A Categorical Approach to Structuring and Promoting Z Specifications

Abstract

Abstract. In this paper, we study a formalisation of specification structuring mechanisms used in Z. These mechanisms are traditionally understood as syntactic transformations. In contrast, we present a characterisation of Z structuring mechanisms which takes into account the semantic counterpart of their typical syntactic descriptions, based on category theory. Our formal foundation for Z employs well established abstract notions of logical systems. This setting has a degree of abstraction that enables us to understand what is the precise semantic relationship between schemas obtained from a schema operator and the schemas it is applied to, in particular with respect to property preservation.Our formalisation is a powerful setting for capturing structuring mechanisms, even enabling us to formalise promotion. Also, its abstract nature provides the rigour and flexibility needed to characterise extensions of Z and related languages, in particular the heterogeneous ones.

Categorical foundations for structured specifications in Z

“…This is the other main contribution of our proposal. This work extends that presented in [CAPM12], where a basic categorical framework to capture schema conjunction and promotion is introduced. We now rework that framework to obtain a more expressive setting, by extending some definitions to be able to capture other schema operators such as schema disjunction and schema quantification.…”

“…A schema defines a set of typed variables, and provides constraints on these variables. At first sight, the notion of axiomatic theory captures naturally the notion of schema, as illustrated in [Spi84,CAPM12]; however, schemas support several operations over them (conjunction, disjunction, promotion, etc.) that do not have a corresponding formalisation in the category of theories over signatures.…”

Satisfiability Calculus: An Abstract Formulation of Semantic Proof Systems

“…More recently, in [6], all these notions of morphism were investigated in more detail by observing how the direction of the arrows modify its interpretation. In this section we will concentrate only on extending the results presented by Tarlecki in [5], focussing on institution morphisms and comorphisms, since these notions have been used to formalise several concepts arising in software engineering: they are used as the main vehicle for borrowing proofs along logic translation in [5]; for defining heterogeneous development environments for software specifications and designs in [10,51], which provides the foundations of tools like HETS [14] and CafeOBJ [11]; for providing structured specifications in general in [7], and for specific formal languages in [52,53]; for defining proof systems for structured specifications [54,55,56]; and for formalising data and specification refinements in [33,57,52], just to give a few examples.…”

Satisfiability Calculus: The Semantic Counterpart of a Proof Calculus in General Logics

“…More recently, in [26] all these notions of morphism were investigated in detail. In this work we will concentrate only on institution representations (or comorphisms in the terminology introduced by Goguen and Rosu), since this is the notion that we have employed to formalize several concepts arising from software engineering, such as data refinement and dynamic reconfiguration [27,28]. The study of other important kinds of functorial relations between satisfiability calculi are left as future work.…”