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The Arithmetical Hierarchy: A Realizability-Theoretic Perspective

Takayuki Kihara

Abstract

In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical decision problems only plays a role in counting the number of quantifiers, jumps, or mind-changes. In contrast, we reveal that when the realizability interpretation is combined with many-one reducibility, it becomes possible to classify natural arithmetical problems in a very nontrivial way.

The Arithmetical Hierarchy: A Realizability-Theoretic Perspective

Abstract

In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical decision problems only plays a role in counting the number of quantifiers, jumps, or mind-changes. In contrast, we reveal that when the realizability interpretation is combined with many-one reducibility, it becomes possible to classify natural arithmetical problems in a very nontrivial way.

Paper Structure

This paper contains 22 sections, 52 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Quantifier
  4. Realizability
  5. Dual quantifier
  6. Natural complete problems
  7. Countable structure
  8. $\forall\forall^\infty\forall$: Local finiteness
  9. $\forall\forall^\infty$: Local finiteness for codes
  10. $\forall\exists!\forall$: Universality
  11. $\forall\forall^\infty$: Verifiability
  12. $\forall\exists\forall$-dicomplete
  13. $\forall^{\downarrow}\forall^\infty$: Asymptotic behavior
  14. $\forall^\infty\forall\exists$: Global boundedness
  15. $\forall\exists\forall$: Independent alternation of quantifiers
  16. ...and 7 more sections

Key Result

Proposition 2.9

A quantifier-pattern $\bar{\sf Q}$ is $\Sigma_3$ iff there exists a pattern $\bar{\sf Q}'$ in Example exa:Sigma03-absorb such that $\bar{\sf Q}$ is absorbable into $\bar{\sf Q}'$ and vice versa.

Figures (4)

  • Figure 1: Complexity of $\Pi_3$-complete problems
  • Figure 2: Example of absorption relations for $\Sigma_3$ quantier-patterns.
  • Figure 3: The many-one classification of $\Sigma_3$- and $\Pi_3$-patterns
  • Figure 4: (left) The $\equiv_{\sf m}$-degrees of $\Sigma_3$-patterns; (center) The $\equiv_{\sf m}$-degrees of $\Pi_3$-patterns; (right) The $\equiv_{\sf dm}$-degrees of $\Pi_3$-patterns.

Theorems & Definitions (135)

  • Definition 1.1: see Definition \ref{['def:many-one-formula-def']} for the rigorous definition
  • Definition 1.2: see also Definition \ref{['def:di-reducible']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6: Arithmetical hierarchy
  • Proposition 2.9
  • proof
  • ...and 125 more