TY - GEN

T1 - Arithmetic circuits

T2 - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

AU - Gupta, Ankit

AU - Kamath, Pritish

AU - Kayal, Neeraj

AU - Saptharishi, Ramprasad

PY - 2013

Y1 - 2013

N2 - We show that, over ℚ, if an n-variate polynomial of degree d = n O(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O( √ d log n log d log s)) (respectively of size exp(O( √ d log n log s)). In particular this yields a ΣΠΣ circuit of size exp(O( √ d· log d)) computing the d × d determinant Detd. It also means that if we can prove a lower bound of exp(ω( √ d · log d)) on the size of any ΣΠΣ-circuit computing the d × d permanent Permd then we get superpolynomial lower bounds for the size of any arithmetic branching program computing Permd. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Detd or Permd, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)).

AB - We show that, over ℚ, if an n-variate polynomial of degree d = n O(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O( √ d log n log d log s)) (respectively of size exp(O( √ d log n log s)). In particular this yields a ΣΠΣ circuit of size exp(O( √ d· log d)) computing the d × d determinant Detd. It also means that if we can prove a lower bound of exp(ω( √ d · log d)) on the size of any ΣΠΣ-circuit computing the d × d permanent Permd then we get superpolynomial lower bounds for the size of any arithmetic branching program computing Permd. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Detd or Permd, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)).

UR - http://www.scopus.com/inward/record.url?scp=84893442229&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2013.68

DO - 10.1109/FOCS.2013.68

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AN - SCOPUS:84893442229

SN - 9780769551357

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 578

EP - 587

BT - Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

Y2 - 27 October 2013 through 29 October 2013

ER -