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Computable Approximations of Semicomputable Graphs

Vedran Čačić, Matea Čelar, Marko Horvat, Zvonko Iljazović

TL;DR

This work advances the understanding of when semicomputable graphs in computable metric spaces are computable by showing they admit computable subgraphs that closely approximate them, obtained by cutting off small subarcs at uncomputable endpoints. The authors establish a computable arc neighborhood for points with an $\mathbb{R}$-like local structure and extend the approximation framework to both compact and non-compact graphs as well as finitely connected 1-manifolds. Central contributions include a constructive method using chains to generate computable neighborhoods, and a principled procedure to replace uncomputable endpoints with computable ones to yield computable graphs whose endpoints are computable. The results enable effective approximation in Hausdorff distance, enable computable type conclusions for endpoints, and provide practical tools for bridging semicomputability and computability in graph-like topological spaces, with implications for effective topology and computable analysis of 1-manifolds.

Abstract

In this work, we study the computability of topological graphs, which are obtained by gluing arcs and rays together at their endpoints. We prove that every semicomputable graph in a computable metric space can be approximated, with arbitrary precision, by its computable subgraph with computable endpoints.

Computable Approximations of Semicomputable Graphs

TL;DR

This work advances the understanding of when semicomputable graphs in computable metric spaces are computable by showing they admit computable subgraphs that closely approximate them, obtained by cutting off small subarcs at uncomputable endpoints. The authors establish a computable arc neighborhood for points with an -like local structure and extend the approximation framework to both compact and non-compact graphs as well as finitely connected 1-manifolds. Central contributions include a constructive method using chains to generate computable neighborhoods, and a principled procedure to replace uncomputable endpoints with computable ones to yield computable graphs whose endpoints are computable. The results enable effective approximation in Hausdorff distance, enable computable type conclusions for endpoints, and provide practical tools for bridging semicomputability and computability in graph-like topological spaces, with implications for effective topology and computable analysis of 1-manifolds.

Abstract

In this work, we study the computability of topological graphs, which are obtained by gluing arcs and rays together at their endpoints. We prove that every semicomputable graph in a computable metric space can be approximated, with arbitrary precision, by its computable subgraph with computable endpoints.

Paper Structure

This paper contains 11 sections, 20 theorems, 80 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Computable metric spaces
  4. Chains
  5. Graphs
  6. Auxiliary results
  7. Existence of a computable neighbourhood
  8. Computable approximations of semicomputable sets
  9. Compact graphs
  10. Non-compact graphs
  11. 1-manifolds

Key Result

Lemma 2.6

compintpoint Let $X$ be a set and let $(C_0,\dots,C_m)$ and $(D_0,\dots,D_{m'})$ be two quasi-chains in $X$ such that $(D_0,\dots,D_{m'})$ refines $(C_0,\dots,C_m)$. Suppose that $i,j,k\in\left\{0,\dots,m\right\}\!$ and $p,q\in \left\{0,\dots,m'\right\}\!$ are such that Then there exists $r\in\left\{0,\dots,m'\right\}$ such that $p<r<q$ and $D_r\subseteq C_k$.

Figures (5)

  • Figure 1: A graph. Filled circles denote endpoints and arrows denote rays.
  • Figure 2: The choice of points $\tilde{a}$ and $\tilde{b}$ from $S'$.
  • Figure 3: The sets $T_n$ and $S_n$ for $n = b_i \in B$
  • Figure 4: Points along an arc as in the proof of Lemma \ref{['lm:izrezivanje']}. The highlighted arc is computable.
  • Figure 5: A semicomputable compact graph. Empty circles denote uncomputable points and filled circles denote computable points. The highlighted subset is a computable compact graph.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 30 more