A coalgebraic perspective on logical interpretations

Abstract

In Computer Science stepwise refinement of algebraic specifications is a well-known formal methodology for rigorous program development. This paper illustrates how techniques from Algebraic Logic, in particular that of interpretation, understood as a multifunction that preserves and reflects logical consequence, capture a number of relevant transformations in the context of software design, reuse, and adaptation, difficult to deal with in classical approaches. Examples include data encapsulation and the decomposition of operations into atomic transactions. But if interpretations open such a new research avenue in program refinement, (conceptual) tools are needed to reason about them. In this line, the paper’s main contribution is a study of the correspondence between logical interpretations and morphisms of a particular kind of coalgebras. This opens way to the use of coalgebraic constructions, such as simulation and bisimulation, in the study of interpretations between (abstract) logics.