On the Sensitivity of Cyclically-Invariant Boolean Functions

In this paper we construct a cyclically invariant Boolean function whose sensitivity is Θ(n1/3 ). This result answers two previously published questions. Tur´an (1984) asked if any Boolean function, invariant under some transitive group of permutations, has sensitivity Ω( √ n). Kenyon and Kutin (2004) asked whether for a “nice” function the product of 0-sensitivity and 1-sensitivity is Ω(n). Our function answers both questions in the negative. We also prove that for minterm-transitive functions (a natural class of Boolean functions including our example) the sensitivity is Ω(n1/3 ). Hence for this class of functions sensitivity and block sensitivity are polynomially related.

• Type: Conference
• Year: 2005
• Where: COCO
• Authors: Sourav Chakraborty
• Keywords: Boolean Functions, COCO 2005, Invariant Boolean Function, Transitive Group