Computable Approximations of Semicomputable Graphs

Marko Horvat; Matea \v{C}elar; Vedran \v{C}a\v{c}i\'c; Zvonko Iljazovi\'c

arxiv: 2411.13672 · v4 · submitted 2024-11-20 · 💻 cs.LO

Computable Approximations of Semicomputable Graphs

Vedran \v{C}a\v{c}i\'c , Matea \v{C}elar , Marko Horvat , Zvonko Iljazovi\'c This is my paper

Pith reviewed 2026-05-23 08:15 UTC · model grok-4.3

classification 💻 cs.LO

keywords semicomputable graphscomputable graphscomputable metric spacestopological graphscomputable approximationsarcs and rayseffective topologygluing constructions

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The pith

Every semicomputable graph in a computable metric space can be approximated arbitrarily precisely by a computable subgraph with computable endpoints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that topological graphs formed by gluing arcs and rays at endpoints, when only semicomputable in a computable metric space, still admit computable subgraphs that match them to any desired degree of accuracy. A sympathetic reader cares because this separates the existence of effective approximations from full computability of the entire structure, allowing partial algorithmic access to objects that are not fully computable. The result applies directly to the gluing construction and shows the approximations can preserve computable endpoints. This matters in settings where one needs to compute or visualize graph-like objects built from effective pieces without requiring the whole graph to be computable from the start.

Core claim

The central claim is that for any semicomputable graph obtained by gluing arcs and rays together at their endpoints inside a computable metric space, and for any positive precision, there exists a computable subgraph with computable endpoints that approximates the original graph to within that precision.

What carries the argument

The computable subgraph with computable endpoints, which serves as the approximating object that can be constructed to arbitrary precision from the semicomputable graph.

If this is right

Any desired level of approximation is achievable by some computable subgraph.
The endpoints of the approximating subgraph can always be taken to be computable points.
The approximation works uniformly for graphs assembled from arcs and rays by endpoint identification.
The result holds inside any computable metric space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar approximation results might hold for other semicomputable topological objects built from effective pieces, such as surfaces or higher-dimensional complexes.
The technique could support effective enumeration or search procedures that locate computable pieces inside larger semicomputable structures.
One could test whether the same approximation property transfers when the underlying space is only computably Hausdorff rather than fully metric.

Load-bearing premise

The definitions of semicomputable graphs via gluing of arcs and rays in computable metric spaces allow the construction of a computable approximating subgraph.

What would settle it

A concrete semicomputable graph in some computable metric space such that for some fixed positive distance no computable subgraph with computable endpoints lies within that distance of the original graph.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves that semicomputable graphs from glued arcs and rays in computable metric spaces can be approximated by computable subgraphs with computable endpoints.

read the letter

The central result is that every semicomputable graph in a computable metric space has a computable subgraph that approximates it to any desired precision, with the endpoints themselves computable. This is stated directly as an existence theorem for graphs built by gluing arcs and rays at endpoints. The claim is specific and does not appear to rest on circular reductions or fitted parameters. The abstract presents it as a new statement rather than a restatement of earlier work on computable metric spaces. The paper does a clean job of isolating this approximation property inside the framework of effective topology. It organizes the problem for this restricted class of objects without adding extra machinery. The argument is described as a direct proof, which keeps the circularity burden low. The main soft spot is that only the abstract is visible here, so the actual construction of the approximating subgraph and the precise handling of the gluing operation cannot be inspected for hidden assumptions in the definitions. If those definitions turn out to be standard and the proof goes through without extra conditions, the result holds up. If the gluing introduces non-computable choices at the endpoints, that would need explicit handling. This is a narrow but technically focused piece for people already working in computable analysis and effective descriptive set theory. A reader who knows computable metric spaces will see the incremental value right away. It is not broad enough to interest a general audience in logic or topology. The claim is precise enough and the subfield active enough that the paper should go to peer review rather than a desk reject. The referee can check the construction details and the citation placement against prior results on semicomputable sets.

Referee Report

0 major / 1 minor

Summary. The paper studies the computability of topological graphs formed by gluing arcs and rays at endpoints in computable metric spaces. It claims to prove that every semicomputable graph can be approximated to arbitrary precision by a computable subgraph whose endpoints are computable.

Significance. If the result holds, it would establish a general approximation theorem linking semicomputable and computable graphs in computable metric spaces, potentially useful for effective topology and computable analysis. The existence claim is direct rather than relying on reductions or self-referential constructions.

minor comments (1)

The provided text contains only the abstract; no definitions of semicomputable graphs, computable metric spaces, gluing constructions, or proof details are visible, preventing technical evaluation of the central claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the summary provided. The report indicates an uncertain recommendation but does not enumerate any specific major comments or concerns about the proof or claims. We are available to provide further clarification on the approximation result if requested.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct existence theorem: every semicomputable graph in a computable metric space admits an arbitrary-precision approximation by a computable subgraph with computable endpoints. The provided abstract and claim description contain no fitted parameters renamed as predictions, no self-definitional reductions, no load-bearing self-citations, and no ansatz smuggled via prior work. The result is presented as a standard proof in computable topology rather than a construction that reduces to its own inputs by definition or statistical forcing. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no free parameters, axioms, or invented entities can be extracted from the provided text.

pith-pipeline@v0.9.0 · 5583 in / 1081 out tokens · 16705 ms · 2026-05-23T08:15:36.416593+00:00 · methodology

Computable Approximations of Semicomputable Graphs

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)