The topological view on distributed computing represents a task T as a relation Δ between the complex ℐ of its inputs and the complex 𝒪 of its outputs. A cornerstone result in the field is an elegant computability characterization of the solvability of colorless tasks in terms of ℐ, 𝒪 and Δ. Essentially, a colorless task is wait-free solvable if and only if there is a continuous map from the geometric realization of ℐ to that of 𝒪 that respects Δ. This paper makes headway towards providing an analogous characterization for general tasks, which are not necessarily colorless, by concentrating on the case of three-process inputless tasks. Our key contribution is identifying local articulation points as an obstacle for the solvability of general tasks, and defining a topological deformation on the output complex of a task T, which eliminates these points by splitting them, to obtain a new task T', with an adjusted relation Δ' between the input complex ℐ and an output complex 𝒪' without articulation points. We obtain a new characterization of wait-free solvability of three-process general tasks: T is wait-free solvable if and only if there is a continuous map from the geometric realization of ℐ to that of 𝒪' that respects Δ'.
@InProceedings{attiya_et_al:LIPIcs.DISC.2024.42, author = {Attiya, Hagit and Fraigniaud, Pierre and Paz, Ami and Rajsbaum, Sergio}, title = {{Brief Announcement: Solvability of Three-Process General Tasks}}, booktitle = {38th International Symposium on Distributed Computing (DISC 2024)}, pages = {42:1--42:7}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-352-2}, ISSN = {1868-8969}, year = {2024}, volume = {319}, editor = {Alistarh, Dan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2024.42}, URN = {urn:nbn:de:0030-drops-212700}, doi = {10.4230/LIPIcs.DISC.2024.42}, annote = {Keywords: Wait-free computing, lower bounds, topology} }
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