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URN: urn:nbn:de:0030-drops-142812
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Separating ABPs and Some Structured Formulas in the Non-Commutative Setting

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Abstract

The motivating question for this work is a long standing open problem, posed by Nisan [Noam Nisan, 1991], regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question remains open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 2011]) to define abecedarian polynomials and models that naturally compute them.
Our main contribution is a possible new approach towards resolving the VF_{nc} vs VBP_{nc} question, via lower bounds against abecedarian formulas. In particular, we show the following.
There is an explicit n²-variate degree d abecedarian polynomial f_{n,d}(𝐱) such that
- f_{n, d}(𝐱) can be computed by an abecedarian ABP of size O(nd);
- any abecedarian formula computing f_{n, log n}(𝐱) must have size at least n^{Ω(log log n)}.
We also show that a super-polynomial lower bound against abecedarian formulas for f_{log n, n}(𝐱) would separate the powers of formulas and ABPs in the non-commutative setting.

BibTeX - Entry

@InProceedings{chatterjee:LIPIcs.CCC.2021.7,
  author =	{Chatterjee, Prerona},
  title =	{{Separating ABPs and Some Structured Formulas in the Non-Commutative Setting}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{7:1--7:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14281},
  URN =		{urn:nbn:de:0030-drops-142812},
  doi =		{10.4230/LIPIcs.CCC.2021.7},
  annote =	{Keywords: Non-Commutative Formulas, Lower Bound, Separating ABPs and Formulas}
}

Keywords: Non-Commutative Formulas, Lower Bound, Separating ABPs and Formulas
Seminar: 36th Computational Complexity Conference (CCC 2021)
Issue date: 2021
Date of publication: 08.07.2021


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