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Fixed-Polynomial Size Circuit Bounds

Lance Fortnow, Rahul Santhanam, Ryan Williams
2009 2009 24th Annual IEEE Conference on Computational Complexity  
We explore questions about 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  ... 
doi:10.1109/ccc.2009.21 dblp:conf/coco/FortnowSW09 fatcat:ogzwc5rxirgkhmyb3krxo2xby4

On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant [article]

Hervé Fournier, Sylvain Perifel, Rémi de Verclos
2013 arXiv   pre-print
Then we investigate the link between 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.  ... 
arXiv:1304.5910v1 fatcat:3oqw2tsvo5gmbmrvyldkmxjxp4

On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant

Hervé Fournier, Sylvain Perifel, Rémi de Verclos
2015 Information and Computation  
We consider the problem of 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.  ... 
doi:10.1016/j.ic.2014.09.006 fatcat:y4xtt46yejh3llo7vkwp4fjyzy

On Fixed-Polynomial Size Circuit Lower Bounds for Uniform Polynomials in the Sense of Valiant [chapter]

Hervé Fournier, Sylvain Perifel, Rémi de Verclos
2013 Lecture Notes in Computer Science  
We consider the problem of 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.  ... 
doi:10.1007/978-3-642-40313-2_39 fatcat:uzwykwdxqbbjjfx27j6zkiw4ee

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields [chapter]

Suryajith Chillara, Partha Mukhopadhyay
2014 Lecture Notes in Computer Science  
, and Kumar-Saraf), where strong depth 4 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.  ... 
doi:10.1007/978-3-662-44465-8_16 fatcat:y6rw6dvrevdpzkh5krnj4vgyka

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields [article]

Suryajith Chillara, Partha Mukhopadhyay
2013 arXiv   pre-print
To the best of our knowledge, the 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.  ... 
arXiv:1401.0189v1 fatcat:qeb2ot7mrraqnnoi2tcteslz6e

On the limits of depth reduction at depth 3 over small finite fields

Suryajith Chillara, Partha Mukhopadhyay
2017 Information and Computation  
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) .  ...  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.  ... 
doi:10.1016/j.ic.2017.04.007 fatcat:ys6nzr3lxfd3tgkkxwsffvphla

NEXP Does Not Have Non-uniform Quasipolynomial-Size ACC Circuits of o(loglogn) Depth [chapter]

Fengming Wang
2011 Lecture Notes in Computer Science  
ACC m 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.  ... 
doi:10.1007/978-3-642-20877-5_17 fatcat:lxkowswvgvc2bbzj6dgnf232qu

Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach [article]

Suryajith Chillara, Partha Mukhopadhyay
2013 arXiv   pre-print
In this paper, we identify a simple combinatorial property such that any 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.  ... 
arXiv:1308.1640v4 fatcat:es3qwpss5zhgdgd5dvnxczmehu

Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems [article]

D. M. Stull
2017 arXiv   pre-print
Specifically, we prove that if Σ_2E does not have 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.  ... 
arXiv:1701.04428v1 fatcat:jf5fce3nijasjeso62ol4gmj2e

Parity, circuits, and the polynomial-time hierarchy

Merrick Furst, James B. Saxe, Michael Sipser
1984 Mathematical Systems Theory  
A super-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  ... 
doi:10.1007/bf01744431 fatcat:ryq53ji6ejb5rnjodgbnxbhg3u

Proving Circuit Lower Bounds in High Uniform Classes

Akinori KAWACHI
2014 Interdisciplinary Information Sciences  
For example, it was shown by Ajtai [3] and Furst, Saxe and Sipser [21] that no AC 0 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).  ... 
doi:10.4036/iis.2014.1 fatcat:l3w445j5vngsvh665qugvuhxx4

Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

Zeev Dvir, Amir Shpilka, Amir Yehudayoff
2010 SIAM journal on computing (Print)  
That is, either NEXP ⊆ P/poly or the Permanent is not computable by 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.  ... 
doi:10.1137/080735850 fatcat:gonpk6xthzgdzldbrgsdqowlha

Hardness-randomness tradeoffs for bounded depth arithmetic circuits

Zeev Dvir, Amir Shpilka, Amir Yehudayoff
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 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.  ... 
doi:10.1145/1374376.1374482 dblp:conf/stoc/DvirSY08 fatcat:2meotxjq4zff7bbpqbqkqvekuq

On Lower Bounds for Constant Width Arithmetic Circuits [article]

V. Arvind, Pushkar S. Joglekar, Srikanth Srinivasan
2009 arXiv   pre-print
For every k > 1, we provide an explicit 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?  ... 
arXiv:0907.3780v2 fatcat:s54xuq2nwba45pjcnw23bkflhi
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