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Luca Reggio (UCL)

Dates:26 April 2023
Times:15:15 - 16:15
What is it:Seminar
Organiser:Department of Mathematics
Who is it for:University staff, External researchers, Adults, Alumni, Current University students
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  • Department of Mathematics

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  • In category "Seminar"
  • In group "(Maths) Maths seminar series"
  • In group "(Maths) Logic"
  • By Department of Mathematics

Title: Logical resources and arboreal adjunctions

Abstract:

I will give an overview of some recent work on applying categorical methods in finite model theory and descriptive complexity.

A key idea of a research program started by Abramsky, Dawar et al. in 2017 is that various forms of model comparison game, central to finite model theory, can be encoded as `game comonads', i.e. resource-indexed comonads on the category of relational structures. For example, the following ingredients can be captured in a syntax-free way: Ehrenfeucht-Fraïssé games, fragments of first-order logic with bounded quantifier rank, and the combinatorial parameter of tree-depth. This approach covers also finite variable fragments, modal and guarded fragments, and more.

The pattern of game comonads has been axiomatised at a general level in terms of `arboreal adjunctions', i.e. adjunctions between an extensional category (typically, in the examples, a category of relational structures) and a resource-indexed family of `arboreal categories'. If time allows, I will illustrate an application of this axiomatic framework to the study of `equi-resource' homomorphism preservation theorems in model theory (exemplified by Rossman's equirank homomorphism preservation theorem) and discuss recent work on identifying the expressive power of arboreal adjunctions.

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