Specifying with defaults: Compositional semantics

We present an abstract specification theory that formalizes non-monotonic composition constructs from specification languages providing explicit non-monotonic mechanisms as a specification facility. This theory generalizes the institutional framework from Goguen and Burstall by adding defeasibility mechanisms to a given institution. The denotation of a specification module consists of defaults (formulas organized by priority) that are assumed to be true in the absence of explicit information to the contrary. In other words defaults are assumed to be true unless they are overridden by other defaults of higher priority. Formulas that cannot be overriden are called axioms. Such structures of axioms and prioritized defaults are called hierarchic specifications. The abstract specification theory of hierarchic specifications formalizes specification operations that use overriding (i.e. contradiction) of previously specified properties. The specification process improves in modularity and reusability when such mechanisms are available. This theory gives formal grounds to “the requirement to re-use specification modules as far as possible, i.e. not only to include the same components in different contexts, but also to prefer modification of given parts over new definitions. To reduce development costs, software should be designed in a “differential” way select a module from the library, refine it by adding new functions, and modify it by overriding some old ones”[9]. The abstract specification theory of hierarchic specifications consists in formalizing, independently of the underlying logic, the structuring operations of hierarchic specifications. These operations are defined both on the syntactical and semantical levels by canonical constructions on corresponding syntactical and semantical categories, and account for the modular construction of hierarchic specifications by combining, reusing and modifying (with overriding) previously specified modules.