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Countability constraints in order-theoretic approaches to computability

Pedro Hack, Daniel A. Braun, Sebastian Gottwald

TL;DR

It is shown how computability can be introduced in some order structures via countability order density and multi-utility constraints.

Abstract

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.

Countability constraints in order-theoretic approaches to computability

TL;DR

It is shown how computability can be introduced in some order structures via countability order density and multi-utility constraints.

Abstract

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.
Paper Structure (23 sections, 43 theorems, 61 equations, 2 figures)

This paper contains 23 sections, 43 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computability via ordered sets
  3. Examples
  4. The Cantor domain
  5. The interval domain
  6. Majorization
  7. Order density and weak bases
  8. Uniform computability via ordered sets
  9. Order density and bases
  10. Continuous Debreu separable dcpos
  11. Conditionally connected Debreu upper separable dcpos
  12. Relating order density with completeness and continuity
  13. Completeness
  14. Continuity
  15. Functional countability restrictions
  16. ...and 8 more sections

Key Result

Proposition 1

If $n\geq 2$, then $\mathbb{Q}^n \cap \Lambda^n$ is an effective weak basis for majorization.

Figures (2)

  • Figure 1: Representation of a dcpo, defined in Proposition \ref{['weak basis not debreu sep']}, with a countable weak basis and no countable Debreu dense subset. In particular, we show $A \coloneqq [0,1]$, $B \coloneqq [2,3]$ and how $x,y,z \in A$, $x<y<z$, are related to $x+2,y+2,z+2 \in B$. Notice an arrow from an element $w$ to an element $t$ represents $w \prec t$.
  • Figure 2: Representation of a dcpo, defined in Proposition \ref{['cont Deb upper sep but not w-cont']}, which is Debreu upper separable and has no countable weak basis. In particular, we show $\Sigma^\omega$ and $\Sigma^*$ for $\Sigma=\{0,1\}$ and how $010101..,000000..,010000.. \in \Sigma^\omega$ are related to $0101, 0, 01 \in \Sigma^*$. Notice an arrow from an element $w$ to an element $t$ represents $w \prec t$.

Theorems & Definitions (107)

  • Definition 1: Computable functions and recursively enumerable sets turing1937computablerogers1987theory
  • Definition 2: Finite map hack2022computation
  • Definition 3: Partial order bridges2013representations
  • Definition 4: Direct set and least upper bound abramsky1994domain
  • Definition 5: Weak basis hack2022computation
  • Definition 6: Effective weak basis hack2022computation
  • Definition 7: Computable element hack2022computation
  • Proposition 1
  • Lemma 1
  • Definition 8: Order density properties
  • ...and 97 more