TY - GEN

T1 - Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in

AU - Karnin, Zohar S.

AU - Mukhopadhyay, Partha

AU - Shpilka, Amir

AU - Volkovich, Ilya

PY - 2010

Y1 - 2010

N2 - We give the first sub-exponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 circuits with a plus gate at the top (also known as ΣΠΣΠ circuits) and has a running time of exp(poly(log(n),log(s),k)) where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. In [AV08], it was shown that derandomizing polynomial identity testing for general ΣΠΣΠ circuits implies a derandomization of polynomial identity testing in general arithmetic circuits. Prior to this work sub-exponential time deterministic algorithms were known for depth-3 circuits with small top fan-in and for very restricted versions of depth-4 circuits. The main ingredient in our proof is a new structural theorem for multilinear ΣΠΣΠ(k) circuits. Roughly, this theorem shows that any nonzero multilinear ΣΠΣΠ(k) circuit contains an 'embedded' nonzero multilinear ΣΠΣ(k) circuit. Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for multilinear ΣΠΣΠ(k) circuits.

AB - We give the first sub-exponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 circuits with a plus gate at the top (also known as ΣΠΣΠ circuits) and has a running time of exp(poly(log(n),log(s),k)) where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. In [AV08], it was shown that derandomizing polynomial identity testing for general ΣΠΣΠ circuits implies a derandomization of polynomial identity testing in general arithmetic circuits. Prior to this work sub-exponential time deterministic algorithms were known for depth-3 circuits with small top fan-in and for very restricted versions of depth-4 circuits. The main ingredient in our proof is a new structural theorem for multilinear ΣΠΣΠ(k) circuits. Roughly, this theorem shows that any nonzero multilinear ΣΠΣΠ(k) circuit contains an 'embedded' nonzero multilinear ΣΠΣ(k) circuit. Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for multilinear ΣΠΣΠ(k) circuits.

KW - arithmetic circuits

KW - bounded depth circuits

KW - derandomization

KW - identity testing

KW - multilinear circuits

UR - http://www.scopus.com/inward/record.url?scp=77954701873&partnerID=8YFLogxK

U2 - 10.1145/1806689.1806779

DO - 10.1145/1806689.1806779

M3 - פרסום בספר כנס

AN - SCOPUS:77954701873

SN - 9781605588179

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 649

EP - 657

BT - STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing

Y2 - 5 June 2010 through 8 June 2010

ER -