Modal Logic and Algebraic Specifications

The established approaches to the semantics of algebraic (equational) specifications are based on a category-theoretic perspective. When possible interpretations are viewed as a category, the extreme points---the initial and final algebras---present themselves as natural candidates for the canonical interpretation. However, neither choice provides a satisfactory solution for incomplete specifications of abstract data types---the initial algebra is not abstract enough and the final algebra often does not exist. We argue that in much of the work on algebraic specifications, the categorical viewpoint is simply a convenient technical device to semantically capture the modalities of necessity and possibility. It is actually more natural to consider the semantic problem from the perspective of modal logic, gathering possible interpretations into a Kripke model. When necessity and possibility are added as modal operators in the logical language, a new candidate for the canonical interpretati..