Lovász-Type Theorems and Game Comonads

Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We...

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Published in:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 1 - 13
Main Authors: Dawar, Anuj, Jakl, Tomas, Reggio, Luca
Format: Conference Proceeding
Language:English
Published: IEEE 29.06.2021
Subjects:
Computational modeling
Computer science
Forestry
Games
Semantics
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Abstract Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lovász' theorem: the result by Dvořák (2010) that characterises equivalence of graphs in the k-dimensional Weisfeiler-Leman equivalence by homomorphism counts from graphs of tree-width at most k, and the result of Grohe (2020) characterising equivalence with respect to first-order logic with counting and quantifier depth k in terms of homomorphism counts from graphs of tree-depth at most k. The connection of our categorical formulation with these results is obtained by means of the game comonads of Abramsky et al. We also present a novel application to homomorphism counts in modal logic.
AbstractList Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lovász' theorem: the result by Dvořák (2010) that characterises equivalence of graphs in the k-dimensional Weisfeiler-Leman equivalence by homomorphism counts from graphs of tree-width at most k, and the result of Grohe (2020) characterising equivalence with respect to first-order logic with counting and quantifier depth k in terms of homomorphism counts from graphs of tree-depth at most k. The connection of our categorical formulation with these results is obtained by means of the game comonads of Abramsky et al. We also present a novel application to homomorphism counts in modal logic.
Author Jakl, Tomas
Reggio, Luca
Dawar, Anuj
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  givenname: Luca
  surname: Reggio
  fullname: Reggio, Luca
  email: luca.reggio@cs.ox.ac.uk
  organization: University of Oxford,Department of Computer Science,UK
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Snippet Lovász (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the...
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SubjectTerms Computational modeling
Computer science
Forestry
Games
Semantics
Title Lovász-Type Theorems and Game Comonads
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