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Fixed-Polynomial Size Circuit Bounds

2009
*
2009 24th Annual IEEE Conference on Computational Complexity
*

We explore questions about

doi:10.1109/ccc.2009.21
dblp:conf/coco/FortnowSW09
fatcat:ogzwc5rxirgkhmyb3krxo2xby4
*fixed*-*polynomial**size**circuit*lower*bounds*around and beyond the algebrization barrier. We find several connections, including ... In 1982, Kannan showed that Σ P 2 does not have n k -*sized**circuits*for any k. Do smaller classes also admit such*circuit*lower*bounds*? ... In fact, this kind of translation result, showing that a*fixed**polynomial**circuit*lower*bound*for a class implies a*fixed**polynomial**circuit*lower*bound*for a language in the class with*polynomial*-*size*...##
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On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant
[article]

2013
*
arXiv
*
pre-print

Then we investigate the link between

arXiv:1304.5910v1
fatcat:3oqw2tsvo5gmbmrvyldkmxjxp4
*fixed*-*polynomial**size**circuit**bounds*in the Boolean and arithmetic settings. ... In positive characteristic p, uniform*polynomials*in VNP have*circuits*of*fixed*-*polynomial**size*if and only if both VP=VNP over F_p and Mod_pP has*circuits*of*fixed*-*polynomial**size*. ... We show how*fixed*-*polynomial**circuit**size*lower*bound*on uniform VNP is connected to various questions in Boolean complexity. ...##
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On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant

2015
*
Information and Computation
*

We consider the problem of

doi:10.1016/j.ic.2014.09.006
fatcat:y4xtt46yejh3llo7vkwp4fjyzy
*fixed*-*polynomial*lower*bounds*on the*size*of arithmetic*circuits*computing uniform families of*polynomials*. ... Then we investigate the link between*fixed*-*polynomial**size**circuit**bounds*in the Boolean and arithmetic settings. ... We show how*fixed*-*polynomial**circuit**size*lower*bound*on uniform VNP is connected to various questions in Boolean complexity. ...##
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On Fixed-Polynomial Size Circuit Lower Bounds for Uniform Polynomials in the Sense of Valiant
[chapter]

2013
*
Lecture Notes in Computer Science
*

We consider the problem of

doi:10.1007/978-3-642-40313-2_39
fatcat:uzwykwdxqbbjjfx27j6zkiw4ee
*fixed*-*polynomial*lower*bounds*on the*size*of arithmetic*circuits*computing uniform families of*polynomials*. ... Then we investigate the link between*fixed*-*polynomial**size**circuit**bounds*in the Boolean and arithmetic settings. ... We show how*fixed*-*polynomial**circuit**size*lower*bound*on uniform VNP is connected to various questions in Boolean complexity. ...##
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On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
[chapter]

2014
*
Lecture Notes in Computer Science
*

, and Kumar-Saraf), where strong depth 4

doi:10.1007/978-3-662-44465-8_16
fatcat:y6rw6dvrevdpzkh5krnj4vgyka
*circuit**size*lower*bounds*are shown. ... In this paper, for an explicit*polynomial*in VP (over*fixed*-*size*finite fields), we prove that any ΣΠΣ*circuit*computing it must be of*size*2 Ω(n log n) . ... Conclusion In this paper, over*fixed*-*size*finite fields, we show a tight lower*bound*on the*size*of ΣΠΣ*circuits*computing a*polynomial*in VP. ...##
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On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
[article]

2013
*
arXiv
*
pre-print

To the best of our knowledge, the

arXiv:1401.0189v1
fatcat:qeb2ot7mrraqnnoi2tcteslz6e
*polynomial*NW_n,ϵ(X) is the first example of an explicit*polynomial*in VNP such that it requires 2^Ω(√(n) n)*size*depth four*circuits*, but no known matching upper*bound*... In this paper, we prove that over*fixed*-*size*finite fields, any ΣΠΣ*circuit*for computing the iterated matrix multiplication*polynomial*of n generic matrices of*size*n× n, must be of*size*2^Ω(n n). ... closed set of monomials. 7 Depth 4*Circuit**Size*Lower*Bound*for NW n,ǫ (X)*Polynomial*In this section we prove the depth 4*size*lower*bound*for the NW n,ǫ (X)*polynomial*. ...##
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On the limits of depth reduction at depth 3 over small finite fields

2017
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Information and Computation
*

In this paper, for an explicit

doi:10.1016/j.ic.2017.04.007
fatcat:ys6nzr3lxfd3tgkkxwsffvphla
*polynomial*in VP (over*fixed*-*size*finite fields), we prove that any ΣΠΣ*circuit*computing it must be of*size*2 Ω(n log n) . ...*size*lower*bounds*are shown [KSS13, KLSS14, KS13b, KS14]. ... the task will be to find a ΣΠΣ*circuit*of*size*2 o(n log n) for the determinant*polynomial*. ...##
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NEXP Does Not Have Non-uniform Quasipolynomial-Size ACC Circuits of o(loglogn) Depth
[chapter]

2011
*
Lecture Notes in Computer Science
*

ACC m

doi:10.1007/978-3-642-20877-5_17
fatcat:lxkowswvgvc2bbzj6dgnf232qu
*circuits*are*circuits*consisting of unbounded fan-in AND, OR and MOD m gates and unary NOT gates, where m is a*fixed*integer. ... We show that there exists a language in non-deterministic exponential time which can not be computed by any non-uniform family of ACC m*circuits*of quasipolynomial*size*and o(log log n) depth, where m ... a*circuit*of quasi-*polynomial**size*. ...##
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Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach
[article]

2013
*
arXiv
*
pre-print

In this paper, we identify a simple combinatorial property such that any

arXiv:1308.1640v4
fatcat:es3qwpss5zhgdgd5dvnxczmehu
*polynomial*f that satisfies the property would achieve similar*circuit**size*lower*bound*for depth-4*circuits*. ...*polynomial*(which is in VP) also requires 2^Ω(√(n) n)*size*depth-4*circuits*. ... Lower*bounds*on the*size*of depth-4*circuits*computing NW(X) and IMM n,d (X) Now we derive the depth-4*circuit**size*lower*bound*for NW(X)*polynomial*by a simple application of Theorem 3. Proof. ...##
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Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems
[article]

2017
*
arXiv
*
pre-print

Specifically, we prove that if Σ_2E does not have

arXiv:1701.04428v1
fatcat:jf5fce3nijasjeso62ol4gmj2e
*polynomial**size*non-deterministic*circuits*, then Σ_2SubEXP does not have*fixed**polynomial**size*non-deterministic*circuits*. ... We show that augmented Arthur-Merlin protocols with one bit of advice do not have*fixed**polynomial**size*non-deterministic*circuits*. ... Cai was able to strengthen this further by showing that S 2 P does not have*fixed**polynomial**size*deterministic*circuits*[4] . Vinodchandran proved*fixed*n k*circuit*lower*bounds*for the class PP. ...##
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Parity, circuits, and the polynomial-time hierarchy

1984
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Mathematical Systems Theory
*

A super-

doi:10.1007/bf01744431
fatcat:ryq53ji6ejb5rnjodgbnxbhg3u
*polynomial*lower*bound*is given for the*size*of*circuits*of*fixed*depth computing the parity function. ... Introducing the notion of*polynomial*-*size*, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. ... Steve Fortune first suggested that the depth 3 case might be tractable, Jim Byrd found the proof showing that parity is constant-depth,*polynomial*-*size*reducible to transitive closure, and Charles Leiserson ...##
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Proving Circuit Lower Bounds in High Uniform Classes

2014
*
Interdisciplinary Information Sciences
*

For example, it was shown by Ajtai [3] and Furst, Saxe and Sipser [21] that no AC 0

doi:10.4036/iis.2014.1
fatcat:l3w445j5vngsvh665qugvuhxx4
*circuit*family, which consists of*polynomial*-*size**circuits*of constant depth with unbounded fan-in gates, solves the ... Also, proving exponential*circuit*lower*bounds*in the class E :¼ TIMEð2 OðnÞ Þ implies the full derandomization of BPP, which is a class of problems solved by*polynomial*-time probabilistic Turing machines ... If not, we are done (since NP has*fixed**polynomial**circuit*lower*bounds*). ...##
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Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

2010
*
SIAM journal on computing (Print)
*

That is, either NEXP ⊆ P/poly or the Permanent is not computable by

doi:10.1137/080735850
fatcat:gonpk6xthzgdzldbrgsdqowlha
*polynomial**size**bounded*depth arithmetic*circuits*. ... In this paper we show that lower*bounds*for*bounded*depth arithmetic*circuits*imply derandomization of*polynomial*identity testing for*bounded*depth arithmetic*circuits*. ... Before doing so we will need to*bound*the*size*/depth of the*circuit*computing g(z, w). Proof. ...##
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Hardness-randomness tradeoffs for bounded depth arithmetic circuits

2008
*
Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08
*

That is, either NEXP ⊆ P/poly or the Permanent is not computable by

doi:10.1145/1374376.1374482
dblp:conf/stoc/DvirSY08
fatcat:2meotxjq4zff7bbpqbqkqvekuq
*polynomial**size**bounded*depth arithmetic*circuits*. ... In this paper we show that lower*bounds*for*bounded*depth arithmetic*circuits*imply derandomization of*polynomial*identity testing for*bounded*depth arithmetic*circuits*. ... Before doing so we will need to*bound*the*size*/depth of the*circuit*computing g(z, w). Proof. ...##
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On Lower Bounds for Constant Width Arithmetic Circuits
[article]

2009
*
arXiv
*
pre-print

For every k > 1, we provide an explicit

arXiv:0907.3780v2
fatcat:s54xuq2nwba45pjcnw23bkflhi
*polynomial*that can be computed by a linear-*sized*monotone*circuit*of width 2k but has no subexponential-*sized*monotone*circuit*of width k. ... It follows, from the definition of the*polynomial*, that the constant-width and the constant-depth hierarchies of monotone arithmetic*circuits*are infinite, both in the commutative and the noncommutative ... Can we extend Nisan's lower*bound*arguments to prove*size*lower*bounds*for noncommutative*bounded*-width*circuits*? ...
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