Pith. Sign in

RECORD

First order sentences about random graphs: small number of alternations

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Aleksandr Matushkin, Maksim Zhukovskii

Combinatorics · Submitted 23 Jan 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID1701.06517 (version 3)
Pith IDNPSLAXWK, Pith's permanent record of this paper and its review
CitationAleksandr Matushkin, Maksim Zhukovskii. "First order sentences about random graphs: small number of alternations." arXiv:1701.06517 (2017). https://pith.science/paper/NPSLAXWK
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,965 input tokens.

What the abstract itself states, as probabilities: claims first 0.13 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Spectrum of a first order sentence is the set of all $\alpha$ such that $G(n, n^{-\alpha})$ does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Spectrum of a first order sentence is the set of all $\alpha$ such that $G(n, n^{-\alpha})$ does not obey zero-one law w.r.t. this sentence. background 1.00 0.08 0.04 asserted 0.99
s1 We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. result 0.99 0.95 0.72 asserted 1.00
s2 We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5. result 1.00 0.91 0.13 asserted 0.80

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Spectrum of a first order sentence is the set of all $\alpha$ such that $G(n, n^{-\alpha})$ does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.