Going Deep and Going Wide: Counting Logic and Homomorphism Indistinguishability over Graphs of Bounded Treedepth and Treewidth

08/11/2023
by   Eva Fluck, et al.
0

We study the expressive power of first-order logic with counting quantifiers, especially the k-variable and quantifier-rank-q fragment 𝖢^k_q, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same 𝖢^k_q-sentences if and only if they are homomorphism indistinguishable over the class 𝒯^k_q of graphs admitting a k-pebble forest cover of depth q. Their proof builds on the categorical framework of game comonads developed by Abramsky, Dawar, and Wang (2017). We reprove their result using elementary techniques inspired by Dvořák (2010). Using these techniques we also give a characterisation of guarded counting logic. Our main focus, however, is to provide a graph theoretic analysis of the graph class 𝒯^k_q. This allows us to separate 𝒯^k_q from the intersection of the graph class 𝒯𝒲_k-1, that is graphs of treewidth less or equal k-1, and 𝒯𝒟_q, that is graphs of treedepth at most q if q is sufficiently larger than k. We are able to lift this separation to the semantic separation of the respective homomorphism indistinguishability relations. A part of this separation is to prove that the class 𝒯𝒟_q is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/18/2020

Counting Bounded Tree Depth Homomorphisms

We prove that graphs G, G' satisfy the same sentences of first-order log...
research
08/10/2023

Limitations of Game Comonads via Homomorphism Indistinguishability

Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-v...
research
03/20/2023

Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation

We introduce the 2-sorted counting logic GC^k that expresses properties ...
research
10/11/2022

Descriptive complexity of the generalized spectra of graphs

Two graphs are cospectral if their respective adjacency matrices have th...
research
03/14/2023

Logarithmic Weisfeiler–Leman and Treewidth

In this paper, we show that the (3k+4)-dimensional Weisfeiler–Leman algo...
research
10/18/2022

First Order Logic on Pathwidth Revisited Again

Courcelle's celebrated theorem states that all MSO-expressible propertie...
research
08/02/2022

On the Expressiveness of a Logic of Separated Relations

We compare the model-theoretic expressiveness of the existential fragmen...

Please sign up or login with your details

Forgot password? Click here to reset