The traditional Byzantine quorum-system model assumes a pre-existing, global agreement on the set of quorums (typically defined as the sets consisting of more than two-thirds of the participants). This assumption is problematic in permissionless systems, which strive to allow anyone to join or leave the system dynamically. While proof-of-stake permissionless systems like Ethereum require newly joining participants to register into the system, other permissionless systems like the Ripple Ledger or the Stellar network allow participants to join the system without synchronization by forgoing agreement on the set of quorums. This results in what we call a heterogeneous quorum system, where each participant has its own, personal set of quorums. An important question is to determine under what condition is it possible to solve synchronization problems like reliable broadcast or consensus in a heterogeneous quorum system. In this work, we show that the traditional quorum intersection and quorum availability conditions are not sufficient in heterogeneous quorum systems. Moreover, we propose quorum subsumption, a new condition which, together with quorum availability and quorum intersection, is sufficient to allow solving reliable broadcast and consensus. Finally, we propose protocols for reliable broadcast and consensus in heterogeneous quorum systems that satisfy quorum subsumption. In particular, we present a practical consensus protocol called Satrapy which in contrast to abstract consensus protocols uses finite state and messages.
Two mobile agents, modeled as points in the plane moving at speed 1, have to get at a distance at most 1 from each other. This task is known as approach or rendezvous in the plane. An adversary initially places both agents at distinct points, called their bases, at distance at most D, and wakes them up at possibly different times. Each of the agents has a fuel tank that allows them to traverse a trajectory of length D, and can be replenished at the base of the agent. The algorithm of each agent consists of a series of actions which are either moves at a chosen distance in a chosen direction or staying idle for a chosen period of time. For a given instance of the approach task, the execution time of an approach algorithm is the length of the period between the start of the later agent and the moment of approach. Our goal is to design approach algorithms with optimal time complexity . We consider two independent coherence assumptions. One of them is time coherence, i.e., agents start simultaneously, and the other is orientation coherence: agents have compatible compasses, showing the same North direction. Our main result is establishing optimal time complexity of the approach problem with restricted fuel tanks. It turns out that this optimal complexity heavily depends on the above coherence assumptions. If both of them are satisfied then approach can be performed in time O(D^2) and we show that this complexity is optimal. If any of the two coherence assumptions is missing then approach can be performed in time O(D^2√(D)) and we prove that this order of magnitude cannot be improved. Our main technical contributions are lower bounds showing that, for each of the considered scenarios, our fairly natural approach algorithms are, in fact, optimal.
In order to develop solutions that perform actions as early as possible, analysis of distributed algorithms using epistemic logic has generally concentrated on “full-information protocols”, which may be inefficient with respect to space and computation time. The paper reconsiders the epistemic analysis of the problem of Simultaneous Byzantine Agreement with respect to weaker, but more practical, exchanges of information. This paper first clarifies some issues concerning both the specification of this problem and the knowledge based program characterizing its solution. One of these differences concerns the distinction between the notions of “nonfaulty” and “not yet failed”, on which there are variances in the literature. A second difference in the literature concerns the use of common knowledge versus common belief in the knowledge based program. The paper identifies situations where these notions are equivalent, but establishes that the version of the knowledge based program using common belief of the nonfaulty agent is more general. It is then shown that, when implemented relative to a given failure model and an information exchange protocol satisfying certain conditions, the common belief based knowledge based program yields a protocol that is optimal relative to solutions using the same information exchange. Conditions are also identified under which this implementation is also an optimum, but an example is provided that shows this does not hold in general.
Several mobile agents, modelled as deterministic automata, navigate in an infinite line in synchronous rounds. All agents start in the same round. In each round, an agent can move to one of the two neighboring nodes, or stay idle. Agents have distinct labels which are integers from the set {1,…, L}. They start in teams, and all agents in a team have the same starting node. The adversary decides the compositions of teams, and their starting nodes. Whenever an agent enters a node, it sees the entry port number and the states of all collocated agents; this information forms the input of the agent on the basis of which it transits to the next state and decides the current action. The aim is for all agents to gather at the same node and stop. Gathering is feasible, if this task can be accomplished for any decisions of the adversary, and its time is the worst-case number of rounds from the start till gathering. We consider the feasibility and time complexity of gathering teams of agents, and give a complete solution of this problem. It turns out that both feasibility and complexity of gathering depend on the sizes of teams. We first concentrate on the case when all teams have the same size x. For the oriented line, gathering is impossible if x=1, and it can be accomplished in time O(D), for x>1, where D is the distance between the starting nodes of the most distant teams. This complexity is of course optimal. For the unoriented line, the situation is different. For x=1, gathering is also impossible, but for x=2, the optimal time of gathering is Θ(Dlog L), and for x≥ 3, the optimal time of gathering is Θ(D). In the case when there are teams of different sizes, we show that gathering is always possible in time O(D), even for the unoriented line. This complexity is of course optimal.
We study the computational power that oblivious robots operating in the plane have under sequential schedulers. We show that this power is much stronger than the obvious capacity these schedulers offer of breaking symmetry, and thus to create a leader. More precisely, we consider the class of pattern formation problems, and focus on the most general problem in this class, Universal Pattern Formation (UPF), which requires the robots to form any pattern given in input, starting from any initial configurations (where robots may occupy the same point). We first show that UPF is unsolvable under ℱ𝒮𝒴𝒩𝒞 , even if the robots are endowed with additional strong capabilities (multiplicity detection, rigid movement, agreement on coordinate systems, presence of a unique leader). On the other hand, we prove that, except for point formation (Gathering), UPF is solvable under any sequential scheduler without any additional assumptions. We then turn our attention to the Gathering problem, and prove that weak multiplicity detection is necessary and sufficient for solvability under sequential schedulers. The obtained results show that the computational power of the robots under ℱ𝒮𝒴𝒩𝒞 (where Gathering is solvable without any multiplicity detection) and that under sequential schedulers are orthogonal.
