Effectively computable obstructions are associated to a distributed decision task (/spl Iscr/,/spl Oscr/,/spl Delta/) in the asynchronous, wait-free, read-write shared-memory model. The key new ingredient of this work is the association of a simplicial complex /spl Tscr/, the task complex, to the input-output relation d. The task determines a simplicial map /spl alpha/ from /spl Tscr/ to the input complex /spl Iscr/. The existence of a wait-free protocol solving the task implies that the map /spl alpha//sub */ induced in homology must surject, and thus elements of H/sub */(/spl Iscr/) that are not in the image of /spl alpha//sub */, are obstructions to solvability of the task. These obstructions are effectively computable when using suitable homology theories, such as mod-2 simplicial homology. We also extend Herlihy and Shavit's Theorem on Spans to the case of protocols that are anonymous relative to the action of a group, provided the action is suitably rigid. For such rigid actions, the quotients of the input complex and the task complex by the group are well-behaved, and obstructions to anonymous solvability of the task are obtained analogously, using the homology of the quotient complexes.
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