License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.09421.7
URN: urn:nbn:de:0030-drops-24126
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2412/
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Mahajan, Meena ; Rao B. V. , Raghavendra

Small space analogues of Valiant's classes and the limitations of skew formula

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09421.MahajanMeena.Paper.2412.pdf (0.3 MB)

Abstract

In the uniform circuit model of computation, the width of a boolean
circuit exactly characterises the ``space'' complexity of the
computed function. Looking for a similar relationship in Valiant's
algebraic model of computation, we propose width of an arithmetic
circuit as a possible measure of space. We introduce the class
VL as an algebraic variant of deterministic log-space L. In
the uniform setting, we show that our definition coincides with that
of VPSPACE at polynomial width.

Further, to define algebraic variants of non-deterministic
space-bounded classes, we introduce the notion of ``read-once''
certificates for arithmetic circuits. We show that polynomial-size
algebraic branching programs can be expressed as a read-once
exponential sum over polynomials in VL, ie
$mbox{VBP}inSigma^R cdotmbox{VL}$.
We also show that $Sigma^R cdot mbox{VBP} =mbox{VBP}$, ie
VBPs are stable under read-once exponential sums. Further, we
show that read-once exponential sums over a restricted class of
constant-width arithmetic circuits are within VQP, and this is the
largest known such subclass of poly-log-width circuits with this
property.

We also study the power of skew formulas and show that exponential
sums of a skew formula cannot represent the determinant polynomial.




BibTeX - Entry

@InProceedings{mahajan_et_al:DagSemProc.09421.7,
  author =	{Mahajan, Meena and Rao B. V., Raghavendra},
  title =	{{Small space analogues of Valiant's classes and the limitations of   skew formula}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--23},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2010/2412},
  URN =		{urn:nbn:de:0030-drops-24126},
  doi =		{10.4230/DagSemProc.09421.7},
  annote =	{Keywords: Algebraic circuits, space bounds, circuit width, nondeterministic circuits, skew formulas}
}

Keywords: Algebraic circuits, space bounds, circuit width, nondeterministic circuits, skew formulas
Collection: 09421 - Algebraic Methods in Computational Complexity
Issue Date: 2010
Date of publication: 19.01.2010

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