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In Byzantine Agreement (BA), there is a set of $n$ parties, from which up to $t$ can act byzantine. All honest parties must eventually decide on a common value (agreement), which must belong to a set determined by the inputs (validity). Depending on the use case, this set can grow or shrink, leading to various possible desiderata collectively known as validity conditions. Varying the validity property requirement can affect the regime under which BA is solvable. We study how the selected validity property impacts BA solvability in the network-agnostic model, where the network can either be synchronous with up to $t_s$ byzantine parties or asynchronous with up to $t_a \leq t_s$ byzantine parties. We show that for any non-trivial validity property the condition $2t_s + t_a < n$ is necessary for BA to be solvable, even with cryptographic setup. Noteworthy, specializing this claim to $t_a = 0$ gives that $t < n / 2$ is required when one expects a purely synchronous protocol to also work in asynchrony when there are no corruptions. This is especially surprising given that for some validity properties $t < n$ are known to be achievable without the last stipulation. Thereafter, we give necessary and sufficient conditions for a validity property to render BA solvable, both for the case with cryptographic setup and for the one without. This traces the precise boundary of solvability in the network-agnostic model for every validity property. Our proof of sufficiency provides a universal protocol, that achieves BA for a given validity property whenever the provided conditions are satisfied., Comment: Working paper

The distributed computing literature considers multiple options for modeling communication. Most simply, communication is categorized as either synchronous or asynchronous. Synchronous communication assumes that messages get delivered within a publicly known timeframe and that parties' clocks are synchronized. Asynchronous communication, on the other hand, only assumes that messages get delivered eventually. A more nuanced approach, or a middle ground between the two extremes, is given by the partially synchronous model, which is arguably the most realistic option. This model comes in two commonly considered flavors: (i) The Global Stabilization Time (GST) model: after an (unknown) amount of time, the network becomes synchronous. This captures scenarios where network issues are transient. (ii) The Unknown Latency (UL) model: the network is, in fact, synchronous, but the message delay bound is unknown. This work formally establishes that any time-agnostic property that can be achieved by a protocol in the UL model can also be achieved by a (possibly different) protocol in the GST model. By time-agnostic, we mean properties that can depend on the order in which events happen but not on time as measured by the parties. Most properties considered in distributed computing are time-agnostic. The converse was already known, even without the time-agnostic requirement, so our result shows that the two network conditions are, under one sensible assumption, equally demanding.

Fractures of the proximal humerus are a frequent ailment in the senior population. One concern with such a case that presented in our emergency room was the risk of varus collapse following open reduction and internal fixation (ORIF) in a patient with marked calcar comminution. The surgical method used in this case of a NEER four-part fracture with medial column instability proved effective in terms of good surgery outcome, patient satisfaction, good range of motion, and low or no pain at post-surgical follow-ups. ORIF with a locking plate and fibular strut graft proved to be a viable alternative to solo plating or shoulder arthroplasty. We conducted upper extremity patient-reported functional outcome scales (ASES, CONSTANT, and Neer's "limited goals") with this particular patient. At the last visit, two years after the first surgical intervention, the patient had a normal range of motion, was pain-free, and was generally satisfied with the results of the surgery. We concluded that ORIF of the proximal humerus with locking plate and fibular strut graft was, in our case, a good solution for preventing varus malalignment in a patient with severe comminution of medial calcar., Competing Interests: The authors have declared that no competing interests exist., (Copyright © 2024, Pop et al.)