The Pebble-Relation Comonad in Finite Model Theory
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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 Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable logic with counting quantifiers using a recently developed categorical generalisation.
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