We present an abstract equational framework for the specification of systems having bothobservational and computational features. Our approach is based on a clear separation between the twocategories of features, and uses algebra, respectively coalgebra to formalise them. This yields acoalgebraically-defined notion of observational indistinguishability, as well as an algebraically-definednotion of reachability under computations. The relationship between the computations yielding new systemstates and the observations that can be made about these states is specified using liftings of thecoalgebraic 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 theassociated notions of satisfaction abstracting away observationally indistinguishable, respectivelyunreachable states, and with the resulting prooftechniques employing coinduction, respectively induction.

This paper gives a semantical underpinning for a many-sorted modallogic associated with certain dynamical systems, like transitionsystems, automata or classes in object-oriented languages. Thesesystems will be described as coalgebras of so-called polynomialfunctors, built up from constants and identities, using products,coproducts and powersets. The semantical account involves Booleanalgebras with operators indexed by polynomial functors, called MBAOs,for Many-sorted Boolean Algebras with Operators, combiningstandard (categorical) models of modal logic and of many-sortedpredicate logic. In this setting we will see Lindenbaum MBAO models asinitial objects, and canonical coalgebraic models of maximallyconsistent sets of formulas as final objects. They will be used to(re)prove completeness results, and Hennessey-Milner stylecharacterisation results for the modal logic, first established byRößiger.

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

Coalgebras for endofunctors ${\mathcal C}\rightarrow{\mathcalC}$ 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 ${\mathcal C}^{op} \times {\mathcalC}\rightarrow{\mathcal C}$ . 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.