Volatility of Boolean functions
Preprint, 2014

We study the volatility of the output of a Boolean function when the in- put bits undergo a natural dynamics. For n = 1, 2, . . ., let fn : {0, 1}mn → {0, 1} be a Boolean function and X(n)(t) = (X1(t), . . . , Xmn (t))t∈[0,∞) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jumpfrom0to1withratepn ∈[0,1]andfrom1to0withrateqn =1−pn. Our object of study will be Cn which is the number of state changes of fn(X(n)(t)) as a function of t during [0, 1]. We say that the family {fn}n≥1 is volatile if Cn → ∞ in distribution as n → ∞ and say that {fn}n≥1 is tame if {Cn}n≥1 is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitiv- ity and noise stability. In addition, we study the question of lameness which means that P(Cn = 0) → 1 as n → ∞. Finally, we investigate these prop- erties for the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees in various regimes.

noise sensitivity

Boolean function

noise stability


Johan Jonasson

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Jeffrey Steif

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik