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# Shrinkage of Decision Lists and DNF Formulas

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## Acknowledgements

I am grateful to the anonymous referees of ITCS 2021 for their valuable comments and to the authors of [Lovett et al., 2020], Shachar Lovett, Kewen Wu and Jiapeng Zhang, for stimulating conversations related to this work.

## Cite As

Benjamin Rossman. Shrinkage of Decision Lists and DNF Formulas. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.70

## Abstract

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).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Circuit complexity
##### Keywords
• shrinkage
• decision lists
• DNF formulas

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