# Topics in Modal Logic (Fall 2019)

This page concerns the course `Topics in Modal Logic', taught at the University of Amsterdam from October - December 2019.

## Practicalities

### Staff

• Lecturer: Yde Venema (y dot venema at uva dot nl, phone: 525 5299)

### Dates & location:

• Classes run from 29 October until 12 December; there will be 13/14 classes in total.
• There are two classes weekly, on Tuesdays from 13.00 - 14.45, and on Thursdays from 15.00 - 16.45, both in room F2.19 (ILLC building, Science Park).
• There is an exam on Tuesday 17 December, from 09.00 - 12.00, in room SP G0.05 (Science Park).

• Grading is primarily through homework assignments, and a written exam at the end of the course.
For the later part of the course additional/alternative requirements may apply (such as working out lecture notes).
See the separate page on grading for more details.

## Course Description

Modal languages are simple yet expressive and flexible tools for describing all kinds of relational structures. Thus modal logic finds applications in many disciplines such as computer science, mathematics, linguistics or economics. Notwithstanding this enormous diversity in appearance and application area, modal logics have a great number of properties in common. This common mathematical backbone form the content of this course, the exact topics change from year to year.

This year, the course will be devoted to connections between coalgebra and modal logic. We will provide an introduction to the notion of a coalgebra and its connection with modal logic. In a nutshell, we will see how:

• universal coalgebra is a unifying theory for many state-based evolving structures including deterministic finite automata, Kripke structures and non-wellfounded sets;
• universal coalgebra can be used to unify many different branches of modal logic under the umbrella of coalgebraic modal logic.

Course Content Here is a tentative list of topics to be covered:

• introduction to coalgebra: definition and examples
• coalgebraic modal logic: examples
• final coalgebras and coinduction
• behavioral equivalence and bisimilarity
• covarieties and structural operations
• Moss' coalgebraic modality
• logics based on predicate liftings
• one-step coalgebraic logic
• properties and desiderata of coalgebraic logics
• finite models: filtration and finite model property
• completeness for coalgebraic modal logics
• coalgebraic fixpoint logics and coalgebra automata
• topological coalgebras and Stone-type dualities
• algebra & coalgebra: analogies & dualities

### Prerequisites

The course is an advanced master course, and we assume that students possess some mathematical maturity; some basic knowledge of algebra and topology will be handy.

We do presuppose some basic skills and background knowledge on modal logic:

• required: familiarity with the syntax and semantics of modal languages, and the notion of bisimulation. More precisely, we build on the material covered in the first weeks of the course Introduction to Modal Logic, corresponding to the sections 1.1-1.3, 2.1-2.3 of the Modal Logic book.
• recommended (but not strictly necessary): previous exposure to the completeness proof of modal logic, and in particular, to the notion of the canonical frame.
No previous exposure to category theory is assumed.
Comments, complaints, questions: mail Yde Venema