Lecturers: Alexandra Silva and Sam Staton.

General information on this course can be found in the studiegids.

Contents of this course webpage


Dates, Time and Location

Course Material

We will be follow Bart Jacobs' draft book, available here. We might give additional material, such as slides, hand-outs, and research papers on coalgebra. All material will be made available electronically via this webpage, or distributed during class. We also recommend the tutorial on coalgebras and coinduction by Bart Jacobs and Jan Rutten. See here.

Lectures Overview

Note: The course will be taught in English.
  1. Lecture 1, Tuesday 2 Sep, 15:30-17:30, in LIN4
  2. Lecture 2, Tuesday 9 Sep, 15:30-17:30, in HG01.028
  3. Lecture 3, Tuesday 16 Sep, 15:30-17:30, in HG01.028
  4. Lecture 4, Tuesday 23 Sep, 15:30-17:30, in HG01.028
  5. Lecture 5, Tuesday 30 Sep, 15:30-17:30, in HG01.028
  6. Lecture 6, Tuesday 7 October, 15:30-17:30, in HG01.028
  7. Lecture 7, Tuesday 14 October, 15:30-17:30, in HG01.028
  8. Lecture 8, Tuesday 21 October, 15:30-17:30, in HG01.028
  9. Lecture 9, Tuesday 11 November, 15:30-17:30, in HG01.028
  10. Lecture 10, Thursday 13 November, 10:30-12:30, in HG00.307
  11. Lecture 11, Tuesday 25 November, 15:30-17:30, in HG01.028.
  12. Lecture 12, Tuesday 2 December, 15:30-17:30, in HG01.028.
  13. Lecture 13, Tuesday 9 December, 15:30-17:30, in HG01.028.
  14. Lecture 14, Tuesday 16 December, 15:30-17:30, in HG01.028.

Course Description and Prerequisites

Course description:

State-based systems are used widely in computer science to model concrete systems such as digital hardware, software programs, and distributed systems. Coalgebra is a unifying framework for studying the behaviour of state-based systems. In coalgebra, a system is viewed as a black box where knowledge of the system's state can only be obtained by observing the external behaviour. In particular, two states s and t are considered equivalent if whenever we run the system starting in state s, the observed behaviour is the same as when we run the system in starting in state t. The type of observations and transitions in the external behaviour is specified by the system type. The theory of universal coalgebra provides general definitions of observable behaviour and bisimilarity that can be instantiated for concrete system types, as well as a powerful and fascinating reasoning principle called coinduction (a notion that is dual to the well known induction principle).

This course is an introduction to coalgebra. The course starts by studying how various types of systems can be modelled as coalgebras, and how coinduction can be applied to them. These systems include basic datatypes, such as infinite streams and trees, and many types of automata (deterministic, nondeterministic, probabilistic, ...). Next, a number of fundamental notions such as language equivalence of automata, bisimilarity of processes, and determinisation of nondeterministic automata, will be treated coalgebraically. The coalgebraic framework will then be used to obtain generalisations of these constructions to other types of systems.

Coalgebra is a rather recent field of research, existing for a mere two decades, and it is attracting an enthusiastic, ever-growing community. Being relatively young, it still has many elementary and exciting research questions to offer.

Prerequisites: We only assume basic knowledge of automata, formal languages and propositional logic, for example, as covered in the courses Talen en Automaten, Discrete Wiskunde, Beweren en Bewijzen, en Semantiek en Correctheid. With respect to category theory and modal logic, the course will be self-contained: Only basic definitions will be needed, and these will be introduced as part of the course.


The final grade will be the 0.3 * H1+ 0.3 * H2 + 0.4 * E, where H1 and H2 are the grade given for the two exntended homework assignments and E is the grade given for the final exam.