Lattice-linearity was introduced as a way to model problems using predicates that induce a lattice among the global states (Garg, SPAA 2020). A key property of \textit{the predicate} representing such problems is that it induces \textit{one} lattice in the state space. An algorithm that emerges from such a predicate guarantees the execution to be correct even if nodes execute asynchronously. However, many interesting problems do not exhibit lattice-linearity. This issue was somewhat alleviated with the introduction of eventually lattice-linear algorithms (Gupta and Kulkarni, SSS 2021). They induce \textit{single} or \textit{multiple} lattices in \textit{a subset of the state space} even when the problem cannot be defined by a predicate under which the global states form a lattice. This paper focuses on analyzing and differentiating between lattice-linear problems and algorithms. We introduce \textit{fully lattice-linear algorithms}. These algorithms partition the \textit{entire} reachable state space into \textit{one or more lattices}, and as a result, ensure that the execution remains correct even if nodes execute asynchronously. For demonstration, we present lattice-linear self-stabilizing algorithms for minimal dominating set (MDS), graph colouring (GC), minimal vertex cover (MVC) and maximal independent set (MIS) problems. The algorithms for MDS, MVC and MIS converge in $n$ moves and the algorithm for GC converges in $n+2m$ moves. These algorithms preserve this time complexity while allowing the nodes to execute asynchronously. They present an improvement to the existing algorithms present in the literature. Our work also demonstrates that to allow asynchrony, a more relaxed data structure can be allowed (called $\prec$-lattice in this paper, where the meet of a pair of global states may not be defined), rather than a distributive lattice as assumed by Garg.

翻译：格线性作为一种建模问题的方法，通过谓词在全局状态间诱导格结构而被引入（Garg, SPAA 2020）。表示此类问题的谓词的一个关键性质是，它能在状态空间中诱导出唯一的格。基于此类谓词衍生的算法，即使节点异步执行，也能保证执行的正确性。然而，许多重要问题并不具备格线性。这一问题随着最终格线性算法的引入（Gupta and Kulkarni, SSS 2021）得到一定程度的缓解。即使问题无法通过使全局状态形成格的谓词来定义，这类算法也能在状态空间的子集中诱导出单个或多个格。本文重点分析和区分格线性问题与算法。我们引入了完全格线性算法。这类算法将整个可达状态空间划分为一个或多个格，从而确保即使节点异步执行，执行过程也能保持正确性。为作演示，我们提出了针对最小支配集、图着色、最小顶点覆盖和最大独立集问题的格线性自稳定算法。最小支配集、最小顶点覆盖和最大独立集算法在n步内收敛，图着色算法在n+2m步内收敛。这些算法在允许节点异步执行的同时，仍保持了该时间复杂度。它们对现有文献中的算法作出了改进。我们的工作还表明，为允许异步性，可以采用更宽松的数据结构（本文称为≺-格，其中一对全局状态的下确界可能未定义），而不必如Garg所假设的那样要求分配格。