Temporal theories as modularisation units for concurrent system specification

Abstract

In this paper, we bring together the use of temporal logic for specifying concurrent systems, in the tradition initiated by A. Pnueli, and the use of tools from category theory as a means for structuring specifications as combinations of theories in the style developed by R. Burstall and J. Goguen. As a result, we obtain a framework in which systems of interconnected components can be described by assembling the specifications of their components around a diagram, using theory morphisms to specify how the components interact. This view of temporal theories as specification units naturally brings modularity to the description and analysis of systems. Moreover, it becomes possible to import into the area of formal development of reactive systems the wide body of specification techniques that have been defined for structuring specifications independently of the underlying logic, and that have been applied with great success in the area of Abstract Data Types. Finally, as a discipline of design, we use the object-oriented paradigm according to which components keep private data and interact by sharing actions, with a view towards providing formal tools for the specification of concurrent objects.

Preservation and reflection in specification

“…Signatures and signature morphisms constitute a category SIGN. ❚ As in [8], a signature <A,Γ> provides the (non-logical) symbols of the language used for specifying a system: A is the set of attribute symbols and Γ is the set of action symbols. In order to simplify the presentation we only consider Boolean attributes.…”

“…Moreover, the condition V(w')=V(w), for every w'∈R(⊥)(w), means that a locus is a model in which the attributes remain unchanged whenever the component remains idle, i.e. attributes are local (encapsulation) [8].…”

“…The "guilty" party in this context is usually the class of liveness properties (e.g., [2]). In the context of an algebraic approach to specification [8], this means that a functorial relationship between a category of programs (or lower-level specifications) and a category of specifications cannot be established [6].…”

Verifiable and executable logic specifications of concurrent objects in $$\mathcal{L}_\pi$$

“…But unrestricted use of TL often leads to unimplementable specifications, and typical approaches pick some restricted class of formulas as dynamic axioms. For instance in [4], one finds safety formulas like ~ A A* ~ Xr and liveness formulas like qo ~ FA* -where A* denotes the occurrence of action A, and X and F are respectively the next and eventually operators of (linear) TL. Concerning liveness properties, in our asynchronous model we will not be able to assert such a strong assertion, but just something like the weaker ~o ~ XmA* where mA* asserts the existence of a pending request for action A.…”

Revisiting the Categorical Approach to Systems*

“…These categorical principles have been used to formalise different mathematical models of system behaviour [13,24,25,14], their logical specifications [2,7] and their realisations as parallel programs [5].…”