## A Causal Semantics for CCS via Rewriting Logic (2000)

Venue: | Theoretical Computer Science |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Degano00acausal,

author = {Pierpaolo Degano and Fabio Gadducci and Corrado Priami},

title = {A Causal Semantics for CCS via Rewriting Logic},

journal = {Theoretical Computer Science},

year = {2000},

volume = {275},

pages = {200--2}

}

### OpenURL

### Abstract

We consider two operational semantics for ccs dened in the literature: the rst exploits Proved Transition Systems (pts) and the second Rewriting Logic (rl). We show that the interleaving interpretation of both semantics agree, in that they dene the same transitions and exhibit the same nondeterministic structure. In addition, we study causality in ccs computations. We recall its treatment via pts, exhibiting the notion of causality presented in the literature, and we show how to recast it in the rl semantics via suitable axioms. Also in this case, the two semantics agree. Contents 1 Introduction 2 2 Some notions on Process Algebras 3 2.1 The Calculus of Communicating Systems 4 2.2 Proved Transition System 6 2.3 Causality and Concurrency 7 ? Research partly supported by the Italian CNR Progetto Strategico Modelli e Metodi per la Matematica e l'Ingegneria and MURST Progetto Tecniche Formali per la Specica, l'Analisi, la Verica, la Sintesi e la Trasformazione di Sistemi Software. ...

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Citation Context ...and equipping it with a semantics able to describe non standard aspects of concurrent computations, in particular causality. The language we choose is ccs, Milner's Calculus for Communicating Systems =-=[24]-=-, and the starting points are the semantics proposed in [16] and in particular in [21]. Roughly, a ccs process is a term of an algebra over a set of process constructors, under the assumption that the... |

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Citation Context ...gs of their proofs. This very concrete representation of process behaviour was used to describe qualitative and quantitative non-interleaving aspects of a variety of calculi, e.g., for the -calculus [=-=26]-=-, a calculus based on naming fairly more complex than ccs [14]. Indeed, simple relabellings of proved computations yield the main semantics presented in the literature, in particular the (interleaving... |

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Citation Context ...s transition system into a rewriting theory requires fairly sophisticated algebraic techniques. As asrst step we recall the settheoretical construction of the term algebra for order-sorted signatures [17]. Actually, we will consider the very limited case where there are only two sorts 1 and 2 , with 1 2 ; however this fragment is sucient for our purposes. Let be such an order-sorted signat... |

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Citation Context ...computations, in particular causality. The language we choose is ccs, Milner's Calculus for Communicating Systems [24], and the starting points are the semantics proposed in [16] and in particular in =-=[21]-=-. Roughly, a ccs process is a term of an algebra over a set of process constructors, under the assumption that the operators represent basic features of a concurrent system, among which communication ... |

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Citation Context ...ons 23 5 Conclusions 26 References 27 1 Introduction Concurrency is maybe the basic aspect of the operational interpretation of rewriting logic. And as Jose Meseguer says in his lecture at concur'96 [=-=23]-=-, . . . my main emphasis in this talk will be on rewriting logic as a semantic framework for concurrency. . . . The goal is . . . to express as faithfully as possible each model [of concurrency] on it... |

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Citation Context ...ntics for ccs. Among those proposals, most similar in presentation to the original interleaving semantics are those provided via transition systems that express the causality between transitions (see =-=[3,8,9,12,19-=-] to mention only a few references) . Remarkably, all these semantics agree on the notion of causality they dene and conservatively extend the original interleaving semantics. Wesrst provide ccs with ... |

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Citation Context ...ntics for ccs. Among those proposals, most similar in presentation to the original interleaving semantics are those provided via transition systems that express the causality between transitions (see =-=[3,8,9,12,19-=-] to mention only a few references) . Remarkably, all these semantics agree on the notion of causality they dene and conservatively extend the original interleaving semantics. Wesrst provide ccs with ... |

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Citation Context ...rministic context. In fact, while the others are more specic, thesrst axiom schemata of Table 7 could be considered analogous to the conditions required for permutation equivalence on term rewriting [=-=2]-=-, since the rewrites occur on disjoint parts of a process. For general results on the relationships between rewriting logic and permutation equivalence, see [7,15,20]. 4.3 Lifting to concurrent comput... |

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Citation Context ...emma 3.1, because they have sort SP . The same argument works for the exchange axiom. 2 The statement above is reminiscent of the standard decomposition property for sequents reported in Lemma 3.6 of =-=[22-=-] (but see also Proposition 21 of [5] and Proposition 4.1 of [15]): the main dierence is its uniqueness, due to the structure of the proof terms entailed by R ccs . Uniqueness is not a trivial propert... |

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Citation Context ...ntics for ccs. Among those proposals, most similar in presentation to the original interleaving semantics are those provided via transition systems that express the causality between transitions (see =-=[3,8,9,12,19-=-] to mention only a few references) . Remarkably, all these semantics agree on the notion of causality they dene and conservatively extend the original interleaving semantics. Wesrst provide ccs with ... |

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Citation Context ...ss behaviour was used to describe qualitative and quantitative non-interleaving aspects of a variety of calculi, e.g., for the -calculus [26], a calculus based on naming fairly more complex than ccs [=-=14]-=-. Indeed, simple relabellings of proved computations yield the main semantics presented in the literature, in particular the (interleaving and the) causal one which is of main interest here. It will b... |

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Citation Context ...To introduce our concurrent ccs rewrite theory, we exploit the relation between \parallel" transitions in a concurrency diamond. The interested reader may wish to compare our approach with the on=-=e of [11]-=- that exploits category theory to characterize some analogous notions of concurrent computations of Petri nets. Or more appropriately, the set of axioms over rewriting logic proposed in [20], in order... |

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Citation Context ... set of axioms that equate those computations diering from the temporal ordering in which two, or more, concurrent transitions may occur. Our starting point has been the Proved Transition Systems of [=-=12,13]-=-, whose enhanced labels have an algebraic structure, that re ects the structure of processes, just as proof terms do. Actually, our set of axioms can be split into two distinct parts. Thesrst is direc... |

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Citation Context ...r the exchange axiom. 2 The statement above is reminiscent of the standard decomposition property for sequents reported in Lemma 3.6 of [22] (but see also Proposition 21 of [5] and Proposition 4.1 of =-=[15-=-]): the main dierence is its uniqueness, due to the structure of the proof terms entailed by R ccs . Uniqueness is not a trivial property, and it is not in general valid for a generic rewriting theory... |

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Citation Context ...e for building an algebra of labels, each element of the term algebra encoding a justication of a rewrite. Out of all the possible dierent ways to introduce such a signature, we follow the lines of [6]. Definition 3.2 (rewriting sequents) Let R = h; E; L; Ri be a rewrite theory; and let be the signature containing all the labels d as suitable operators, with the corresponding arity and sort give... |

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Citation Context ...rmutation equivalence on term rewriting [2], since the rewrites occur on disjoint parts of a process. For general results on the relationships between rewriting logic and permutation equivalence, see =-=[7,15,2-=-0]. 4.3 Lifting to concurrent computations In this section we lift Theorem 4.2 to the truly concurrent semantics as dened on proved transition systems, showing that the notion of concurrency of Deniti... |

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Citation Context ...ard aspects of concurrent computations, in particular causality. The language we choose is ccs, Milner's Calculus for Communicating Systems [24], and the starting points are the semantics proposed in =-=[16]-=- and in particular in [21]. Roughly, a ccs process is a term of an algebra over a set of process constructors, under the assumption that the operators represent basic features of a concurrent system, ... |

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Citation Context ...he one of [11] that exploits category theory to characterize some analogous notions of concurrent computations of Petri nets. Or more appropriately, the set of axioms over rewriting logic proposed in =-=[-=-20], in order to axiomatize permutation equivalence in the framework of -calculus. Definition 4.1 The process theory associated with ccs is the equational specication h eccs ; E s i, where E s is the ... |

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Citation Context ... the -calculus given in [20]. Acknowledgments. We are grateful to Georgia Carabetta for the many interesting discussions: the present paper is based on a previous work by her and thesrst two authors [=-=4]-=-, where a weaker correspondence between sequents and proved transitions was established. We are also grateful to one of the referees, whose remarks helped us in improving the overall presentation of o... |

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Citation Context ... The same argument works for the exchange axiom. 2 The statement above is reminiscent of the standard decomposition property for sequents reported in Lemma 3.6 of [22] (but see also Proposition 21 of =-=[5-=-] and Proposition 4.1 of [15]): the main dierence is its uniqueness, due to the structure of the proof terms entailed by R ccs . Uniqueness is not a trivial property, and it is not in general valid fo... |

1 |
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Citation Context ...ons in a weakly equivalent one without, see [25]. Furthermore, the general version of ! only requires a more careful denition of causality than the one given here. The interested reader maysnd it in [=-=10]-=-. Future work will aim at combining our notion of concurrent computation with some behavioural equivalence. In other words, we would like to characterize 26 bisimulation in Rewriting Logic. However, t... |

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1 | A non-interleaving semantics from CCS based on proved transitions, Fund - Boudol, Castellani |