Modal Logic and Algebraic Specifications

The established approaches to the semantics of algebraic (equational) speciications are based on a category-theoretic perspective. When possible interpretations are viewed as a category, the extreme points|the initial and nal algebras|present themselves as natural candidates for the canonical interpretation. However, neither choice provides a satisfactory solution for incomplete speciications of abstract data types|the initial algebra is not abstract enough and the nal algebra often does not exist. We argue that in much of the work on algebraic speciications, 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 interpretation|which we call the optimal algebra 1 |arises naturally. The optimal algebra turns out to be a natural generalization of the nal algebra, and provides a satisfactory semantics in situations where the spirit of nal algebra semantics is desired but a nal algebra does not exist. Optimal semantics has a topological avor. Our Kripke models are topological spaces in a natural way. In most (but not all) of the interesting cases, the Baire Category Theorem holds for the topology of a Kripke model, in which case the optimal semantics validates exactly those equational properties which hold in dense open subsets of the Kripke model. In analogy to many similar situations, we may regard these as properties that hold almost everywhere in the model. 1 The term \optimal algebra" was suggested to us by Vaughan Pratt, who also independently arrived at some of the related intuitions. Francesco Parisi-Pressice has informed us that in unpublished work, he considered optimality several years ago, also as a generalization of nal algebra semantics.