We present methods to compute least fixed points of multiple monotone inflationary functions in parallel and distributed settings. While the classic Knaster-Tarski theorem addresses a single function with sequential iteration, modern computing systems require parallel execution with overwrite semantics, non-atomic updates, and stale reads. We prove three convergence theorems under progressively relaxed synchronization: (1) Interleaving semantics with fair scheduling, (2) Parallel execution with update-only-on-change semantics (processes write only on those coordinates whose values change), and (3) Distributed execution with bounded staleness (updates propagate within $T$ rounds) and $i$-locality (each process modifies only its own component). Our approach differs from prior work in fundamental ways: Cousot-Cousot's chaotic iteration uses join-based merges that preserve information. Instead, we use coordinate-wise overwriting. Bertsekas's asynchronous methods assume contractions. We use coordinate-wise overwriting with structural constraints (locality, bounded staleness) instead. Applications include parallel and distributed algorithms for the transitive closure, stable marriage, shortest paths, and fair division with subsidy problems. Our results provide the first exact least-fixed-point convergence guarantees for overwrite-based parallel updates without join operations or contraction assumptions.

翻译：本文提出了在并行与分布式环境下计算多个单调递增函数最小不动点的方法。经典的Knaster-Tarski定理针对的是单函数顺序迭代场景，而现代计算系统需要支持覆盖语义、非原子更新和过期读取的并行执行。我们在逐步放宽同步要求的条件下证明了三个收敛定理：(1) 公平调度下的交错语义，(2) 仅当值变化时更新的并行执行语义（进程仅写入值发生变化的坐标），以及(3) 具有有限过期性（更新在$T$轮内传播）和$i$-局部性（每个进程仅修改自身分量）的分布式执行语义。本方法与已有研究存在根本差异：Cousot-Cousot的混沌迭代采用基于连接运算的合并策略以保持信息完整性，而我们采用坐标级覆盖机制；Bertsekas的异步方法假设收缩映射条件，而我们采用具有结构约束（局部性、有限过期性）的坐标级覆盖机制。应用场景包括传递闭包、稳定婚姻、最短路径及带补贴的公平分配等问题的并行与分布式算法。本研究首次为基于覆盖机制的并行更新提供了精确的最小不动点收敛保证，无需连接运算或收缩映射假设。