A Refinement Calculus for Tuple Spaces

It is fairly accepted that the realization of complex systems must be accomplished step by step from the initial specification, through a sequence of intermediate phases, to the final program. These development steps, linking a preliminary version, or description, of the program to a more detailed one, are usually called refinement steps, while the intermediate stages of a refinement process are called levels of abstraction. A refinement calculus is a means to support this modus operandi in program development, allowing to link different levels of abstraction: it introduces a precise relation between intermediate descriptions, and the rules to check whether the relation is satisfied. Tuple space languages are concurrent languages, that foster the definition of autonomous entities of computation (the processes), and offer mechanisms for their synchronization and communication. In particular, they represent one of the most acknowledged models of coordination. Tuple space languages are based on the idea that a dynamic collection of tuples can act as shared state of concurrent processes, and play the role of coordination media among them. To build a refinement calculus for tuple spaces, we address three points, in this paper: 1) We single out a specification language, a variation of first order temporal logic. Temporal relations between propositional formulae are not expressive enough to describe relations between tuple spaces, which are multisets of atoms. The specification language, called Oikos-tl, includes three new temporal operators that enhance the expressive power of the logic, permitting to directly link state transitions and state configurations. The semantics of the specification language is formally defined, and a set of useful properties for refinement are shown. 2) We introduce a reference language for tuple spaces, dubbed TuSpRel, and define its axiomatic and operational semantics. We need the former to derive properties, the latter to describe the allowed computations of a system. We relate these descriptions, and guarantee that using the axiomatic semantics we can derive properties, which are correct and complete with respect to the operational behaviour. The non-deterministic features of tuple space languages make this result new, and more complex than in other programming paradigms. One of the contributions of our work is the idea to derive weakest preconditions exploiting the demonic strict choice in non-deterministic selection. The transition system defining the operational semantics is based on the new notion of enabling precondition, which exploits the angelic strict choice. 3) To build the refinement calculus, we take a compositional approach. We first consider the basic statements of the language, and say under which conditions they satisfy a property, then compose these proofs to derive that a system refines a specification. Finally, in the refinement calculus definition, we extend to tuple space languages the ability to exploit logic formulae to specify the behaviour of unrefined modules: in the intermediate steps, a system is only partially written in the programming language, and the still unrefined features are described by logical formulae