Shrinkage of Decision Lists and DNF Formulas

12/09/2020

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by   Benjamin Rossman, et al.

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We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the p-random restriction 𝐑_p for all values of p ∈ [0,1]. For a function f with domain {0,1}^n, let DL(f) denote the minimum size of a decision list that computes f. We show that 𝔼[ DL(f↾𝐑_p) ] ≤DL(f)^log_2/(1-p)(1+p/1-p). For example, this bound is √(DL(f)) when p = √(5)-2 ≈ 0.24. For Boolean functions f, we obtain the same shrinkage bound with respect to DNF formula size plus 1 (i.e., replacing DL(·) with DNF(·)+1 on both sides of the inequality).