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Shrinkage of Decision Lists and DNF Formulas

Author Benjamin Rossman

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Benjamin Rossman
  • Duke University, Durham, NC, USA

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.

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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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