Shrinkage of Decision Lists and DNF Formulas
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We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the p-random restriction π_p for all values of p β [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 πΌ[ DL(fβΎπ_p) ] β€DL(f)^log_2/(1-p)(1+p/1-p). For example, this bound is β(DL(f)) when p = β(5)-2 β 0.24. For Boolean functions f, we obtain the same shrinkage bound with respect to DNF formula size plus 1 (i.e., replacing DL(Β·) with DNF(Β·)+1 on both sides of the inequality).
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