Game comonads, introduced by Abramsky, Dawar and Wang, and developed by Abramsky and Shah, give a categorical semantics for model comparison games. We present an axiomatic account of Feferman-Vaught-Mostowski (FVM) composition theorems within the game comonad framework, parameterized by the model comparison game. In a uniform way, we produce compositionality results for the logic in question, and its positive existential and counting quantifier variants. Secondly, we extend game comonads to the second order setting, specifically in the case of Monadic Second Order (MSO) logic. We then generalize our FVM theorems to the second order case. We conclude with an abstract formulation of Courcelle's algorithmic meta-theorem, exploiting our earlier developments. This is instantiated to recover well-known bounded tree-width and bounded clique-width Courcelle theorems for MSO on graphs.

翻译：由 Abramsky 、 Dawar 和 Wang 介绍的游戏共鸣由 Abramsky 和 Abramsky 和 Shah 开发, 给模型比较游戏提供了明确的语义。 我们在游戏共奏框架中展示了 Feferman-Vaught- Mostowski (FVM) 构成理论的不言自明的解说, 由模型比较游戏参数来参数。 我们以统一的方式为相关逻辑及其积极存在和计数变量生成了构成结果。 其次, 我们将游戏共奏扩展至第二顺序设置, 特别是莫纳迪第二秩序( MSO) 逻辑。 我们然后将我们的FVM 理论概括到第二个顺序。 我们以Courcelle 的算法元理论的抽象配方来结束, 利用我们早期的开发。 这是为了在图形中恢复有名的、 捆绑的树木维度和捆绑的Courcelle 圆形。