We present an abstract equational framework for the specification of systems having both observational and computational features. Our approach is based on a clear separation between the two categories of features, and uses algebra, respectively coalgebra to formalise them. This yields a coalgebraically-defined notion of observational indistinguishability, as well as an algebraically-defined notion of reachability under computations. The relationship between the computations yielding new system states and the observations that can be made about these states is specified using liftings of the coalgebraic structure of state spaces to a coalgebraic structure on computations over these state spaces. Also, correctness properties of system behaviour are formalised using equational sentences, with the associated notions of satisfaction abstracting away observationally indistinguishable, respectively unreachable states, and with the resulting proof techniques employing coinduction, respectively induction.

This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of many-sorted predicate logic. In this setting we will see Lindenbaum MBAO models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as final objects. They will be used to (re)prove completeness results, and Hennessey-Milner style characterisation results for the modal logic, first established by Rößiger.

Coalgebras have been proposed as formal basis for the semantics of objects in the sense of object-oriented programming. This paper shows that this semantics provides a smooth interpretation for subtyping, a central notion in object-oriented programming. We show that different characterisations of behavioural subtyping found in the literature can conveniently be expressed in coalgebraic terms. We also investigate the subtle difference between behavioural subtyping and refinement.

Coalgebras for endofunctors ${\mathcal C}\rightarrow{\mathcal C}$ can be used to model classes of object-oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors${\mathcal C}^{op} \times {\mathcal C}\rightarrow{\mathcal C}$. This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation and invariants for coalgebras of extended polynomial functors and proves many standard results.