On the classification of finite Boolean functions up to fairness

Two parties, P₁ and P₂, wish to jointly compute some function f(x, y) where P₁ only knows x, whereas P₂ only knows y. Furthermore, and most importantly, the parties wish to reveal only what the output suggests. Function f is said to be computable with complete fairness if there exists a protocol computing f such that whenever one of the parties obtains the correct output, then both of them do. The only protocol known to compute functions with complete fairness is the one of Gordon et al (STOC 2008). The functions in question are finite, Boolean, and the output is shared by both parties. The classification of such functions up to fairness may be a first step towards the classification of all functionalities up to fairness. Recently, Asharov (TCC 2014) identifies two families of functions that are computable with fairness using the protocol of Gordon et al and another family for which the protocol (potentially) falls short. Surprisingly, these families account for almost all finite Boolean functions. In this paper, we expand our understanding of what can be computed fairly with the protocol of Gordon et al. In particular, we fully describe which functions the protocol computes fairly and which it (potentially) does not. Furthermore, we present a new class of functions for which fair computation is outright impossible. Finally, we confirm and expand Asharov’s observation regarding the fairness of finite Boolean functions: almost all functions f: X×Y → {0, 1} for which |X| ≠ |Y| are fair, whereas almost all functions for which |X| = |Y| are not.