Game Comonads & Generalised Quantifiers

TL;DR

This paper extends game comonads to encompass logics with generalized quantifiers, establishing a categorical semantics for these logics and introducing a new notion of tree decomposition linked to the comonad structure.

Contribution

It introduces a graded comonad framework for logics with generalized quantifiers, connecting isomorphisms to Duplicator strategies and defining a new tree decomposition concept.

Findings

Categorical semantics for logics with generalized quantifiers.

A graded comonad capturing Duplicator winning strategies.

A novel tree decomposition arising from the comonad construction.

Abstract

Game comonads, introduced by Abramsky, Dawar and Wang and developed by Abramsky and Shah, give an interesting categorical semantics to some Spoiler-Duplicator games that are common in finite model theory. In particular they expose connections between one-sided and two-sided games, and parameters such as treewidth and treedepth and corresponding notions of decomposition. In the present paper, we expand the realm of game comonads to logics with generalised quantifiers. In particular, we introduce a comonad graded by two parameters $n \leq k$ such that isomorphisms in the resulting Kleisli category are exactly Duplicator winning strategies in Hella's $n$-bijection game with $k$ pebbles. We define a one-sided version of this game which allows us to provide a categorical semantics for a number of logics with generalised quantifiers. We also give a novel notion of tree decomposition that emerges from the construction.