The pioneering work of Hewitt and Baker on the foundations of concurrency during the seventies has inspired the development of a promising object-based framework for understanding open distributed systems, the actor model. So far, theoretical research on actors has focused on identifying the basic primitives of the model and on characterising the operational behaviour of distributed programming languages in terms of actor components. In this paper, we show that the actor model can also be used as a faithful basis for rigorously designing open distributed systems. We argue that a proof-theoretic approach is better suited to this purpose. An abstract data type like axiomatisation of the actor primitives is proposed to support composing and reasoning from specifications of actor communities within a temporal logical system.

1 Introduction The current, and quite real, technical problems blocking software compositionality in practice could benefit from formal foundations providing clear semantics, new architecture and software composition methods, and better languages. The choice of foundations is however a delicate matter, because the proposed semantic framework should explictly address and support key features such as openness, distribution, object-orientation, compositionality, and mobility. Rewriting logic is a promising semantic framework supporting such key features and offering simple and rigorous formal foundations for software composition and evolution. It also provides a wide-spectrum approach in which architectural descriptions, executable formal specifications, and distributed and mobile code can be related, composed, transformed, and integrated in a rigorous way. Reflection is a key feature of rewriting logic of great usefulness for software composition, dynamicity, and mobility.

The syntax and semantics of actors and -agents is first defined separately, using a uniform, "unbiased" approach. New coordination primitives are the added to the union of the two calculi which allow actors and -agents to cooperate.

Data Type Specification, in combination with modal logics for formalizing the process of building systems from interconnected components. This combination of logical and categorical techniques has also been applied to parallel program design languages in the style of UNITY [14] and IP [41], providing semantics for modularization techniques based on the notion of superposition. This has resulted in the development of a programming design language called Community [33]. Two formalisms that provide explicit support for object systems and can reason about their rewriting logic specifications have been recently developed. One is a version of the modal -calculus proposed by Lechner [48, 49] for reasoning about object-oriented Maude specifications. Another is Denker's objectoriented distributed temporal logic DTL + [24, 22], that extends the DTL and D 1 distributed object temporal logics of Ehrich and Denker [30, 23, 29]. Lechner [48, 49] uses her version of the modal -calculus to identif...

Abstract. This paper discusses event-based semantics in the context of the emerging concept of Cyber Physical Systems and describes two related formal models concerning policy-based coordination and Interactive Agents. 1