Lower bound on wait-free counting

A counting protocol (mod m) consists of shared memory bits - referred to as the counter - and of a procedure for incrementing the counter value by 1 (mod m). The procedure may be executed by many processes concurrently. It is required to satisfy a very weak correctness requirement, namely: the counter is required to show a correct value only in quiescent states - states in which no process is incrementing the counter. Special cases of counting protocols are `counting networks' [AHS91] and `concurrent counters' [MTY92]. We consider the problem of implementing a wait-free counting protocol, assuming that the basic atomic operation of a process is a read-modify-write on a single bit. Let flip(Pr) be the maximum number of times a single increment operation changes the counter bits in a counting protocol Pr. Our main result is: In any wait-free counting protocol Pr which counts modulo m, m divides 2^flip(Pr). Thus, flip(Pr)≥log m and m is a power of 2. This result provides interesting generalizations of lower bounds and impossibility results for counting and smoothing networks.