有限模型理论中的卵石关系共单子
The Pebble-Relation Comonad in Finite Model Theory
AI总结：
本文提出卵石关系共单子，用范畴论方法刻画路径宽，并证明其coKleisli态射与k变量无穷逻辑限制合取片段的真值保持等价，进而得到联系路径宽与带计数量词逻辑的Lovász型定理。
摘要：
由Abramsky、Dawar和Wang引入的卵石共单子为有限模型理论中的k-卵石博弈提供了范畴论解释。卵石共单子的coKleisli范畴刻画了无穷k变量逻辑的不同片段和扩张下的等价关系。此外，该卵石共单子上的余代数刻画了树宽并对应于树分解。本文引入卵石关系共单子，它刻画了路径宽且其余代数对应于路径分解。我们进一步证明，在该共单子中存在coKleisli态射等价于在k变量无穷逻辑的限制合取片段中保持真值。我们使用Dalmau的卵石关系博弈和一个等价的全合一卵石博弈来证明这一点。然后，通过全合一卵石博弈的双射版本，我们对相应的coKleisli同构给出了类似的处理。最后，作为推论，我们证明了一个新的Lovász型定理，将路径宽与带计数量词的k变量无穷逻辑的限制合取片段联系起来。
英文摘要：
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, which characterises pathwidth and whose coalgebras correspond to path decompositions. We further show that the existence of a coKleisli morphism in this comonad is equivalent to truth preservation in the restricted conjunction fragment of k-variable infinitary logic. We do this using Dalmau's pebble-relation game and an equivalent all-in-one pebble game. We then provide a similar treatment to the corresponding coKleisli isomorphisms via a bijective version of the all-in-one pebble game. Finally, we show as a consequence a new Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable infinitary logic with counting quantifiers.
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