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DOI: 10.4230/LIPIcs.ITP.2025.9

Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups

Authors: Eric Wang, Arohee Bhoja, Cayden Codel, and Noah G. Singer

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract

Graded unipotent Chevalley groups are an important family of groups on matrices with polynomial entries over a finite field. Using the Lean theorem prover, we verify that three such groups, namely, the A₃- and the two B₃-type groups, satisfy a useful group-theoretic condition. Specifically, these groups are defined by a set of equations called Steinberg relations, and we prove that a certain canonical "smaller" set of Steinberg relations suffices to derive the rest. Our work is motivated by an application for building topologically-interesting objects called higher-dimensional expanders (HDXs). In the past decade, HDXs have formed the basis for many new results in theoretical computer science, such as in quantum error correction and in property testing. Yet despite the increasing prevalence of HDXs, only two methods of constructing them are known. One such method builds an HDX from groups that satisfy the aforementioned property, and the Chevalley groups we use are (essentially) the only ones currently known to satisfy it.

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Eric Wang, Arohee Bhoja, Cayden Codel, and Noah G. Singer. Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wang_et_al:LIPIcs.ITP.2025.9,
  author =	{Wang, Eric and Bhoja, Arohee and Codel, Cayden and Singer, Noah G.},
  title =	{{Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.9},
  URN =		{urn:nbn:de:0030-drops-246071},
  doi =		{10.4230/LIPIcs.ITP.2025.9},
  annote =	{Keywords: Group presentations, term rewriting, metaprogramming, proof automation, the Lean theorem prover}
}

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APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.15

Oblivious Algorithms for the Max-kAND Problem

Authors: Noah G. Singer

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract

Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-kAND problem. This is a class of simple, combinatorial algorithms which round each variable with probability depending only on a quantity called the variable’s bias. Our definition generalizes a class of algorithms defined by Feige and Jozeph (Algorithmica '15) for Max-DICUT, a special case of Max-2AND. For each oblivious algorithm, we design a so-called factor-revealing linear program (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all k, oblivious algorithms for Max-kAND provably outperform a special subclass of algorithms we call "superoblivious" algorithms. Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every k, O(log n)-space sketching algorithms for Max-kAND known to be optimal in o(√n)-space can be beaten in (a) O(log n)-space under a random-ordering assumption, and (b) O(n^{1-1/k} D^{1/k}) space under a maximum-degree-D assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.

Cite as

Noah G. Singer. Oblivious Algorithms for the Max-kAND Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{singer:LIPIcs.APPROX/RANDOM.2023.15,
  author =	{Singer, Noah G.},
  title =	{{Oblivious Algorithms for the Max-kAND Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.15},
  URN =		{urn:nbn:de:0030-drops-188409},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.15},
  annote =	{Keywords: streaming algorithm, approximation algorithm, constraint satisfaction problem (CSP), factor-revealing linear program}
}

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Documents authored by Singer, Noah G.

Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups

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Oblivious Algorithms for the Max-kAND Problem

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