TY - GEN
T1 - Linear systems over composite moduli
AU - Chattopadhyay, Arkadev
AU - Wigderson, Avi
PY - 2009
Y1 - 2009
N2 - We study solution sets to systems of generalized linear equations of the form ℓi(x1,x2, middot;middot;middot;, xn) ∈ Ai (mod m) where ℓ1,⋯, ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of ℤm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. exp (- Ω(n)), with the boolean function MODq, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constant-depth circuits with modular gates. We derive the first exponential lower bound on the size of depth-three circuits of type MAJ o AND o MODAm (i.e. having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MOD m gates at the base) computing the function MODq. This settles an open problem of Beigel and Maciel [5], for the case of such modulus m. Our technique makes use of the work of Bourgain [6] on estimating exponential sums involving a low-degree polynomial and ideas involving matrix rigidity from the work of Grigoriev and Razborov [15] on arithmetic circuits over finite fields.
AB - We study solution sets to systems of generalized linear equations of the form ℓi(x1,x2, middot;middot;middot;, xn) ∈ Ai (mod m) where ℓ1,⋯, ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of ℤm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. exp (- Ω(n)), with the boolean function MODq, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constant-depth circuits with modular gates. We derive the first exponential lower bound on the size of depth-three circuits of type MAJ o AND o MODAm (i.e. having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MOD m gates at the base) computing the function MODq. This settles an open problem of Beigel and Maciel [5], for the case of such modulus m. Our technique makes use of the work of Bourgain [6] on estimating exponential sums involving a low-degree polynomial and ideas involving matrix rigidity from the work of Grigoriev and Razborov [15] on arithmetic circuits over finite fields.
KW - Boolean circuit complexity
KW - Constant-depth circuits
KW - Exponential sums
KW - Matrix rigidity
KW - Modular gates
UR - http://www.scopus.com/inward/record.url?scp=77952341043&partnerID=8YFLogxK
U2 - 10.1109/FOCS.2009.17
DO - 10.1109/FOCS.2009.17
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AN - SCOPUS:77952341043
SN - 9780769538501
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 43
EP - 52
BT - Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
T2 - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
Y2 - 25 October 2009 through 27 October 2009
ER -