-(revised Version)

It is proved that a large class of distributed tasks cannot,be solved in the presence of faulty processors. This class contains 'tasks who~ unsolvability in presence of faults is known (the conSensus task and its variants, cf. [4]) as well as some new tasks (e.g. constructing a spanning tree). In particular, we introduce the notion of th~ decision graph of a taslC, and show t1)at every problem whose decision graph is disconnected cannot be solved in the presence of one faulty processor, by reducing the 1insolvability 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 ofOne faulty processor if in any cxeCution it satisfies (1) aU non faulty processors eventually halt, and (2) if no processor is faulty, it solves the problem. Hence, the unsolvability ofa problem in this model implies its unsolvability in other models appearing in the literature.