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Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers

Authors Chengyu Lin, Shengyu Zhang

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Chengyu Lin
Shengyu Zhang

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Chengyu Lin and Shengyu Zhang. Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.51

Abstract

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for f(x and y) and f(x xor y) with monotone functions f, where and and xor are bit-wise AND and XOR , respectively. In this paper, we extend these results to functions f which alternate values for a relatively small number of times on any monotone path from 0^n to 1^n. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.
Keywords
  • Analysis of Boolean functions
  • Sensitivity Conjecture
  • Log-rank Conjecture
  • Alternating Number

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