Shrinkage of de Morgan formulae under restriction

Michael S. Paterson*, Uri Zwick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations

Abstract

It is shown that a random restriction leaving only a fraction ϵ of the input variables unassigned reduces the expected de Morgan formula size of the induced function by a factor of O(E(5−√2)/2) = O(ϵ1.63). (A de Morgan, or unate, formula is a formula over the basis {∧, ∨, ¬}.) This improves a long‐standing result of O(ϵ1.5) by Subbotovskaya and a recent improvement to O(ϵ(21−√73)/8) = O(ϵ1.55) by Nisan and Impagliazzo. The New exponent yields an increased lower bound of n(7−√3)/2−o(1) = Ω(n2.63) for the de Morgan formula size of a function in P defined by Andreev. This is the largest formula size lower bound known, even for functions in NP. © 1993 John Wiley & Sons, Inc.

Original languageEnglish
Pages (from-to)135-150
Number of pages16
JournalRandom Structures and Algorithms
Volume4
Issue number2
DOIs
StatePublished - 1993

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