We introduce the notion of an ACP process algebra and the notion of a meadow enriched ACP process algebra. The former notion originates from the models of the axiom system ACP. The latter notion is a simple generalization of the former notion to processes in which data are involved, the mathematical structure of data being a meadow. Moreover, for all associative operators from the signature of meadow enriched ACP process algebras that are not of an auxiliary nature, we introduce variable-binding operators as generalizations. These variable-binding operators, which give rise to comprehended terms, have the property that they can always be eliminated. Thus, we obtain a process calculus whose terms can be interpreted in all meadow enriched ACP process algebras. Use of the variable-binding operators can have a major impact on the size of terms.
Deep Dive into A process calculus with finitary comprehended terms.
We introduce the notion of an ACP process algebra and the notion of a meadow enriched ACP process algebra. The former notion originates from the models of the axiom system ACP. The latter notion is a simple generalization of the former notion to processes in which data are involved, the mathematical structure of data being a meadow. Moreover, for all associative operators from the signature of meadow enriched ACP process algebras that are not of an auxiliary nature, we introduce variable-binding operators as generalizations. These variable-binding operators, which give rise to comprehended terms, have the property that they can always be eliminated. Thus, we obtain a process calculus whose terms can be interpreted in all meadow enriched ACP process algebras. Use of the variable-binding operators can have a major impact on the size of terms.
In many formalisms proposed for the description and analysis of processes in which data are involved, algebraic specifications of the data types concerned have to be given over and over again. This is also the case with the principal ACP-based formalisms proposed for the description and analysis of processes in which data are involved, to wit µCRL [16,17] and PSF [23]. There is a mismatch between the process specification part and the data specification part of these formalisms. Firstly, there is a choice of one built-in type of processes, whereas there is a choice of all types of data that can be specified algebraically. Secondly, the semantics of the data specification part is its initial algebra in the case of PSF and its class of minimal Boolean preserving algebras in the case of µCRL, whereas the semantics of the process specification part is a model based on transition systems and bisimulation equivalence. Sticking to this mismatch, no lasting axiomatizations in the style of ACP has emerged for process algebras that have to do with processes in which data are involved.
Our first main objective is to obtain a lasting axiomatization in the style of ACP for process algebras that have to do with processes in which data are involved. To achieve this objective, we first introduce the notion of an ACP process algebra and then the notion of a meadow enriched ACP process algebra.
ACP process algebras are essentially models of the axiom system ACP. Meadow enriched ACP process algebras are data enriched ACP process algebras in which the mathematical structure for data is a meadow. Meadows were defined for the first time in [13]. The prime example of a meadow is the rational number field with the multiplicative inverse operation made total by imposing that the multiplicative inverse of zero is zero. Although the notion of a meadow enriched ACP process algebra is a simple generalization of the notion of an ACP process algebra, it is an interesting one: there is a multitude of finite and infinite meadows and meadows obviate the need for Boolean values and operations on data that yield Boolean values to deal with conditions on data.
In the work on ACP, the emphasis has always been on axiom systems. In this paper, we put the emphasis on algebras. That is, ACP process algebras are looked upon in the same way as groups, rings, fields, etc. are looked upon in universal algebra (see e.g. [14]). The set of equations that are taken to characterize ACP process algebras is a revision of the axiom system ACP. The revision is primarily a matter of streamlining. However, it also involves a minor generalization that allows for the generalization to meadow enriched ACP process algebras to proceed smoothly.
In µCRL and PSF, we find variable-binding operators generalizing associative operators of ACP. Our second main objective is to determine to what extent such variable-binding operators fit in with meadow enriched ACP process algebras. To achieve this objective, we introduce, for all associative operators from the signature of meadow enriched ACP process algebras that are not of an auxiliary nature, variable-binding operators as generalizations.
These variable-binding operators, which give rise to comprehended terms, have the property that they can always be eliminated. That is, for each comprehended term, we can derive from axioms concerning the variable-binding operators that the comprehended term is equal to a term over the signature of meadow enriched ACP process algebras. Those axioms are axioms of a calculus because the distinction between free and bound variables is essential in derivations. The terms of this process calculus are interpreted in meadow enriched ACP process algebras.
Full elimination of all variable-binding operators occurring in a comprehended term can lead to a combinatorial explosion. We show that a combinatorial explosion can be prevented if variable-binding operators that bind variables with a two-valued range are still permitted in the resulting term. We also show that in the latter case the size of the resulting term can be further reduced if we add an identity element for sequential composition to meadow enriched ACP process algebras. Moreover, we demonstrate that there is an alternative to introducing variable-binding operators for several associative operators on processes if we add a sort of process sequences and suitable operators on process sequences to meadow enriched ACP process algebras.
For readability, it is imprecisely said above that the mathematical structure for data in meadow enriched ACP process algebras is a meadow. It is actually a signed meadow, i.e. a meadow expanded with a signum operation. In the presence of a signum operation, the ordering on the elements of a meadow that corresponds with the usual ordering on the elements of a field becomes definable.
This paper is organized as follows. First, we give a brief summary of signed meadows (Section 2). Next, we introduce th
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