Finite and algorithmic model theory

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Finite and algorithmic model theory

Lehrveranstaltung mit SWS 2/2/0 (Vorlesung/Übung/Praktikum) in SS 2021

Dozent

Tutor

Umfang (SWS)

• 2/2/0

Module

Leistungskontrolle

• Mündliche Prüfung

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
 Vorlesung Introduction & Compactness method DS5, 13. April 2021 in Videokonferenz Datei Übung Q&A DS3, 14. April 2021 in Videokonferenz Vorlesung Zero-One Laws of FO DS5, 20. April 2021 in Videokonferenz Datei Übung Exercises on Compactness DS3, 21. April 2021 in Videokonferenz Datei Vorlesung The missing proofs on 0-1 law + Intro to EF Games DS5, 27. April 2021 in Videokonferenz Datei Übung Exercises on Zero-One Laws DS3, 28. April 2021 in APB E005 Datei Vorlesung E-F Games DS5, 4. Mai 2021 in APB E005 Datei Entfällt NO EXERCISES DS3, 5. Mai 2021 in APB E005 Vorlesung Proving E-F Games DS5, 11. Mai 2021 in APB E005 Datei Übung Exercises on EF games DS3, 12. Mai 2021 in APB E005 Datei Vorlesung Locality DS5, 18. Mai 2021 in APB E005 Datei Übung Exercises on E-F games and locality DS3, 19. Mai 2021 in APB E005 Datei Entfällt No lecture DS5, 25. Mai 2021 in APB E005 Entfällt No exercises DS3, 26. Mai 2021 in APB E005 Vorlesung Tilings and undecidability of FO DS5, 1. Juni 2021 in APB E005 Datei Übung Exercises on pebble games, Hanf locality and MSO DS3, 2. Juni 2021 in APB E005 Datei Vorlesung Model checking on graphs with bounded degree and NP-completeness of FO1 and C1 DS5, 8. Juni 2021 in APB E005 Datei Vorlesung Decidable fragment of first-order logic, C1 and introduction to FO2 DS5, 15. Juni 2021 in APB E005 Datei Übung Exercises on decidability/undecidability DS3, 16. Juni 2021 in APB E005 Datei Vorlesung The last lecture [recap] DS5, 20. Juni 2021 in APB E005 Vorlesung Finite model property for FO2 DS5, 22. Juni 2021 in APB E005 Datei Übung Exercises on C1 and related logics DS3, 23. Juni 2021 in APB E005 Datei Vorlesung Guarded Fragment: Part 1 DS5, 29. Juni 2021 in APB E005 Datei Übung Exercises on FO2 and related logics DS3, 30. Juni 2021 in APB E005 Übung Exercises on FO2 and related logics: Part II DS3, 7. Juli 2021 in APB E005 Datei Vorlesung Guarded Fragment: Part 2 DS5, 13. Juli 2021 in APB E005 Datei