem in a given degree.
3. In relation to the halting problem, Post's theorem emphasizes that some recursively enumerable sets cannot be effectively determined, which aligns with the undecidability results established by Turing.
4. The construction of particular recursively enumerable sets often relies on the priority method, illustrating how Post's theorem can be applied in practical scenarios within recursion theory.
5. Post's theorem has implications for understanding hyperarithmetical sets and functions, revealing deeper layers of complexity within the arithmetic hierarchy.

Review Questions

• How does Post's theorem illustrate the limitations of computation and decision problems?
  • Post's theorem demonstrates the limitations of computation by showing that there exist recursively enumerable sets which cannot be effectively enumerated or decided. This means there are problems for which no algorithm exists that can provide a definitive yes or no answer for every input. This realization underlines the boundaries of what can be computed and leads to a deeper understanding of undecidability in recursion theory.
• Discuss how Post's theorem relates to Turing degrees and what it implies about complete sets for recursively enumerable degrees.
  • Post's theorem establishes a connection between recursively enumerable sets and Turing degrees by proving that there are complete sets within these degrees. This means that certain problems encapsulate the complexity of all problems in their degree class. Consequently, these complete sets serve as benchmarks for comparing the difficulty of various decision problems, highlighting how they can be equally unsolvable despite differing contexts.
• Evaluate the significance of Post's theorem in relation to hyperarithmetical sets and functions and its broader impact on recursion theory.
  • Post's theorem plays a crucial role in understanding hyperarithmetical sets and functions by illustrating how they fit into a broader framework of computability. The results derived from Post's theorem indicate that even at higher levels of the arithmetical hierarchy, issues related to enumeration and decision remain prevalent. This impact extends beyond individual results to shape our understanding of complexity within mathematical logic, emphasizing the intricate relationships between different classes of problems in recursion theory.

Related terms

Recursively Enumerable Set: A set of natural numbers for which there exists a Turing machine that will list all the elements of the set, though it may not halt for non-members.

Turing Degree: A measure of the level of unsolvability of sets of natural numbers, representing the equivalence classes of sets under Turing reducibility.

Priority Method: A technique used to construct recursively enumerable sets with specific properties by assigning priorities to conditions and ensuring that higher-priority conditions are satisfied over lower ones.