We introduce c-Lattice Aggregation, a fault-tolerant reconstruction problem for distributed verification under crash and Byzantine failures. In our setting, n asynchronous processes supervise a concurrent execution I: each process holds a local sample, and must collaboratively reconstruct I from partial, potentially overlapping observations. A protocol solves c-Lattice Aggregation if at least c correct processes output the complete execution I, while all correct outputs are comparable and bounded by I. This strengthens Lattice Agreement [Attiya, Herlihy and Rachman, 1995] and Byzantine Lattice Agreement [Di Luna et al., 2020; Zheng and Garg, 2020]. We parameterize inputs by a redundancy parameter x -- every element of I appears in at least x initial samples -- and establish tight feasibility thresholds. Under crash failures with at most t faulty processes, Lattice Aggregation is solvable if and only if x >= t + 1. Under Byzantine failures with t < n/3, c-Lattice Aggregation is solvable if and only if x >= 2t + c. All bounds are tight: we present matching algorithms based on SCD-broadcast [Imbs et al., 2018; Khanchandani and Wattenhofer, 2024] and indistinguishability-based lower bounds. Finally, we define globally dependent languages -- those for which no partial view can certify correctness, including consensus, linearizability, k-set agreement, and leader election -- and prove that soundness of any monitoring system is achievable if and only if c-Lattice Aggregation is solved, yielding the first complete characterization of fault-tolerant verification under Byzantine failures.

翻译：我们提出c-格聚合（c-Lattice Aggregation），这是一个在崩溃和拜占庭故障下用于分布式验证的容错重构问题。在我们的设定中，n个异步进程监督一个并发执行I：每个进程持有一个局部样本，必须基于部分且可能重叠的观测协作重构出I。如果一个协议能确保至少c个正确进程输出完整的执行I，并且所有正确输出可比较且被I所界定，则该协议解决了c-格聚合。这强化了格协议（Lattice Agreement）[Attiya, Herlihy and Rachman, 1995]和拜占庭格协议（Byzantine Lattice Agreement）[Di Luna et al., 2020; Zheng and Garg, 2020]。我们将输入参数化为冗余参数x——I中的每个元素至少出现在x个初始样本中——并建立严格的可解阈值。在至多t个进程发生崩溃故障的情况下，当且仅当x≥t+1时，格聚合是可解的。在t<n/3的拜占庭故障下，当且仅当x≥2t+c时，c-格聚合是可解的。所有界限均为紧的：我们基于SCD广播[Imbs et al., 2018; Khanchandani and Wattenhofer, 2024]提出了匹配的算法，并推导了基于不可区分性的下界。最后，我们定义了全局依赖语言（globally dependent languages）——即任何局部视图都无法验证其正确性，包括共识、线性化性、k-集合共识和领导者选举——并证明任何监控系统的可靠性可达成当且仅当c-格聚合问题被解决，从而首次给出了拜占庭故障下容错验证的完整刻画。