Glossary

9.4 Hyperarithmetical reducibility and degrees

4 min read•august 20, 2024

extends the arithmetic hierarchy, offering a finer measure of complexity than . It captures computations using oracles from the hyperarithmetic hierarchy, providing a stronger notion of relative between sets of natural numbers.

classify sets based on their hyperarithmetical complexity. The degree structure is more intricate than Turing degrees, with uncountably many degrees forming a dense partial order. This framework enables deeper analysis of computational complexity in various mathematical domains.

Definition of hyperarithmetical reducibility

  • Hyperarithmetical reducibility is a notion of relative computability between sets of natural numbers that extends the arithmetic hierarchy
  • Captures the idea of computation using an oracle for a set in the hyperarithmetic hierarchy
  • Provides a finer-grained measure of complexity compared to Turing reducibility

Comparison to Turing reducibility

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  • Hyperarithmetical reducibility is a stronger notion than Turing reducibility
  • Every hyperarithmetically reducible set is also Turing reducible, but the converse does not hold
  • There exist sets that are Turing reducible but not hyperarithmetically reducible (e.g., the hyperjump of the empty set)

Hyperarithmetical hierarchy

  • The is a transfinite extension of the arithmetic hierarchy
  • Defined using and the
  • Each level of the hierarchy corresponds to a recursive ordinal and contains sets computable from the hyperjump iterated up to that ordinal

Hyperarithmetical sets

  • A set is hyperarithmetical if it belongs to the hyperarithmetical hierarchy
  • Equivalently, a set is hyperarithmetical if it is Turing reducible to a hyperarithmetical set
  • The form a strict subset of the Δ11\Delta^1_1 sets in the analytical hierarchy

Properties of hyperarithmetical reducibility

  • Hyperarithmetical reducibility satisfies several important properties that make it a well-behaved reducibility notion

Reflexivity

  • Every set is hyperarithmetically reducible to itself
  • Formally, for any set AA, we have A≤hAA \leq_h A, where ≤h\leq_h denotes hyperarithmetical reducibility

Transitivity

  • If a set AA is hyperarithmetically reducible to a set BB, and BB is hyperarithmetically reducible to a set CC, then AA is hyperarithmetically reducible to CC
  • Formally, if A≤hBA \leq_h B and B≤hCB \leq_h C, then A≤hCA \leq_h C

Antisymmetry

  • If a set AA is hyperarithmetically reducible to a set BB, and BB is hyperarithmetically reducible to AA, then AA and BB have the same hyperarithmetical degree
  • Formally, if A≤hBA \leq_h B and B≤hAB \leq_h A, then A≡hBA \equiv_h B, where ≡h\equiv_h denotes hyperarithmetical equivalence

Hyperarithmetical degrees

  • Hyperarithmetical degrees provide a way to classify sets based on their hyperarithmetical complexity

Definition of hyperarithmetical degree

  • The hyperarithmetical degree of a set AA, denoted by deg⁡h(A)\deg_h(A), is the equivalence class of sets that are hyperarithmetically equivalent to AA
  • Two sets AA and BB have the same hyperarithmetical degree if and only if A≤hBA \leq_h B and B≤hAB \leq_h A

Degree structure vs Turing degrees

  • The structure of hyperarithmetical degrees is more complex than the structure of Turing degrees
  • There are uncountably many hyperarithmetical degrees, while there are only countably many Turing degrees
  • The hyperarithmetical degrees form a dense partial order with no minimal or maximal elements

Hyperarithmetical jump operator

  • The hyperarithmetical jump of a set AA, denoted by AhA^h, is the set of indices of computable infinitary formulas true in (Lω1CK,A)(L_{\omega_1^{CK}}, A)
  • The hyperarithmetical jump operator is not monotone with respect to hyperarithmetical reducibility, unlike the Turing jump operator
  • The hyperjump of the empty set, ∅h\emptyset^h, is a natural example of a non-hyperarithmetical set

Relationship to other reducibilities

  • Hyperarithmetical reducibility can be compared to other notions of reducibility in computability theory

Comparison to arithmetic reducibility

  • Arithmetic reducibility is a weaker notion than hyperarithmetical reducibility
  • Every hyperarithmetically reducible set is also arithmetically reducible, but the converse does not hold
  • The form a substructure of the hyperarithmetical degrees

Comparison to constructible reducibility

  • Constructible reducibility, based on the constructible hierarchy LL, is a stronger notion than hyperarithmetical reducibility
  • Every constructibly reducible set is also hyperarithmetically reducible, but the converse does not hold
  • The constructible degrees form a finer hierarchy than the hyperarithmetical degrees

Hyperarithmetical sets and functions

  • Hyperarithmetical sets and functions have several important properties and characterizations

Closure properties

  • The hyperarithmetical sets are closed under complement, union, intersection, and projection
  • The hyperarithmetical functions are closed under composition, primitive recursion, and minimization

Existence of non-hyperarithmetical sets

  • There exist sets that are not hyperarithmetical, such as the hyperjump of the empty set ∅h\emptyset^h
  • The existence of non-hyperarithmetical sets follows from diagonalization arguments and the fact that the hyperarithmetical sets are countable

Characterizations via computable infinitary formulas

  • A set is hyperarithmetical if and only if it is definable by a computable infinitary formula in the language of second-order arithmetic
  • This characterization provides a logical description of the hyperarithmetical sets and connects them to the study of computable infinitary logic

Applications and examples

  • Hyperarithmetical sets and reducibility have applications in various areas of mathematics

Hyperarithmetical sets in analysis

  • Many important sets in real analysis, such as the set of computable real numbers and the set of differentiable , are hyperarithmetical
  • The study of hyperarithmetical sets provides a foundation for computable analysis and the investigation of computability in continuous settings

Hyperarithmetical sets in algebra

  • Hyperarithmetical sets play a role in computable algebra, particularly in the study of computable structures and their properties
  • Examples include the set of computable algebraically closed fields and the set of computable torsion-free abelian groups

Hyperarithmetical sets in topology

  • In computable topology, hyperarithmetical sets are used to classify the complexity of computable topological spaces and their properties
  • The set of computable metric spaces and the set of computable compact subsets of Euclidean space are examples of hyperarithmetical sets in topology

Key Terms to Review (20)

Alan Turing: Alan Turing was a British mathematician and logician, widely regarded as one of the fathers of computer science. His pioneering work laid the foundations for modern computing, particularly through his concepts of algorithms, computation, and the development of the Turing machine, which provides a formal framework for understanding computability and recursion.
Arithmetic degrees: Arithmetic degrees are a way to classify sets of natural numbers based on their definability in the framework of arithmetic, specifically through the use of recursive functions and relations. They provide a measure of complexity for sets, allowing mathematicians to discuss and compare the intricacies of various problems and their solutions in terms of arithmetic reducibility. This concept connects to hyperarithmetical reducibility, where sets can be categorized not just by their complexity but also by how they can be transformed into one another using certain operations.
Computability: Computability refers to the ability to determine whether a problem can be solved by an algorithm or a computational process. This concept is central in understanding the limits of what can be computed, which connects directly to different types of functions, their classifications, and various theoretical frameworks that explore what it means for a function to be computable or non-computable.
Computable functions: Computable functions are functions for which there exists a finite procedure or algorithm that can produce the function's output for any valid input in a finite amount of time. These functions can be represented by algorithms and are foundational in the study of recursion and computability theory, linking to the concepts of the arithmetical hierarchy, hyperarithmetical sets, and their reducibility.
Degree of unsolvability: The degree of unsolvability refers to the classification of decision problems based on their level of solvability by Turing machines. It provides a way to measure how 'unsolvable' certain problems are in comparison to others, revealing a hierarchy among problems based on their computational difficulty. This concept is crucial in understanding the structure of decision problems, the limitations of computation, and the relationships between various degrees of unsolvability.
Enumerability: Enumerability refers to the property of a set that allows its elements to be listed or counted in a sequence, either finite or infinite. In the context of recursion theory, enumerability is crucial in understanding the relationship between various classes of functions and sets, particularly how some sets can be effectively listed while others cannot. This concept plays a significant role in distinguishing between recursive and recursively enumerable sets, as well as understanding hyperarithmetical reducibility and degrees.
Hyperarithmetical degrees: Hyperarithmetical degrees are a classification of sets of natural numbers that can be defined through hyperarithmetical operations, extending the concept of arithmetical degrees. These degrees capture the complexity of sets that can be computed or defined through various levels of recursion and ordinal numbers, particularly those associated with the hyperarithmetical hierarchy. They serve as a bridge between the computable and non-computable realms, highlighting the limits of algorithmic methods in set theory.
Hyperarithmetical hierarchy: The hyperarithmetical hierarchy is a classification of sets and functions based on their definability in terms of recursive functions and ordinals, extending the arithmetical hierarchy. It provides a framework to study the complexity of sets and functions that are not merely computable but can be analyzed through transfinite recursion and ordinal numbers, allowing for deeper insights into their structure and relationships.
Hyperarithmetical reducibility: Hyperarithmetical reducibility is a concept in recursion theory that describes a specific type of relationship between sets of natural numbers, indicating that one set can be effectively transformed into another through hyperarithmetical means. This notion helps categorize sets into degrees of complexity, particularly focusing on those that can be defined by higher-level operations than arithmetic, showcasing the hierarchy of computability beyond mere recursive functions.
Hyperarithmetical sets: Hyperarithmetical sets are a class of sets in the realm of recursion theory that extend beyond the arithmetical hierarchy, defined by their properties of being definable in a certain logical framework involving transfinite induction up to the ordinal $eta_1$. They serve as a bridge between computability and descriptive set theory, helping to explore deeper relationships between various levels of definability and computability.
Hyperjump operator: The hyperjump operator is a fundamental concept in recursion theory that extends the notion of the Turing jump. It allows for the construction of sets that are hyperarithmetical, thereby exploring the degrees of unsolvability and providing a means to compare the complexity of decision problems. The hyperjump operator connects to various levels of definability within the arithmetical hierarchy and plays a crucial role in analyzing hyperarithmetical reducibility.
Kurt Gödel: Kurt Gödel was an Austrian-American logician, mathematician, and philosopher, best known for his groundbreaking work on the incompleteness theorems. His contributions have had a profound impact on various fields, including the limitations of formal systems, computability, and set theory.
Moscovici's Theorem: Moscovici's Theorem is a result in the theory of recursive functions that describes the relationship between hyperarithmetical sets and their degrees of unsolvability. This theorem provides a framework to understand how certain sets can be classified based on their hyperarithmetical reducibility, which is crucial for analyzing the complexity of problems in recursion theory.
Post's theorem: 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.
Priority method: The priority method is a technique used in recursion theory to construct sets or degrees of unsolvability by systematically assigning priorities to different requirements that must be met. This method allows for the creation of computably enumerable sets while ensuring that certain conditions are satisfied, making it a crucial tool in addressing problems like Post's problem and understanding hyperarithmetical reducibility and degrees.
Recursive ordinals: Recursive ordinals are a specific type of ordinal numbers that can be defined or generated by recursive processes. These ordinals serve as important benchmarks in understanding the hierarchy of computable functions and the limits of algorithmic processes. Their significance extends to various concepts like the hyperarithmetical hierarchy, well-orderings, and ordinal notations, making them foundational in the study of mathematical logic and computation.
Recursive sets: Recursive sets are collections of natural numbers that can be defined by a total recursive function, meaning there exists a procedure that can determine membership in the set within a finite amount of time. This concept is significant because recursive sets help categorize problems based on their computability and decidability, linking them to notions of hyperarithmetical sets and functions as well as hyperarithmetical reducibility and degrees.
Sacrificing technique: Sacrificing technique refers to a method of computation in recursive function theory where one aspect of a problem is intentionally compromised to gain a clearer understanding or simpler resolution of another aspect. This approach is particularly useful when dealing with complex structures and hierarchies, allowing mathematicians and computer scientists to focus on essential components without getting lost in intricate details. It can lead to insights about hyperarithmetical reducibility and degrees by clarifying the relationships between different levels of complexity in recursive functions.
Semidecidable sets: Semidecidable sets, also known as recursively enumerable sets, are collections of decision problems for which a Turing machine will halt and accept if a solution exists but may either reject or run indefinitely if no solution exists. This concept is essential in understanding the limits of computation and the classification of problems based on their solvability. Semidecidable sets play a critical role in discussions around hyperarithmetical reducibility and degrees, as they help to classify the complexity of decision problems based on how easily they can be enumerated or recognized by computational processes.
Turing reducibility: Turing reducibility is a concept in computability theory that describes how one problem can be solved using the solution to another problem, with the allowance of Turing machines. If a problem A is Turing reducible to a problem B, it means that there exists an algorithm that can solve A using an algorithm that solves B as a subroutine. This relationship helps to categorize problems based on their computational complexity and establishes connections to the hyperarithmetical hierarchy and degrees of unsolvability.
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