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On some computational properties of open sets

Dag Normann, Sam Sanders

TL;DR

The paper investigates the computational content of open-set representations by introducing the Ω_C functional, which decides the nonemptiness of closed sets in a Cantor-like space, and developing the Ω_C-cluster that links Ω_C to a wide range of classical mathematical notions such as semi-continuity, the Urysohn lemma, the Tietze extension, and Lebesgue measure. It demonstrates cluster theorems showing many mainstream functionals are computationally equivalent to Ω_C under Kleene's $ ext{exists}^2$, analyzes the lameness of Ω_C (Ω_C+$ ext{exists}^2$ yields the same reals as $ ext{exists}^2$), and investigates the computational landscape between Ω and Ω_C, Cantor-type results, and the Lebesgue measure, including a detailed cluster for measure-related functionals. The work also develops a lambda-calculus formulation for S1-S9, correcting a prior model via computation trees and fixed-point operators to handle partial objects, and provides an operational semantics that aligns with the denotational view. Together, these results map the boundaries of computability for open-set representations and related analysis concepts, highlighting both the relative weakness of Ω_C and the rich web of equivalences that connect disparate areas of analysis to higher-order computation. The findings offer a framework for understanding the cost of computing open-set codes and open avenues for comparing foundational functionals across mathematics.

Abstract

Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their 'codes' or 'representations'. In this paper, we study how hard it is to compute, given an arbitrary open set of reals, the most common representation, i.e. a countable set of open intervals. We work in Kleene's higher-order computability theory, which was historically based on the S1-S9 schemes and which now has an intuitive lambda calculus formulation due to the authors. We establish many computational equivalences between on one hand the 'structure' functional that converts open sets to the aforementioned representation, and on the other hand functionals arising from mainstream mathematics, like basic properties of semi-continuous functions, the Urysohn lemma, and the Tietze extension theorem. We also compare these functionals to known operations on regulated and bounded variation functions, and the Lebesgue measure restricted to closed sets. We obtain a number of natural computational equivalences for the latter involving theorems from mainstream mathematics.

On some computational properties of open sets

TL;DR

The paper investigates the computational content of open-set representations by introducing the Ω_C functional, which decides the nonemptiness of closed sets in a Cantor-like space, and developing the Ω_C-cluster that links Ω_C to a wide range of classical mathematical notions such as semi-continuity, the Urysohn lemma, the Tietze extension, and Lebesgue measure. It demonstrates cluster theorems showing many mainstream functionals are computationally equivalent to Ω_C under Kleene's , analyzes the lameness of Ω_C (Ω_C+ yields the same reals as ), and investigates the computational landscape between Ω and Ω_C, Cantor-type results, and the Lebesgue measure, including a detailed cluster for measure-related functionals. The work also develops a lambda-calculus formulation for S1-S9, correcting a prior model via computation trees and fixed-point operators to handle partial objects, and provides an operational semantics that aligns with the denotational view. Together, these results map the boundaries of computability for open-set representations and related analysis concepts, highlighting both the relative weakness of Ω_C and the rich web of equivalences that connect disparate areas of analysis to higher-order computation. The findings offer a framework for understanding the cost of computing open-set codes and open avenues for comparing foundational functionals across mathematics.

Abstract

Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their 'codes' or 'representations'. In this paper, we study how hard it is to compute, given an arbitrary open set of reals, the most common representation, i.e. a countable set of open intervals. We work in Kleene's higher-order computability theory, which was historically based on the S1-S9 schemes and which now has an intuitive lambda calculus formulation due to the authors. We establish many computational equivalences between on one hand the 'structure' functional that converts open sets to the aforementioned representation, and on the other hand functionals arising from mainstream mathematics, like basic properties of semi-continuous functions, the Urysohn lemma, and the Tietze extension theorem. We also compare these functionals to known operations on regulated and bounded variation functions, and the Lebesgue measure restricted to closed sets. We obtain a number of natural computational equivalences for the latter involving theorems from mainstream mathematics.
Paper Structure (26 sections, 20 theorems, 34 equations, 1 figure)

This paper contains 26 sections, 20 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and overview
  3. Preliminaries and definitions
  4. Kleene's computability theory
  5. Some comprehension functionals
  6. Some definitions
  7. Populating the $\Omega_{C}$-cluster
  8. Introduction
  9. The cluster theorems
  10. More on the $\Omega_{C}$-cluster
  11. Computing with arbitrary functions
  12. On moduli of continuity
  13. A word on the Suslin functional
  14. The computational landscape around $\Omega_{C}$
  15. Introduction
  16. ...and 11 more sections

Key Result

Theorem 1.11

A function $f : [0, 1] \rightarrow {\mathbb R}$ of bounded variation is the difference of two non-decreasing functions $g, h:[0,1]\rightarrow {\mathbb R}$.

Figures (1)

  • Figure 1: Some relations among our functionals

Theorems & Definitions (67)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4: Open sets and representations
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 57 more