# Finite and algorithmic model theory (22/23)

##### Lehrveranstaltung mit SWS 2/2/0 (Vorlesung/Übung/Praktikum) in WS 2022

Dozent

Tutor

Umfang (SWS)

• 2/2/0

Module

Leistungskontrolle

• Mündliche Prüfung

Check the newest lecture slides and exercise lists! OPAL: https://bildungsportal.sachsen.de/opal/auth/RepositoryEntry/37215338499

Finite and algorithmic model theory (winter semester 2022/23). The lectures and exercise sessions will be given by Bartosz Bednarczyk.

### Course Description

The goal of the lecture is to present a basic mathematical toolkit useful for studying expressivity&complexity of first-order logic and its fragments. It is motivated by applications of logics in computer science (e.g. in formal verification, databases or knowledge representation). The course is recommended to students enjoying theoretical computer science or/and pure mathematics. Note that the course is intended to be relatively advanced.

### Schedule and Location

In-person, blackboard talk (with slides from time to time). Both lectures and exercise sessions will be given by Bartosz Bednarczyk. Exercise sessions will contain material required for the lecture and vice-versa, so I fully recommend attending the exercise sessions (or at list skimming the exercise list before attending the lecture).

Lecture: Wednesdays, 14:50-16:20, APB/E007

Tutorial: Thursday, 13:00-14:50, APB/2026.

### Lecture plan

The expected content of the lecture will be as follows:

1. Inexpressivity via compactness theorem and why it is not appropriate for finite models. Applications of compactness in proving useful model-theoretic properties of FO like interpolation, preservation theorems and so on. Failures of such properties in the finite.

2. Zero-one laws of FO.

3. Ehrenfeucht-Fraïssé games - a basic tool for showing FO-inexpressivity.

4. FO can express only local properties: Hanf locality with applications to fixed-parameter tractability of FO model-checking on graphs of bounded degree.

5. A bit of model theory for the modal logic K.

6. Order-invariant First-Order Logic.

7. Undecidability of the satisfiability problem for FO and related issues. NP-completeness of FO1 and solving finite satisfiability for FO1 with counting quantifiers (a detour through integer programming).

8. The two-variable fragment of FO, model theory, complexity and the finite model property.

9. The guarded fragment of FO, model theory, complexity and the finite model property.

### Opportunities

B. Bednarczyk is happy to provide research-level master's or bachelor's project ideas (of different difficulty levels) and to supervise them. There is a very high chance to offer scholarships to students interested in doing research.

### Prerequisites

Undergraduate-level knowledge of predicate and first-order logic (syntax&semantics of FO), as well as a little from computational complexity (Turing machines, standard (non)deterministic complexity classes and basic knowledge about undecidable problems), is required. Don't worry if you are not fluent with the mentioned material from computational complexity -- it will be possible to organize some extra lessons to cover the essentials and such notions are not required for 80% of the lecture.

### Contact

Please, feel free to contact B. Bednarczyk via email (bartosz.bednarczyk at cs.uni.wroc.pl) if you have any further questions. I promise to reply no later than after 10 hours!
• Erich Grädel et al, Finite Model Theory and Its Applications
• Leonid Libkin, Elements of Finite Model Theory
• Martin Otto, Finite Model Theory — Lecture Notes
• Erich Grädel, Algorithmic Model Theory — Lecture Notes
• Erich Gradel, Egon Börger, Yuri Gurevich, The Classical Decision Problem