How Do Read-Once Formulae Shrink?

Let f be a de Morgan read-once function of n variables. Let f e be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 -e)/2, the value 1 with the same probability, and leaving it unassigned with probability e. We show that f e depends, on the average, on only O (e α n + e n 1/α ) variables, where . This result is asymptotically the tightest possible. It improves a similar result obtained recently by Hastad, Razborov and Yao.

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