# Finite and algorithmic model theory

##### Course with SWS 2/2/0 (lecture/exercise/practical) in SS 2021

Lecturer

Tutor

SWS

• 2/2/0

Modules

Examination method

• Oral exam

Finite and algorithmic model theory (summer semester 2020/21)

### 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.

### 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.

### Schedule and Location

Because of the ongoing COVID-19 pandemic, all lectures and tutorials will be online, via Zoom, respectively on Tuesdays (14:50 - 16:20) and Wednesdays (11:10 - 12:40). The lecturer and the tutor for the course will be Bartosz Bednarczyk, teaching under the supervision of Sebastian Rudolph.

### 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.

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. On the complexity of fixed-variable fragments of FO, undecidability of FO.

6. NP-completeness of FO1 and solving finite satisfiability for FO1 with counting quantifiers via integer programming.

7. NExpTime upper bounds for FO2 and exponential model property of FO2.

8. ExpTime upper bound for GF2 and its tree-model property.

9. AExpSpapce=2ExpTime upper bounds for full GF.

### Opportunities

B. Bednarczyk is happy to provide research-level master's or bachelor's projects ideas (of different difficulty levels) and to supervise them.

### Contact

Please, feel free to contact B. Bednarczyk if you have any further questions.
• 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
 Lecture Introduction & Compactness method DS5, April 13, 2021 in Video conference File Exercise Q&A DS3, April 14, 2021 in Video conference Lecture Zero-One Laws of FO DS5, April 20, 2021 in Video conference File Exercise Exercises on Compactness DS3, April 21, 2021 in Video conference File Lecture The missing proofs on 0-1 law + Intro to EF Games DS5, April 27, 2021 in Video conference File Exercise Exercises on Zero-One Laws DS3, April 28, 2021 in APB E005 File Lecture E-F Games DS5, May 4, 2021 in APB E005 File No session NO EXERCISES DS3, May 5, 2021 in APB E005 Lecture Proving E-F Games DS5, May 11, 2021 in APB E005 File Exercise Exercises on EF games DS3, May 12, 2021 in APB E005 File Lecture Locality DS5, May 18, 2021 in APB E005 File Exercise Exercises on E-F games and locality DS3, May 19, 2021 in APB E005 File No session No lecture DS5, May 25, 2021 in APB E005 No session No exercises DS3, May 26, 2021 in APB E005 Lecture Tilings and undecidability of FO DS5, June 1, 2021 in APB E005 File Exercise Exercises on pebble games, Hanf locality and MSO DS3, June 2, 2021 in APB E005 File Lecture Model checking on graphs with bounded degree and NP-completeness of FO1 and C1 DS5, June 8, 2021 in APB E005 File Lecture Decidable fragment of first-order logic, C1 and introduction to FO2 DS5, June 15, 2021 in APB E005 File Exercise Exercises on decidability/undecidability DS3, June 16, 2021 in APB E005 File Lecture The last lecture [recap] DS5, June 20, 2021 in APB E005 Lecture Finite model property for FO2 DS5, June 22, 2021 in APB E005 File Exercise Exercises on C1 and related logics DS3, June 23, 2021 in APB E005 File Lecture Guarded Fragment: Part 1 DS5, June 29, 2021 in APB E005 File Exercise Exercises on FO2 and related logics DS3, June 30, 2021 in APB E005 Exercise Exercises on FO2 and related logics: Part II DS3, July 7, 2021 in APB E005 File Lecture Guarded Fragment: Part 2 DS5, July 13, 2021 in APB E005 File