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Almost Optimal Testers for Concise Representations

Author Nader H. Bshouty

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Nader H. Bshouty
  • Department of Computer Science, Technion, Haifa, Israel

Acknowledgements

I am extremely grateful to Oded Goldreich for very helpful discussions and comments and for allowing me to include his overview of the first tester in the paper (Section 2). I am also grateful to the reviewers of the earlier version of the manuscript for their helpful comments.

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Nader H. Bshouty. Almost Optimal Testers for Concise Representations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.5

Abstract

We give improved and almost optimal testers for several classes of Boolean functions on n variables that have concise representation in the uniform and distribution-free model. Classes, such as k-Junta, k-Linear, s-Term DNF, s-Term Monotone DNF, r-DNF, Decision List, r-Decision List, size-s Decision Tree, size-s Boolean Formula, size-s Branching Program, s-Sparse Polynomial over the binary field and functions with Fourier Degree at most d.
The approach is new and combines ideas from Diakonikolas et al. [Ilias Diakonikolas et al., 2007], Bshouty [Nader H. Bshouty, 2018], Goldreich et al. [Oded Goldreich et al., 1998], and learning theory. The method can be extended to several other classes of functions over any domain that can be approximated by functions with a small number of relevant variables.

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Probabilistic algorithms
  • Theory of computation → Probabilistic computation
Keywords
  • Property Testing
  • Boolean function
  • Junta

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