### Table 2.2 Axioms for Basic Process Algebra A1

2000

### Table 2.2 Axioms for Basic Process Algebra A1

2000

### Table 1: Transition rules for standard process algebraic constructions

"... In PAGE 4: ... Formally, we require i 62 Act and let Act+ = Act [ fig to be ranged over by a+. The whole operational semantics of CTR is given in Table1 and Table 2 (we assume f(i) = i for any relabelling function f). x _ ?y means maxfx ? y; 0g and ] [ stands for any kind of left end of the interval (] or [).... In PAGE 10: ... First we de ne the inline choice operational semantics as the transition rela- tion ?!I. For that we can take all the rules de ning the \standard quot; semantics ?! in Table1 and Table 2 (with changing syntactically ?! to ?!I) except for two. First of those to be changed is the main rule for delay of the delay pre x, which is to read now as [c; e]:P (d) ?! [c0; e0]:P whenever max(0; c ? d) c0 e0 e ? d.... ..."

### Table 1 lists the axioms of pCRL. Axioms A1{A7 are well known from process algebra. The P-operator and the use of capital X will be explained below.

2001

"... In PAGE 3: ... Table1 : Axioms of pCRL Data types in CRL are algebraically speci ed in the standard way using sorts, functions and axioms. For data sorts we use D; E; : : :, and for data variables of the respective sorts we use d; e; : : :.... ..."

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### Table 1. syntax of a simple process algebraic language Name Axioms and inference rules

"... In PAGE 3: ... The environment may contain mutually recursive process de nitions. The label types Lp are usually left unde ned, and are implicitly understood to be the smallest label types satisfying the static constraints of Table1 . In the application part of the paper we will provide concrete instances of the set of actions Act and the process de nition environment.... In PAGE 3: ... In the application part of the paper we will provide concrete instances of the set of actions Act and the process de nition environment. In addition to the process algebraic combinators introduced by Table1 we will use generalizations for the choice and composition operators. If B denotes a nite set of behaviour expressions then P B and QG B denote the repeated application of `+ apos; and `jjG apos;, respectively, to the elements of B.... In PAGE 4: ... [Mil89]) Fact 1. The relation is a congruence with respect to all the combinators introduced in Table1 and satis es the laws listed in Table 3. u t... In PAGE 5: ...gain we have a standard result (cf. [Mil89]). Fact 2. The relation is a congruence with respect to all the combinators introduced in Table1 except for the choice combinator +, and its generalization P.... In PAGE 5: ... u t Fact 3. The relation trace is a congruence with respect to all the combinators introduced in Table1 and trace. u t... In PAGE 20: ... We start with the equivalence corresponding to Traces!(B) de ned by B1 trace! B2 i Traces!(B1) = Traces!(B2) Fact 5. The relation trace! is a congruence with respect to all the combinators introduced in Table1 and trace! trace. u t Fact 6.... ..."

### Table 4: Axioms for interworking sequencing The structured operational semantics of the interworking sequencing and of the auxiliary opera- tors is given in Table 5. The term deduction system T(IWD quot;(A; EID; E)) consists of the deduction rules of T(BPA ; quot;(A)) and the deduction rules of Table 5. Next, we will formulate some interesting theorems concerning this process algebra. These theorems relate the process algebra IWD quot;(A; EID; E) to the process algebra BPA ; quot;(A) and to the structured operational semantics as given by the term deduction systems. Theorem 2.2.1 (Congruence) Bisimulation equivalence is a congruence for the function sym- bols in the signature of IWD quot;(A; EID; E). Proof It is straightforward to verify that the deduction rules of the term deduction system which 5

1995

"... In PAGE 5: ... L iw y = . Consequently, we also de ne x R iw quot; = . If we apply this in the de nition of the sequencing operator as given in [MvWW93] we get quot; iw quot; = quot; L iw quot; + quot; R iw quot; = + = . This is not what we want and therefore we need the additional operator p as given in Table4 . This operator has also been used by Baeten and Weijland [BW90] in axiomatizing the free merge in... ..."

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### Table 4: Rules for operators from other process algebras. Rules marked y have a symmetric form.

1990

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### Table 3: The basic signature of combinatory process algebra. We shall use the following notational conventions. (a) Binary operators will be written in x.

### Table 4: Process Algebra Formalisms

1997

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