Limitations of Game Comonads via Homomorphism Indistinguishability

TL;DR

This paper demonstrates fundamental limitations of game comonads in capturing linear-algebraic logic equivalences, showing that no finite-rank comonad can characterize invertible-map equivalence through homomorphism indistinguishability.

Contribution

It proves that invertible-map equivalence cannot be characterized by finite-rank comonads or homomorphism counts, revealing a key limitation of the comonad approach in finite model theory.

Findings

No finite-rank comonad characterizes IM-equivalence.

IM-equivalence cannot be distinguished by homomorphism counts.

Results answer an open question about the limitations of game comonads.

Abstract

Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lov\'asz-type theorems for game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.