The effect of random restrictions on formula size

Russell Impagliazzo*, Noam Nisan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations

Abstract

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.

Original languageAmerican English
Pages (from-to)121-133
Number of pages13
JournalRandom Structures and Algorithms
Volume4
Issue number2
DOIs
StatePublished - 1993

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