TY - JOUR

T1 - The effect of random restrictions on formula size

AU - Impagliazzo, Russell

AU - Nisan, Noam

PY - 1993

Y1 - 1993

N2 - We consider the formula size complexity of boolean functions over the base consisting of ∧, ∨, and ¬ gates. In 1961 Subbotovskaya proved that, for any boolean function on n variables, setting all but a randomly chosen ϵn variables to randomly chosen constants, reduces the expected complexity of the induced function by at least a factor. This fact was used by Subbotovskaya to derive a lower bound of Ω(n1.5) for the formula size of the parity function, and more recently by Andreev who exhibited at lower bound of ΩC(n2.5/logO(1)(n)) for a function in P. We present the first improvement of Subbotovskaya's result, showing that the expected formual complexity of a function restricted as above is at most an O(ϵγ) franction of its original complexity, for (Formula Presented.) This allows us to give an improved lower bound of Ω(n2.556…) for Andreev's function. At the time of discovery, this was the best formula lower bound known for any function in NP. © 1993 John Wiley & Sons, Inc.

AB - We consider the formula size complexity of boolean functions over the base consisting of ∧, ∨, and ¬ gates. In 1961 Subbotovskaya proved that, for any boolean function on n variables, setting all but a randomly chosen ϵn variables to randomly chosen constants, reduces the expected complexity of the induced function by at least a factor. This fact was used by Subbotovskaya to derive a lower bound of Ω(n1.5) for the formula size of the parity function, and more recently by Andreev who exhibited at lower bound of ΩC(n2.5/logO(1)(n)) for a function in P. We present the first improvement of Subbotovskaya's result, showing that the expected formual complexity of a function restricted as above is at most an O(ϵγ) franction of its original complexity, for (Formula Presented.) This allows us to give an improved lower bound of Ω(n2.556…) for Andreev's function. At the time of discovery, this was the best formula lower bound known for any function in NP. © 1993 John Wiley & Sons, Inc.

UR - http://www.scopus.com/inward/record.url?scp=84990716822&partnerID=8YFLogxK

U2 - 10.1002/rsa.3240040202

DO - 10.1002/rsa.3240040202

M3 - Article

AN - SCOPUS:84990716822

SN - 1042-9832

VL - 4

SP - 121

EP - 133

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 2

ER -