Extended impossibility results for asynchronous complete networks

Abstract

It is proved that a large class of distributed tasks cannot be solved in the presence of faulty processors. This class contains tasks whose unsolvability in the presence of faults is known (the consensus task and its variants. cf. Fischer et al. (1985)) as well as some new tasks (e.g., constructing a spanning tree). In particular, we introduce the notion of the decision graph of a task, and show that every problem whose decision graph is disconnected cannot be solved in the presence of one faulty processor, by reducing the unsolvability of this problem to the unsolvability of the consensus problem. The notion of unsolvability used here is very weak: We say that a protocol solves a given problem in spite of one faulty processor if in any execution it satisfies (i) all nonfaulty processors eventually halt, and (ii) if no processor is faulty, it solves the problem. Hence, the unsolvability of a problem in this model implies its unsolvability in other models appearing in the literature.