This paper investigates the compositionality of operational models for concurrency induced by labeled transition systems (LTS's). These models are defined on the basis of a metric domain first introduced by de Bakker and Zucker; the domain is a complete metric space consisting of tree-like structures called processes. Transition system specifications (TSS's) define LTS's; the set of states of such a LTS A is the set of terms generated by a signature Σ. For the syntactical operators F contained in Σ, semantic operations (on processes) associated with F are derived from the TSS S by which A is defined, provided that S satisfies certain syntactical restrictions. By means of these operations, the compositionality of the operational model induced by A is established. A similar result was obtained by Rutten from TTS's which define finitely branching LTS's. The main contribution of this paper is generalization of Rutten's result to be applicable to TSS's which are based on applicative languages including recursion, parameterized statements, and value passing, and which define infinitely branching LTS's. A version of typed λ-calculus incorporating µ-notation is employed as a formalism for treating recursion, parameterized statements, and value-passing. Infinitely branching LTS's are needed to treat programming languages including value passing such as CCS.

The semantics of a language for communicating processes is investigated, and three full abstractness results for are established. The language contains atomic actions, termination, inaction, sequential composition, alternative composition, parallel composition, action restriction, and a form of guarded recursion. (The guardedness restriction on recursion is needed to establish one of the full abstractness results.) Three Plotkin-style operational semantics WL, CWL, and IWL of the language are defined. These semantics are linear in that the meaning of each program in any of these semantics is a set of action sequences the program may perform, and are weak in that the action sequences are obtained by ignoring (finte sequences of) internal moves. All the three semantics distinguish divergence (an infinite sequence of internal moves) from deadlock. The semantics IWL differs from the other two in that IWL is a so-called interal action semantics taking into account only internal moves under the assumption that the environment allows no (external) communication actions, and hence, the only possible actions for processes are internal moves, whereas the other two semantics take into account communication actions in addition to internal moves. The two semantics WL and CWL differ each other in that CWL is a so- called completed trace semantics, whereas WL is not. Then, two compositional models RF and IRF for the language are proposed, and the full a bstractness of RF (resp. of IRF) w. r. t. WL and CWL (resp. w. r. t. IWL), as expressed in the following, is established:s1s2(*) S[][S[]is a context of ⇒S[s1]S[s2]],where ,RF,WL, RF,CWL, IRF,IWL. A similar full abstractness result has been established by Bergstra, Klop, and Olderog for a language without recursion or internal moves. Moreover, Rutten investigated the semantics of a language similar to , in the framework of complete metric spaces, and showed that the failures model is fully abstract w. r. t. a strong linear semantics L, where L is strong in that it does not abstract from internal moves. The full abstractness of RF w. r. t. CWL, expressed in (*) with ,RF,CWL, is an extension of the result of Bergstra et al. to a language with recursion and internal moves. Also, the full abstractness of IRF w. r. t. IWL, expressed in (*) with ,IRF,IWL, is an extension of the Rutten's result, to the case of weak linear semantics with divergence.