Modelling Process Algebra

CoCasl [11], a recently developed coalgebraic extension of the algebraic specification language Casl [2], allows for modelling systems in terms of inductive datatypes as well as of co-inductive process types. Here, we demonstrate how to specify process algebras, namely CCS [10] and CSP [8,17], within such an algebraic-coalgebraic framework. It turns out that CoCasl can deal with the fundamental concepts of process algebra in a natural way: The type system of communications, the syntax of processes and their structural operational semantics fit well in the algebraic world of Casl, while the additional coalgebraic constructs of CoCasl cover the various process equivalences (bisimulation, weak bisimulation, observational congruence, and trace equivalence) and provide fully abstract semantic domains. CoCasl hence becomes a meta-framework for studying the semantics and proof theory of reactive systems. Among the various frameworks for the description and modelling of reactive systems, process algebra plays a prominent role. It has proven to be suitable at the level of requirement specification, for formal refinement proofs as well as for writing design specifications. Almost all of the underlying concepts of process algebra can be found in the languages CCS [10] and CSP [8,17]: a type system on the communications; synchronous as well as asynchronous communication; operational semantics; and also various notions of process equivalence like strong and weak bisimulation, observation congruence, and trace equivalence. Thus, when proposing a new framework which aims at the specification of reactive systems in general, it is worthwile to study if these process algebras and their semantic concepts are covered. 1 [email protected] 2 [email protected] 3 [email protected] 4 Research supported by the DFG project Multiple (KR 1191/5-2) 5 Research supported by the DFG project COOFL (Qi 1/4-2) 6 Research supported by the DFG project HasCASL (KR 1191/7-1) This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs Mossakowski, Roggenbach, Schröder CoCasl [11], a recently defined coalgebraic extension of the algebraic specification language Casl [2], allows for modelling systems in terms of inductive datatypes as well as of co-inductive process types. Here, the language Casl provides a many sorted first order logic with partial functions and subsorting, and also features sort generation constraints for generated and free datatypes. Furtheron, Casl allows for initial semantics in terms of a structured free construct. CoCasl extends this language by a basic co-datatype construct, cogeneratedness constraints, and structured cofree specifications; moreover, coalgebraic modal logic is introduced as syntactical sugar. The idea behind the design of CoCasl is to obtain a fruitful synergy of algebraic and coalgebraic system modelling. Specifying the process algebras CCS and CSP provides an extensive case study in this language. Thus, besides beeing a proof of concept for CoCasl, these specifications demonstrate how to integrate different system aspects within one framework. Casl as the algebraic sub-language of CoCasl will deal with the syntactic aspects and the operational semantics of these process algebras, while the equivalence relations on processes like bisimulation, which have been shown to be of coalgebraic nature [3,19,16], will be modelled with the coalgebraic constructs provided by CoCasl. The full specifications are available at [12]. The paper is organized as follows: Section 1 gives an overview of CoCasl. Then the different aspects of the process algebras CCS and CSP are specified in parallel, separating between syntax and semantics: Section 2 describes process algebra syntax in terms of Casl free datatypes; Section 3 provides the structural operational semantics. Finally, various notions of process equivalences are specified in Section 4. 1 An overview of CoCasl The specification language Casl (Common Algebraic Specification Language) has been designed by CoFI, the international Common Framework Initiative for Algebraic Specification and Development [7]. For the language definition cf. [2,4]; a full formal semantics is laid out in [1]. Here, we recall some basic language features as needed for understanding the specifications below. Casl is separated into various levels, including a level of basic specifications and a level of structured specifications. Basic specifications essentially list signature items and axioms in an unstructured way, thus determining a category of first order models. Structured specifications serve to combine such basic specifications into larger specifications in a hierarchical and modular fashion. The structuring operations are independent of the underlying logic and hence apply to language extensions such as CoCasl in the same way as to Casl. At the level of basic specifications, one can declare sorts (keyword sort), operations (keyword op), and predicates (keyword pred) with given input and result sorts. Sorts may be declared to be in a subsorting relation; if s is a subsort of t, then terms of type s may be used wherever terms of type t are