TY - JOUR

T1 - Improving 3 N Circuit Complexity Lower Bounds

AU - Find, Magnus Gausdal

AU - Golovnev, Alexander

AU - Hirsch, Edward A.

AU - Kulikov, Alexander S.

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2023/12

Y1 - 2023/12

N2 - While it can be easily shown by counting that almost allBoolean predicates of n variables have circuit size Ω (2 n/ n) , we haveno example of an NP function requiring even a superlinear number of gates. Moreover, only modest linear lower bounds are known. Untilrecently, the strongest known lower bound was 3 n- o(n) presentedby Blum in 1984. In 2011, Demenkov and Kulikov presented a much simpler proof of the same lower bound, but for a more complicated function—an affine disperser for sublinear dimension. Informally, this is a function that is resistant to any n- o(n) affine substitutions. In 2011,Ben-Sasson and Kopparty gave an explicit construction of such a function. The proof of the lower bound basically goes by showing that for any circuit there exists an affine hyperplane where the function complexitydecreases at least by three gates. In this paper, we prove the following two extensions. 1. A (3+186n-o(n) lower bound for the circuit size of an affinedisperser for sublinear dimension. The proof is based on the gate elimination technique extended with the following three ideas:(i) generalizing the computational model by allowing circuits tocontain cycles, this in turn allows us to perform affine substitutions,(ii) a carefully chosen circuit complexity measure to trackthe progress of the gate elimination process, and (iii) quadraticsubstitutions that may be viewed as delayed affine substitutions. 2. A much simpler proof of a stronger lower bound of 3.11 n for aquadratic disperser. Informally, a quadratic disperser is resistantto sufficiently many substitutions of the form x← p , where pis a polynomial of degree at most two. Currently, there are noconstructions of quadratic dispersers in NP (although there are constructions over large fields, and constructions with weaker parametersover GF(2)). The key ingredient of this proof is theinduction on the size of the underlying quadratic variety insteadof the number of variables as in the previously known proofs.

AB - While it can be easily shown by counting that almost allBoolean predicates of n variables have circuit size Ω (2 n/ n) , we haveno example of an NP function requiring even a superlinear number of gates. Moreover, only modest linear lower bounds are known. Untilrecently, the strongest known lower bound was 3 n- o(n) presentedby Blum in 1984. In 2011, Demenkov and Kulikov presented a much simpler proof of the same lower bound, but for a more complicated function—an affine disperser for sublinear dimension. Informally, this is a function that is resistant to any n- o(n) affine substitutions. In 2011,Ben-Sasson and Kopparty gave an explicit construction of such a function. The proof of the lower bound basically goes by showing that for any circuit there exists an affine hyperplane where the function complexitydecreases at least by three gates. In this paper, we prove the following two extensions. 1. A (3+186n-o(n) lower bound for the circuit size of an affinedisperser for sublinear dimension. The proof is based on the gate elimination technique extended with the following three ideas:(i) generalizing the computational model by allowing circuits tocontain cycles, this in turn allows us to perform affine substitutions,(ii) a carefully chosen circuit complexity measure to trackthe progress of the gate elimination process, and (iii) quadraticsubstitutions that may be viewed as delayed affine substitutions. 2. A much simpler proof of a stronger lower bound of 3.11 n for aquadratic disperser. Informally, a quadratic disperser is resistantto sufficiently many substitutions of the form x← p , where pis a polynomial of degree at most two. Currently, there are noconstructions of quadratic dispersers in NP (although there are constructions over large fields, and constructions with weaker parametersover GF(2)). The key ingredient of this proof is theinduction on the size of the underlying quadratic variety insteadof the number of variables as in the previously known proofs.

KW - 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

KW - Boolean circuits

KW - dispersers

KW - lower bounds

UR - http://www.scopus.com/inward/record.url?scp=85179925507&partnerID=8YFLogxK

U2 - 10.1007/s00037-023-00246-9

DO - 10.1007/s00037-023-00246-9

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AN - SCOPUS:85179925507

SN - 1016-3328

VL - 32

JO - Computational Complexity

JF - Computational Complexity

IS - 2

M1 - 13

ER -