We provide a complete characterization of the solvability/impossibility of deterministic stabilizing consensus in virtually any computing model with benign process and communication faults using point-set topology. Relying on the topologies for infinite executions introduced by Nowak, Schmid and Winkler (JACM, 2024) for terminating consensus, we show that semi-open decision sets and semi-continuous decision functions as introduced by Levin (AMM, 1963) are the appropriate means for this characterization: Unlike the continuous decision functions for terminating consensus, semi-continuous functions do not require the inverse image of an open set to be open and hence allow to map a connected space to a disconnected one. We also show that multi-valued stabilizing consensus with weak and strong validity are equivalent, as is the case for terminating consensus. By applying our results to (variants of) all the known possibilities/impossibilities for stabilizing consensus, we easily provide a topological explanation of these results.

We develop a new lower bound for k-set agreement in synchronous message-passing systems connected by an arbitrary directed communication network, where up to t processes may crash. Our result thus generalizes the ⌊t/k⌋ + 1 lower bound for complete networks in the t-resilient model by Chaudhuri, Herlihy, Lynch, and Tuttle [JACM 2000]. Moreover, it generalizes two lower bounds for oblivious algorithms in synchronous systems connected by an arbitrary undirected communication network known to the processes, namely, the domination number-based lower bound by Castañeda, Fraigniaud, Paz, Rajsbaum, Roy, and Travers [TCS 2021] for failure-free processes, and the radius-based lower bound in the t-resilient model by Fraigniaud, Nguyen, and Paz [STACS 2024]. Our topological proof non-trivially generalizes and extends the connectivity-based approach for the complete network, as presented in the book by Herlihy, Kozlov, and Rajsbaum (2013). It is based on a sequence of shellable carrier maps that, starting from a shellable input complex, determine the evolution of the protocol complex: During the first ⌊t/k⌋ rounds, carrier maps that crash exactly k processes per round are used, which ensure high connectivity of their images. A Sperner’s lemma style argument can thus be used to prove that k-set agreement is still impossible by that round. From round ⌊t/k⌋ + 1 up to our actual lower bound, a novel carrier map is employed, which maintains high connectivity. As a by-product, our proof also provides a strikingly simple lower-bound for k-set agreement in synchronous systems with an arbitrary communication network, where exactly t ≥ 0 processes crash initially, i.e., before taking any step. We demonstrate that the resulting additional agreement overhead can be expressed via an appropriately defined radius of the communication graphs, and show that the usual input pseudosphere complex for k-set agreement can be replaced by an exponentially smaller input complex based on Kuhn triangulations, which we prove to be also shellable.

Abstract We study the problem of solving consensus in synchronous directed dynamic networks, in which communication is controlled by an oblivious message adversary that picks the communication graph to be used in a round from a fixed set of graphs $$\textbf{D}$$ D arbitrarily. In this fundamental model, determining consensus solvability and designing efficient consensus algorithms is surprisingly difficult. Enabled by a decision procedure that is derived from a well-established previous consensus solvability characterization for a given set $$\textbf{D}$$ D , we study, for the first time, the time complexity of solving consensus in this model: We provide both upper and lower bounds for this time complexity, and also relate it to the number of iterations required by the decision procedure. Among other results, we find that reaching consensus under an oblivious message adversary can take exponentially longer than both deciding consensus solvability and broadcasting the input value of some unknown process to all other processes.

We study the problem of solving consensus in synchronous directed dynamic networks, in which communication is controlled by an oblivious message adversary that picks the communication graph to be used in a round from a fixed set of graphs 𝐃 arbitrarily. In this fundamental model, determining consensus solvability and designing efficient consensus algorithms is surprisingly difficult. Enabled by a decision procedure that is derived from a well-established previous consensus solvability characterization for a given set 𝐃, we study, for the first time, the time complexity of solving consensus in this model: We provide both upper and lower bounds for this time complexity, and also relate it to the number of iterations required by the decision procedure. Among other results, we find that reaching consensus under an oblivious message adversary can take exponentially longer than both deciding consensus solvability and broadcasting the input value of some unknown process to all other processes.

The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized distributed tasks that are wait-free solvable and uncovered deep connections with combinatorial topology. We provide an alternative characterization of those tasks by means of the novel concept of continuous tasks, which have an input/output specification that is a continuous function between the geometric realizations of the input and output complex: We state and prove a precise characterization theorem (CACT) for wait-free solvable tasks in terms of continuous tasks. Its proof utilizes a novel chromatic version of a foundational result in algebraic topology, the simplicial approximation theorem, which is also proved in this paper. Apart from the alternative proof of the ACT implied by our CACT, we also demonstrate that continuous tasks have an expressive power that goes beyond classic task specifications, and hence open up a promising venue for future research: For the well-known approximate agreement task, we show that one can easily encode the desired proportion of the occurrence of specific outputs, namely, exact agreement, in the continuous task specification.