Counting Bounded Tree Depth Homomorphisms

@article{Grohe2020CountingBT,
  title={Counting Bounded Tree Depth Homomorphisms},
  author={Martin Grohe},
  journal={Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science},
  year={2020}
}
  • Martin Grohe
  • Published 2020
  • Computer Science, Mathematics
  • Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science
We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G' are homomorphism-indistinguishable over a class F of graphs if for each graph F ϵ F, the number of homomorphisms from F to G equals the number of homomorphisms from F to G'. 
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