pith. sign in

arxiv: 2606.23162 · v1 · pith:37WKRAJRnew · submitted 2026-06-22 · 💻 cs.DS

Sublinear Time Algorithms for Abelian Group Property Testing

Nader H. Bshouty This is my paper

Pith reviewed 2026-06-26 06:27 UTC · model grok-4.3

classification 💻 cs.DS
keywords abelian group property testingsublinear time algorithmsPS-modelFS-modelCayley tableproperty testinggroup axioms
0
0 comments X

The pith

A tester distinguishes whether a binary operation forms an abelian group or is far from any abelian group using Õ(√|G| + 1/ε) time in the partially specified model.

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

The paper develops an algorithm to test if a given binary operation on a finite set G makes (G, *) an abelian group or leaves it ε-far from every abelian group on the same set. The algorithm works in the partially specified model by drawing random elements and reading the operation results only among those elements. This yields a runtime of Õ(√|G| + 1/ε), which is faster than earlier linear-time methods that examined the full group or its table. The same approach extends directly to deciding membership in isomorphism-closed subclasses of abelian groups once a decision procedure for the canonical form Z_{m1} × ⋯ × Z_{mr} is supplied. A reader would care because the result shows that algebraic structure can be verified with far fewer resources than the size of the set when only random samples are available.

Core claim

The central claim is that there exists a tester in the PS-model (and hence in the FS-model) that accepts with high probability when the input is an abelian group and rejects with high probability when the input is ε-far from every abelian group, while using time and queries Õ(√|G| + 1/ε). The same tester, augmented by a subroutine that decides membership of a product of cyclic groups in a fixed isomorphism-closed class, yields testers for the subclasses of rank-at-most-k abelian groups, abelian p-groups, and vector spaces over Z_p, each still running in time T + Õ(√|G| + 1/ε) where T is the cost of the membership decision.

What carries the argument

The sampling procedure that draws random elements of G and obtains the full Cayley table restricted to those elements, then uses the table to verify the group axioms and distance to the nearest abelian group.

If this is right

  • The runtime improves from O(|G|/ε) in the earlier Goldreich-Tauber tester to Õ(√|G| + 1/ε).
  • The same bound applies to testing whether the group belongs to any isomorphism-closed subclass once a T-time decision procedure for the product-of-cyclic-groups form is available.
  • Concrete subclasses such as rank-at-most-k abelian groups, abelian p-groups, and Z_p-vector spaces now admit testers with the same sublinear bound.
  • The tester also improves an earlier Goldreich-Tauber result that used O(|G|^2) time and Õ(|G| + 1/ε) queries.

Where Pith is reading between the lines

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

  • If the PS-model sampling oracle is realizable in practice, the method could accelerate verification of algebraic structure inside large distributed systems where only random peers can be contacted.
  • The reduction to testing on a random subsample suggests that similar square-root speed-ups might exist for property testing of other algebraic objects whose axioms can be localized to small subsets.
  • Extending the technique to non-abelian groups or to rings would require a new way to localize the relevant axioms, which the paper leaves open.

Load-bearing premise

The algorithm requires the ability to draw uniformly random elements from G and to read every operation result among the sampled elements.

What would settle it

Run the tester on an operation table that is known to be ε-far from every abelian group on G; if it accepts with probability greater than 1/3, the correctness guarantee fails.

read the original abstract

In this paper, we study the problems of abelian group property testing in two models. In the partially specified model (PS-model), the algorithm does not know the group size but can access randomly chosen elements of the group, along with the Cayley table of these elements, which provides the result of the binary operation for every pair of selected elements. In the stronger fully specified model (FS-model), the algorithm knows the size of the group and has access to all its elements and the Cayley table. In property testing of abelian group property, given a finite set $G$ and oracle access to a binary operation $*:G^2\to G$, we aim to distinguish whether $(G,*)$ is an abelian group or is $\epsilon$-far from any abelian group over $G$. Using a novel approach, we present a tester in the PS-model (and consequently in the FS-model) that runs in time $\tilde O(\sqrt{|G|}+1/\epsilon)$, improving upon the Goldreich-Tauber tester, which runs in time $O(|G|/\epsilon)$. Additionally, our tester improves another tester by Goldreich and Tauber that runs in time $O(|G|^2)$ and makes $\tilde O(|G|+1/\epsilon)$ queries. We further extend our result to testing subclasses of abelian groups ${\cal G}$ that are closed under isomorphism. Specifically, if one can decide in time $T$ whether an abelian group of the form $Z_{m_1}\times \cdots\times Z_{m_r}$ belongs to ${\cal G}$, then there exists a tester for ${\cal G}$ that runs in time $T+\tilde O(\sqrt{|G|}+1/\epsilon)$ and makes $O(\sqrt{|G|}+1/\epsilon)$ queries. This result gives testers that run in time $O(\sqrt{|G|}+1/\epsilon)$ for subclasses such as abelian groups of rank at most $k$, abelian $p$-groups, and vector spaces over~$Z_p$.

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.

Referee Report

3 major / 2 minor

Summary. The paper studies property testing for abelian groups over a finite set G with oracle access to a binary operation. It introduces the PS-model (random samples plus their full Cayley table) and the stronger FS-model, and claims a tester running in Õ(√|G| + 1/ε) time and O(√|G| + 1/ε) queries that distinguishes abelian groups from those ε-far from any abelian group. The same bound is claimed to hold in the FS-model. The work also gives an extension: if membership in a isomorphism-closed subclass G of abelian groups can be decided in time T for groups presented as Z_{m1} × ⋯ × Z_{mr}, then there is a tester for G running in T + Õ(√|G| + 1/ε) time; this yields O(√|G| + 1/ε)-time testers for rank-at-most-k groups, p-groups, and vector spaces over Z_p. The claimed improvement is over Goldreich–Tauber testers requiring O(|G|/ε) time or O(|G|^2) time.

Significance. If the model definitions and runtime analysis are sound, the result would constitute a substantial improvement in the query and time complexity of algebraic property testing, moving from linear to sublinear dependence on |G|. The explicit reduction from subclass testing to a decision procedure for the invariant-factor presentation is a clean modular contribution. The paper supplies concrete query bounds and an algorithmic construction rather than an existence argument.

major comments (3)
  1. [Abstract and model definition (presumably §2)] Abstract and model definition (presumably §2): the PS-model supplies the entire s×s Cayley table on a sample of size s ≈ √|G|. Under any standard per-entry reading or processing cost, this table alone requires Ω(s²) = Ω(|G|) time or queries. The claimed Õ(√|G| + 1/ε) time bound therefore requires an explicit statement of the computational model and charging rule for table access; without it the central complexity claim is not yet justified.
  2. [§3 (tester construction) and analysis] §3 (tester construction) and analysis: the abstract states that a “novel approach” yields the Õ(√|G|) bound, yet the provided text does not exhibit the algorithm, the sampling procedure, or the analysis showing that the tester only examines O(√|G|) entries of the supplied tables. Because the runtime improvement is the central claim, the absence of this concrete construction and its cost accounting is load-bearing.
  3. [Extension theorem (presumably §4)] Extension theorem (presumably §4): the reduction to a T-time decision procedure for the invariant-factor form is stated, but the query and time overhead of the reduction itself must be shown to be O(√|G| + 1/ε) rather than depending on the number of generators or the bit length of the mi’s. The current statement leaves this overhead implicit.
minor comments (2)
  1. Notation for the two models (PS-model vs. FS-model) should be introduced once with a clear table or bullet list of oracle capabilities rather than being redefined in the abstract and again in the body.
  2. The abstract claims both a time bound and a query bound; these should be stated uniformly in the theorem statements that follow.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight important points about model clarification and exposition that we will address in a revision. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [Abstract and model definition (presumably §2)] Abstract and model definition (presumably §2): the PS-model supplies the entire s×s Cayley table on a sample of size s ≈ √|G|. Under any standard per-entry reading or processing cost, this table alone requires Ω(s²) = Ω(|G|) time or queries. The claimed Õ(√|G| + 1/ε) time bound therefore requires an explicit statement of the computational model and charging rule for table access; without it the central complexity claim is not yet justified.

    Authors: We agree that the computational model requires an explicit statement. In the PS-model the Cayley table on the sampled set is provided via an oracle that returns any requested entry in constant time; the algorithm never reads the entire table and only makes O(√|G|) oracle calls in total. The Õ(√|G|) time bound counts arithmetic operations and oracle accesses under this standard oracle model (analogous to the usual group-operation oracle in algebraic property testing). We will add a dedicated paragraph in Section 2 that defines the model, the oracle interface, and the precise charging rule, making the sublinear bound unambiguous. revision: yes

  2. Referee: [§3 (tester construction) and analysis] §3 (tester construction) and analysis: the abstract states that a “novel approach” yields the Õ(√|G|) bound, yet the provided text does not exhibit the algorithm, the sampling procedure, or the analysis showing that the tester only examines O(√|G|) entries of the supplied tables. Because the runtime improvement is the central claim, the absence of this concrete construction and its cost accounting is load-bearing.

    Authors: Section 3 does contain the sampling procedure (selecting s = Θ(√|G|) random elements) together with the subsequent verification steps that examine only O(√|G|) table entries; the analysis bounds the number of examined entries by a direct counting argument that appears after the algorithm description. We acknowledge that the exposition is terse and that pseudocode would improve readability. In the revision we will insert explicit pseudocode for the tester, expand the paragraph that counts the examined entries, and add a short lemma that isolates the O(√|G|) cost accounting. revision: yes

  3. Referee: [Extension theorem (presumably §4)] Extension theorem (presumably §4): the reduction to a T-time decision procedure for the invariant-factor form is stated, but the query and time overhead of the reduction itself must be shown to be O(√|G| + 1/ε) rather than depending on the number of generators or the bit length of the mi’s. The current statement leaves this overhead implicit.

    Authors: The reduction obtains an invariant-factor presentation from the sampled elements using O(√|G|) group operations and then invokes the T-time decision procedure once. Because the number of generators is at most O(log |G|) and arithmetic on the mi’s is performed in the standard word-RAM model (bit length O(log |G|)), the overhead is absorbed into the Õ(√|G|) term. Nevertheless we agree that an explicit bound is needed. We will add a short lemma in Section 4 that states the precise overhead (O(√|G| + log |G|) operations plus one call to the T-procedure) and confirms that it does not depend on the bit length beyond the logarithmic factor already hidden by the Õ notation. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation is an explicit algorithmic construction

full rationale

The paper presents a novel algorithmic tester for distinguishing abelian groups from those ε-far from any abelian group, with runtime Õ(√|G| + 1/ε) in the PS-model (and thus FS-model). This is derived from an explicit construction that samples elements and uses the provided Cayley table on those elements, improving on the cited Goldreich-Tauber results without any reduction to self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations. The central claim rests on the algorithm's design and analysis rather than tautological mappings or imported uniqueness theorems from the same authors. No enumerated circularity pattern applies, and the result is self-contained as an independent algorithmic contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions of property testing and finite-group theory with no free parameters or invented entities introduced.

axioms (2)
  • domain assumption G is a finite set and * is a binary operation G×G → G.
    Stated in the problem definition of abelian group property testing.
  • domain assumption Uniform random sampling of elements from G is possible.
    Required by the definition of the PS-model.

pith-pipeline@v0.9.1-grok · 5908 in / 1137 out tokens · 21689 ms · 2026-06-26T06:27:17.781182+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 14 canonical work pages

  1. [1]

    Rothblum

    Noga Amir, Oded Goldreich, and Guy N. Rothblum. Doubly sub-linear interactive proofs of proximity.Electron. Colloquium Comput. Complex., TR24-143, 2024. URL: https: //eccc.weizmann.ac.il/report/2024/143,arXiv:TR24-143

    2024

  2. [2]

    Self-testing/correcting with ap- plications to numerical problems.J

    Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with ap- plications to numerical problems.J. Comput. Syst. Sci., 47(3):549–595, 1993. doi: 10.1016/0022-0000(93)90044-W

    work page doi:10.1016/0022-0000(93)90044-w 1993

  3. [3]

    Nader H. Bshouty. Sublinear time algorithms for abelian group isomorphism and basis construction. In Jakub Kozik and Alexander Wolff, editors,SOFSEM 2026: Theory and Practice of Computer Science - 51st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2026, Kraków, Poland, February 9-13, 2026, Proceedings, volum...

    work page doi:10.1007/978-3-032-17801-5 2026

  4. [4]

    Sublinear-time algorithms.Bull

    Artur Czumaj and Christian Sohler. Sublinear-time algorithms.Bull. EATCS, 89:23–47, 2006

    2006

  5. [5]

    Spot-checkers.J

    Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers.J. Comput. Syst. Sci., 60(3):717–751, 2000. URL:https://doi.org/10. 1006/jcss.1999.1692,doi:10.1006/JCSS.1999.1692. 13

    work page doi:10.1006/jcss.1999.1692 2000

  6. [6]

    Verifying groups in linear time

    Shai Evra, Shay Gadot, Ohad Klein, and Ilan Komargodski. Verifying groups in linear time. In65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, Chicago, IL, USA, October 27-30, 2024, pages 2131–2147. IEEE, 2024.doi:10.1109/ FOCS61266.2024.00126

    arXiv 2024

  7. [7]

    A basic lower bound for property testing.CoRR, abs/2403.04999,

    Eldar Fischer. A basic lower bound for property testing.CoRR, abs/2403.04999,

    arXiv

  8. [8]

    URL: https://doi.org/10.48550/arXiv.2403.04999, arXiv:2403.04999, doi: 10.48550/ARXIV.2403.04999

    work page doi:10.48550/arxiv.2403.04999

  9. [9]

    Efficient testing of groups

    Katalin Friedl, Gábor Ivanyos, and Miklos Santha. Efficient testing of groups. InProceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22-24, 2005, pages 157–166, 2005.doi:10.1145/1060590.1060614

    work page doi:10.1145/1060590.1060614 2005

  10. [10]

    Property testing for cyclic groups and beyond.J

    François Le Gall and Yuichi Yoshida. Property testing for cyclic groups and beyond.J. Comb. Optim., 26(4):636–654, 2013. URL:https://doi.org/10.1007/s10878-011-9445-8, doi: 10.1007/S10878-011-9445-8

    work page doi:10.1007/s10878-011-9445-8 2013

  11. [11]

    Transitive-Closure Spanners:

    Oded Goldreich, editor.Property Testing - Current Research and Surveys, volume 6390 of Lecture Notes in Computer Science. Springer, 2010.doi:10.1007/978-3-642-16367-8

    work page doi:10.1007/978-3-642-16367-8 2010

  12. [12]

    Cambridge University Press, 2017

    Oded Goldreich.Introduction to Property Testing. Cambridge University Press, 2017. URL: http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9781107194052, doi: 10.1017/9781108135252

    work page doi:10.1017/9781108135252 2017

  13. [13]

    On testing group properties.Electron

    Oded Goldreich and Laliv Tauber. On testing group properties.Electron. Colloquium Comput. Complex., TR23-214, 2023. URL:https://eccc.weizmann.ac.il/report/2023/ 214,arXiv:TR23-214

    2023

  14. [14]

    Kaltofen and Gilles Villard

    Erich L. Kaltofen and Gilles Villard. On the complexity of computing determinants.Comput. Complex., 13(3-4):91–130, 2005. URL:https://doi.org/10.1007/s00037-004-0185-3, doi:10.1007/S00037-004-0185-3

    work page doi:10.1007/s00037-004-0185-3 2005

  15. [15]

    Polynomial algorithms for computing the smith and hermite normal forms of an integer matrix.SIAM J

    Ravindran Kannan and Achim Bachem. Polynomial algorithms for computing the smith and hermite normal forms of an integer matrix.SIAM J. Comput., 8(4):499–507, 1979. doi:10.1137/0208040

    work page doi:10.1137/0208040 1979

  16. [16]

    Testing commutativity of a group and the power of randomization.LMS J

    Igor Pak. Testing commutativity of a group and the power of randomization.LMS J. Comput. Math., 15:38–43, 2012. URL:https://doi.org/10.1112/s1461157012000046, doi:10.1112/S1461157012000046

    work page doi:10.1112/s1461157012000046 2012

  17. [17]

    Schulman

    Sridhar Rajagopalan and Leonard J. Schulman. Verification of identities.SIAM J. Comput., 29(4):1155–1163, 2000.doi:10.1137/S0097539797325387

    work page doi:10.1137/s0097539797325387 2000

  18. [18]

    Property testing: A learning theory perspective.Foundations and Trends in Machine Learning, 1(3):307–402, 2008.doi:10.1561/2200000004

    Dana Ron. Property testing: A learning theory perspective.Foundations and Trends in Machine Learning, 1(3):307–402, 2008.doi:10.1561/2200000004

    work page doi:10.1561/2200000004 2008

  19. [19]

    Algorithmic and analysis techniques in property testing.Foundations and Trends in Theoretical Computer Science, 5(2):73–205, 2009.doi:10.1561/0400000029

    Dana Ron. Algorithmic and analysis techniques in property testing.Foundations and Trends in Theoretical Computer Science, 5(2):73–205, 2009.doi:10.1561/0400000029

    work page doi:10.1561/0400000029 2009

  20. [20]

    Robust characterizations of polynomials with ap- plications to program testing.SIAM J

    Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with ap- plications to program testing.SIAM J. Comput., 25(2):252–271, 1996. doi:10.1137/ S0097539793255151. 14

    1996