Combinatorial and algebraic topology have both provided useful mathematical tools to the field of distributed computing, giving researchers a rigorous mathematical model in which one can prove elegant theorems about task solvability and computability. An early result in the field is the impossibility of deciding whether an arbitrary task is solvable, using a class of tasks called loop agreement. The first part of this thesis extends the original work by describing a natural way in which loop agreement tasks can be composed in parallel, additionally considering a category theoretic perspective of loop agreement. In previous work, Herlihy and Shavit formulated a topological characterization of wait-free task solvability, called the asynchronous computability theorem. However, the original proof relied on an intricate geometric argument with little insight on algorithmic intuition. This thesis develops a distributed protocol, called the convergence algorithm, for solving a chromatic simplex agreement task on simplicial complexes, and from this result, the original asynchronous computability theorem follows. More significantly though, the convergence algorithm highlights the importance of link-connectivity, a property of simplicial complexes, and its relation to more general models of fault-tolerance beyond wait-freedom. The asynchronous computability theorem is then generalized from wait-freedom to the t-resilient model, which permits up to t failures. To do so, the delayed snapshot is introduced, serving as a building block for t-resilient protocols. Using the convergence algorithm, it is shown that only one delayed snapshot, and hence one synchronization barrier, is ever required in implementing any t-resilient protocol, after which processes may run wait-free. This snapshot protocol is further extended to a model of fault-tolerance in which processes run in the presence of an adversary. It is demonstrated using this snapshot that one synchronization barrier may not suffice for tasks solvable against an adversary.
Saraph, Vikram,
"Fault-Tolerant Distributed Computability"
(2019).
Computer Science Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/rqph-3r36
