Abstract
The asynchronous computability theorem (ACT) uses concepts from combinatorial topology to characterize which tasks have wait-free solutions in read–write memory. A task can be expressed as a relation between two chromatic simplicial complexes. The theorem states that a task has a protocol (algorithm) if and only if there is a certain color-preserving simplicial map compatible with that relation. The original proof of the ACT, given by Herlihy and Shavit (Proceedings of the 25th annual ACM symposium on theory of computing, pp 111–120, 1993; J ACM 46(6):858–923, 1999) relied on an involved geometric argument. Borowsky and Gafni (Proceedings of the 16th annual ACM symposium on principles of distributed computing, pp 189–198, 1997) later proposed an alternative proof based on a distributed algorithmic, termed the “convergence algorithm”. However the description of this algorithm was incomplete, and presented without proof. In this paper, we give the first complete description, along with a proof of correctness.
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Notes
Borowsky and Gafni called the view the “core”.
Borowsky and Gafni called this the “intersection of cores”.
\({{\mathrm{Lk}}}(c, P)\) is a slight abuse of notation. For this notation to make sense, c must be a simplex and P must be a complex. Strictly speaking, while P is a set of processes and not itself a complex, it can be identified with the subcomplex \({{\mathrm{Div}}}(\sigma _P)\), where \(\sigma _P\) is the subset of input simplex \(\Sigma \) colored exactly by processes in P. It is shown in the next section that c is a simplex in \({{\mathrm{Div}}}(\sigma _P)\), so \({{\mathrm{Lk}}}(c, P)\) is shorthand for \({{\mathrm{Lk}}}(c, {{\mathrm{Div}}}(\sigma _P))\).
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Saraph, V., Herlihy, M. & Gafni, E. An algorithmic approach to the asynchronous computability theorem. J Appl. and Comput. Topology 1, 451–474 (2018). https://doi.org/10.1007/s41468-018-0014-4
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DOI: https://doi.org/10.1007/s41468-018-0014-4