On Semantic Constraints in Distributed Systems, Part II: Equivalence-Completions and Their Hierarchies

The notion of completion has been proposed by Francez et al. (1992) to transform a nonequivalence-robust fairness notion to an equivalence-robust one while maintaining several properties of the source. However, a completion may not preserve strong feasibility|a necessary and su cient condition for a completion to be implementable. In this paper, we study the system requirement for a completion to be strongly feasible, and determine the strongest implementable completion for every given fairness notion. Moreover, for most systems we obtain a fairness notion, which we refer to as SG+, such that SG+ is the strongest fairness notion that is both implementable and equivalence-robust. We also provide a comprehensive comparison of SG+ and several well-known fairness notions and their minimal and maximal completions. Finally, we show that, if equivalence-robustness is dropped, then in general it is impossible to de ne a fairness notion that is implementable and stronger than all other implementable fairness notions, unless the system consists of only one interaction. This implies plenty of leeway in the design of fairness notions suitable for various applications.