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Duality for Finite Models: Relating Structure and Power

Periodic Reporting for period 1 - D-FINED (Duality for Finite Models: Relating Structure and Power)

Reporting period: 2020-02-01 to 2022-01-31

This project focusses on the research area of logic and semantics of computation. Finite model theory is the specialisation of model theory to finite structures and has been called “the logic of computer science” since in the latter field the basic models of computation are finite. Many classical results of model theory fail when restricted to finite models. For this reason, finite model theory has developed independently from model theory and the research communities, as well as the techniques, are almost disjoint. Finite model theory exemplifies a strand in the field of logic in computer science focussing on expressiveness and complexity (“Power”), as opposed to the one focussing on semantics and compositionality (“Structure”).

The overall objective of this project is to bridge the gap between the semantics methods of model theory, and the combinatorial and complexity-theoretic methods of finite model theory, i.e. to relate Structure and Power. The three main goals in this direction are to:

- (Objective 1) Develop a structural approach to the study of spaces of finite structures, based on duality and categorical methods.
- (Objective 2) Extend the applicability of these methods from finite structures to tame classes of infinite structures.
- (Objective 3) Relate the duality approach to existing categorical semantics, such as Lawvere's hyperdoctrines and the game comonads recently introduced by Abramsky, Dawar and their collaborators.

This project may contribute to a more unified view of logic in mathematics and computer science, providing new tools for a structural approach to more algorithmic and complexity-oriented areas of logic such as finite model theory.
Objective 1 has been fully met, mainly through the work of Gehrke, Jakl and Reggio on a duality theoretic view on limits of finite structures. The main results were put in a broader context in a subsequent book chapter by the same authors, which offers a gentle introduction to this research area for a wider audience of mathematicians and computer scientists.
Objective 2 has been fully met. In particular, the work by Luca Reggio on polyadic sets and homomorphism counting, and on a duality perspective on the Beth definability property for the dual of compact Hausdorff spaces, provide new tools for the applicability of duality methods beyond finite structures.
Objective 3 has been fully met, especially through the work of Dawar, Jakl and Reggio on game comonads and Lovász-type results in finite model theory, and the work of Abramsky and Reggio on arboreal categories. The first makes the link between homomorphism counting results in finite model theory, game comonads, and polyadic sets—which are closely related to hyperdoctrines in categorical logic. The second provides an axiomatic approach to game comonads, yielding a unified treatment of several important aspects of resource-sensitive logics and paving the way for a systematised view of different categorical semantics for predicate logic.

Given the interdisciplinary nature of the project, an effort has been made to disseminate the results to as wide an audience as possible. Thus, the main results of this project have been presented at both mathematics and computer science conferences. These include two invited talks in international conferences and workshops (BLAST 2021 and the Resources and co-Resources workshop) and nine talks in international conferences (Logic Colloquium, ICALP, LiCS) and local seminars. Further, in August 2022 Jakl and Reggio will teach a course at the ESSLLI Summer School on game comonads, in which some of the results of this project will be presented.
Most of the results outlined above represent a significant move beyond the state of the art and an important step towards a more unified view of logic across mathematics and computer science. We expect that these results will contribute to advances in the areas of finite model theory and database theory, and more broadly of logic in computer science, which would impact society at large.
Talk at the Università degli Studi di Pavia