On the Classification of Finite Boolean Functions up to Fairness

Abstract. Two parties, P1 and P2, wish to jointly compute some func-tion f(x, y) where P1 only knows x, whereas P2 only knows y. Further-more, and most importantly, the parties wish to reveal only what the output suggests. Function f is said to be computable with complete fair-ness 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 func-tions 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 (po-tentially) 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 partic-ular, 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 fi-nite Boolean functions: almost all functions f: X×Y → {0, 1} for whic