Finite 模型理论中的 Pebble- relations Comonad 质模型理论 (The Pebble-Relation Comonad in Finite Model Theory)

The pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a categorical interpretation for the k-pebble games from finite model theory. The coKleisli category of the pebbling comonad specifies equivalences under different fragments and extensions of infinitary k-variable logic. Moreover, the coalgebras over this pebbling comonad characterise treewidth and correspond to tree decompositions. In this paper we introduce the pebble-relation comonad that characterises pathwidth and whose coalgebras correspond to path decompositions. We further show how the coKleisli morphisms of the pebble-relation comonad provide a categorical interpretation to Duplicator's winning strategies in Dalmau's pebble-relation game. We then provide a similar treatment to the corresponding coKleisli isomorphisms via a novel bijective pebble-game with a hidden pebble. Finally, we prove a new Lov\'asz-type theorem relating pathwidth to the restricted conjunction fragment of k-variable logic with counting quantifiers using a recently developed categorical generalisation.

翻译：由 Abramsky 、 Dawar 和 Wang 引入的滚动comonad 。由 Abramsky 、 Dawar 和 Wang 引入的滚动的comonad 提供了来自有限模型理论的Kpeble 游戏的绝对解释。卷动的comonad 的colesli 类别在不同的碎片和无限 k可变逻辑的延伸下指定等同。此外, 由这种卷动的comonad 的comonad 特性和树分解构成的cepble- relation comonad 。在本文中, 我们引入了带有路径和路径分解法对应的 kebbble- comonad 。我们进一步展示了卷动的cokelesli comonad 类别是如何对 Dalmau 的 comond 游戏中Dolmau 的赢取策略进行绝对解释的。我们随后对相应的cleisli li commble- game- game 进行类似的处理, 和隐藏的rebbbbbbbbleble- gangble- gangble 相近的平等化的。最后, 我们证明了新的Lov\\\ chaltal- chal- chal- slateal- slational- chaliztaliztaliztaldaliztaliztaliztaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldalizaldaldaldaldaldaldalizaldaldaldaldaldalizcaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldald 和一种新的公式, 方法, 用了与最近开发的平平平制的平制的平制的平制的平制的平式的平式平式平式的平式的平式平式平式平式平式平

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