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 students will learn how to combine coinduction and induction to derive effective specification and reasoning techniques for automata. 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.