DROPS

Document

DOI: 10.4230/LIPIcs.ITCS.2026.9

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Authors: Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials (f_n(x₁,…, x_n))_n that have poly(n)-sized roABPs such that some irreducible factor of f_n requires roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials (f_n(x₁,…, x_n))_n with poly(n)-sized roABPs such that any super-constant power of f_n does not have roABPs of polynomial size in any order (and f_nⁿ requires exponential size in any order). - Non-closure under symmetric operations: There are symmetric polynomials (f_n(e₁,…, e_n))_n that have roABPs of polynomial size such that f_n(x₁,…, x_n) do not have roABPs of subexponential size. (Here, e₁,…, e_n denote the elementary symmetric polynomials in n variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.

Cite as

Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On Closure Properties of Read-Once Oblivious Algebraic Branching Programs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

Copy BibTex To Clipboard

@InProceedings{andrews_et_al:LIPIcs.ITCS.2026.9,
  author =	{Andrews, Robert and Armand, Jules and Dwivedi, Prateek and Hansen, Magnus Rahbek Dalgaard and Limaye, Nutan and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
  title =	{{On Closure Properties of Read-Once Oblivious Algebraic Branching Programs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.9},
  URN =		{urn:nbn:de:0030-drops-252964},
  doi =		{10.4230/LIPIcs.ITCS.2026.9},
  annote =	{Keywords: Factoring, Closure Properties, Sparsity Bounds, Symmetric Polynomials, roABP, Expander Graphs}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.15

Algebraic Pseudorandomness in VNC⁰

Authors: Robert Andrews

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract

We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are computable in VNC⁰, i.e., can be computed by arithmetic formulas of constant size. Unconditionally, we construct a VNC⁰-computable generator that hits arithmetic circuits of constant depth and polynomial size. We also give conditional constructions, under strong but plausible hardness assumptions, of VNC⁰-computable generators that hit arithmetic formulas and arithmetic branching programs of polynomial size, respectively. As a corollary of our constructions, we derive lower bounds for subsystems of the Geometric Ideal Proof System of Grochow and Pitassi. Constructions of such generators are implicit in prior work of Kayal on lower bounds for the degree of annihilating polynomials. Our main contribution is a construction whose correctness relies on circuit complexity lower bounds rather than degree lower bounds.

Cite as

Robert Andrews. Algebraic Pseudorandomness in VNC⁰. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{andrews:LIPIcs.CCC.2025.15,
  author =	{Andrews, Robert},
  title =	{{Algebraic Pseudorandomness in VNC⁰}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.15},
  URN =		{urn:nbn:de:0030-drops-237092},
  doi =		{10.4230/LIPIcs.CCC.2025.15},
  annote =	{Keywords: Polynomial identity testing, Algebraic circuits, Ideal Proof System}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.37

Algebraic Hardness Versus Randomness in Low Characteristic

Authors: Robert Andrews

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

We show that lower bounds for explicit constant-variate polynomials over fields of characteristic p > 0 are sufficient to derandomize polynomial identity testing over fields of characteristic p. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the notion of hardness to be stronger than the standard syntactic notion of hardness used in algebraic complexity. Our results make no restriction on the characteristic of the field and use standard notions of hardness. We do this by combining the Kabanets-Impagliazzo generator with a white-box procedure to take p-th roots of circuits computing a p-th power over fields of characteristic p. When the number of variables appearing in the circuit is bounded by some constant, this procedure turns out to be efficient, which allows us to bypass difficulties related to factoring circuits in characteristic p. We also combine the Kabanets-Impagliazzo generator with recent "bootstrapping" results in polynomial identity testing to show that a sufficiently-hard family of explicit constant-variate polynomials yields a near-complete derandomization of polynomial identity testing. This result holds over fields of both zero and positive characteristic and complements a recent work of Guo, Kumar, Saptharishi, and Solomon, who obtained a slightly stronger statement over fields of characteristic zero.

Cite as

Robert Andrews. Algebraic Hardness Versus Randomness in Low Characteristic. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 37:1-37:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{andrews:LIPIcs.CCC.2020.37,
  author =	{Andrews, Robert},
  title =	{{Algebraic Hardness Versus Randomness in Low Characteristic}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{37:1--37:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.37},
  URN =		{urn:nbn:de:0030-drops-125895},
  doi =		{10.4230/LIPIcs.CCC.2020.37},
  annote =	{Keywords: Polynomial identity testing, hardness versus randomness, low characteristic}
}

Search Results

Documents authored by Andrews, Robert

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Abstract

Cite as

Algebraic Pseudorandomness in VNC⁰

Abstract

Cite as

Algebraic Hardness Versus Randomness in Low Characteristic

Abstract

Cite as

Thanks for your feedback!

Could not send message