Existential and positive games: a comonadic and axiomatic view

Samson Abramsky; Thomas Laure; Luca Reggio

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arXiv · 2503.01247

Existential and positive games: a comonadic and axiomatic view

Samson Abramsky·Thomas Laure·Luca Reggio

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Abstract

A number of model-comparison games central to (finite) model theory, such as pebble and Ehrenfeucht-Fra\"{i}ss\'{e} games, can be captured as comonads on categories of relational structures. In particular, the coalgebras for these comonads encode in a syntax-free way preservation of resource-indexed logic fragments, such as first-order logic with bounded quantifier rank or a finite number of variables. In this paper, we extend this approach to existential and positive fragments (i.e., without universal quantifiers and without negations, respectively) of first-order and modal logic. We show, both concretely and at the axiomatic level of arboreal categories, that the preservation of existential fragments is characterised by the existence of so-called pathwise embeddings, while positive fragments are captured by a newly introduced notion of positive bisimulation. As an application, we offer a new proof of an equi-resource Lyndon positivity theorem for (multi)modal logic.

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Samson Abramsky, Thomas Laure, Luca Reggio. 2025-03-03. Existential and positive games: a comonadic and axiomatic view. https://arxiv.org/abs/2503.01247

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