Periodic Reporting for period 1 - D-FINED (Duality for Finite Models: Relating Structure and Power)
Período documentado: 2020-02-01 hasta 2022-01-31
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 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.