Topics in Algebraic Logic and Duality Theory
Topics in Algebraic Logic and Duality Theory
Coordinated project, June 2025, ILLC. Coordination: Rodrigo N. Almeida, Simon Lemal.
Topic: Amalgamation, Interpolation and Definability Properties.
Description
In the study of logical systems, there are several desirable properties which make the system more tractable: the finite model property, variants of the disjunction property, admissibility of rules, etc. Amongst these, interpolation is a quite attractive property, concerning a very natural problem: suppose one wants to check whether an entailment of A from B holds; can we do so by restricting the proof search to formulas containing only the symbols occurring in both? If we can, this can make the complexity of proofs, and of problems involving proof checking or model checking, dramatically lower. Nevertheless, unlike some of these other properties, interpolation is a rather rare and sparse property.
In this project we will study interpolation properties as well as their algebraic counterparts: amalgamation properties. We will also zoom in on related questions: some typically weaker properties, like definability type theorems (Beth properties), and some typically stronger properties, like uniform interpolation.
Participants
The following people participated in the project:
- Christopher van Altena.
- Noam Cohen.
- Qian Chen.
- Josef Doyle.
- Tristan Hewitt.
- Zhaorui Hu.
- Estel Koole.
- Bardo Maienborn.
- Joel Maxson.
- Edoardo Menorello.
- Tenyo Takahashi.
- Minxin Wang.
References
- Alexander Chagrov, Michael Zacharyaschev. (2007) Modal Logic.
- Dov Gabbay, Larisa Maksimova. (2007) Interpolation and Definability in Modal and Intuitionistic Logic.
- Silvio Ghilardi, Marek Zawadowski. (2002). Sheaves, Games and Model Completions.
- Sam van Gool, George Metcalfe, Constantine Tsinakis. (2017). Uniform interpolation and Compact Congruences.
- Sam van Gool, Luca Reggio. (2017). An open mapping theorem for finitely copresented Esakia spaces.
- Eva Hoogland. (2001). Definability and Interpolation: Model-theoretic investigations.
- Albert Visser. (1996). Uniform Interpolation and Layered Bisimulation.
Seminar Sheets
There are four seminar sheets prepared for the four seminars interleaved with the lectures. These are the following:
You can find some project presentation ideas here. Given the number of people involved, it is allowed to work in pairs (both for the presentation and the reports). If you have interest on a topic, feel free to approach the instructors about it.
Lecture Schedule
There will be four lectures and four seminars. These will take place in the following schedule; slides of the lectures will be added as lectures take place:
| Date | Topic | Location | Time |
|---|---|---|---|
| 02/06/2025 | Lecture 1: Basics of Interpolation and Amalgamation. | F1.15 (slides) | 09h30-12h00 |
| 04/06/2025 | Seminar 1 | F1.15 | 14h00-16h00 |
| 05/06/2025 | Lecture 2: Beth properties and epimorphism surjectivity. | L2.07 (slides) | 14h00-16h00 |
| 06/06/2025 | Seminar 2 | F1.15 | 14h00-16h00 |
| 10/06/2025 | Lecture 3: Uniform interpolation. (notes) | L2.06 (slides) | 14h00-16h00 |
| 11/06/2025 | Seminar 3 | L2.07 | 14h00-16h00 |
| 12/06/2025 | Lecture 4: Maksimova-style characterizations of logics with interpolation. | F1.15 (slides) | 14h00-16h00 |
| 13/06/2025 | Seminar 4 | F1.15 | 14h00-16h00 |
| 18/06/2025 | Consultation Seminar | A1.22 | 14h00 - 16h00 |
| 25/06/2025 | Consultation Seminar | L2.07 | 14h00 - 16h00 |
| TBC | Guest Lecture | Room TBA | 13h00-15h00 |
| 27/06/2025 | Final Presentations | A1.28 | 14h00-16h00 |