The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.

We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Set-functors straightforwardly gives rise to a transformation of coalgebras for the respective functors. This latter transformation preserves homomorphisms and thus bisimulations. For comparison of probabilistic system types we also need reflection of bisimulation. We build the hierarchy of probabilistic systems by exploiting the new result that the transformation also reflects bisimulation in case the natural transformation is componentwise injective and the first functor preserves weak pullbacks. Additionally, we illustrate the correspondence of concrete and coalgebraic bisimulation in the case of general Segala-type systems.

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

The existence of a final coalgebra is equivalent to the existence of a formal logic with a set (small class) of formulas that has the Hennessy-Milner property of distinguishing coalgebraic states up to bisimilarity. This applies to coalgebras of any functor on the category of sets for which the bisimilarity relation is transitive. There are cases of functors that do have logics with the Hennessy-Milner property, but the only such logics have a proper class of formulas. The main theorem gives a representation of states of the final coalgebra as certain satisfiable sets of formulas. The key technical fact used is that any function between coalgebras that is truth-preserving and has a simple codomain must be a coalgebraic morphism.