## CCS with Hennessy’s merge has no finite equational axiomatization (2005)

Venue: | Theoretical Computer Science |

Citations: | 19 - 16 self |

### BibTeX

@ARTICLE{Aceto05ccswith,

author = {Luca Aceto and Wan Fokkink and Anna Ingólfsdóttir and Bas Luttik},

title = {CCS with Hennessy’s merge has no finite equational axiomatization},

journal = {Theoretical Computer Science},

year = {2005},

volume = {330},

pages = {377--405}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper confirms a conjecture of Bergstra and Klop’s from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner’s Calculus of Communicationg Systems is not finitely based modulo bisimulation equivalence. Thus Hennessy’s merge cannot replace the left merge and communication merge operators proposed by Bergstra and Klop, at least if a finite axiomatization of parallel composition is desired.

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Citation Context ... and allow for some form of synchronization between them. For example, Milner’s CCS offers the binary operator |, whose intended semantics is described by the following classic rules in Plotkin-style =-=[32]-=-: x µ → x ′ x | y µ → x ′ | y y µ → y ′ x | y µ → x | y ′ x α → x ′ , y ¯α → y ′ x | y τ → x ′ | y ′ (In the above rules, the symbol µ stands for an action that a process may perform, α and ¯α are two... |

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Citation Context ... operator is not straightforward. In their seminal paper [22], Hennessy and Milner offered, amongst a wealth of other classic results, a complete equational axiomatization of bisimulation equivalence =-=[31]-=- over the recursion free fragment of CCS. The axiomatization proposed by Hennessy and Milner ibidem dealt with parallel composition using the so-called expansion law—a law that, intuitively, allows on... |

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Citation Context ... the above rules describe the behaviour of the parallel composition operator in very intuitive fashion, the equational characterization of this operator is not straightforward. In their seminal paper =-=[22]-=-, Hennessy and Milner offered, amongst a wealth of other classic results, a complete equational axiomatization of bisimulation equivalence [31] over the recursion free fragment of CCS. The axiomatizat... |

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Citation Context ...bounds with results on equational axiomatizations of various notions of behavioural equivalences or preorders over languages incorporating some notion of parallel composition—see, e.g., the textbooks =-=[7, 20, 26]-=- and the classic papers [8, 22, 25] for general references. Early ω-complete axiomatizations are offered in [19, 28]. More recently, Fokkink and Luttik have shown in [17] that the process algebra PA [... |

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Citation Context ... the parallel composition operator could be axiomatized in bisimulation semantics by means of a finite collection of equations. This question was answered positively by Bergstra and Klop, who gave in =-=[8]-=- a finite equational axiomatization of the merge operator in terms of the auxiliary left merge and communication merge operators. Moller showed 2 (1)sin [29, 30] that strong bisimulation equivalence i... |

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Citation Context ... equational axiomatization of the theory of languages over concatenation and shuffle, solving an open problem raised by Pratt. In proving this result he essentially rediscovered the concept of pomset =-=[33]-=-—a model of concurrency based on partial orders whose algebraic aspects have been investigated by Gischer in [18]—, and proved that the equational theory of series-parallel pomsets coincides with that... |

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Citation Context ...| y x α → x ′ , y ¯α → y ′ x | / y τ → x ′ | y ′ These transition rules give rise to transitions between CCSH terms. The operational semantics for CCSH is thus given by the labelled transition system =-=[24]-=- whose 6sstates are CCSH terms, and whose labelled transitions are those that are provable using the rules. As usual, for each term t and action µ, we write t µ → if t µ → t ′ holds for some term t ′ ... |

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Citation Context ...semantics has led to general algorithms for the generation of (finite) equational axiomatizations for behavioural equivalences from their operational semantics—see, e.g., [3, 6] and the references in =-=[4]-=- for further details. Parallel composition appears as the shuffle operator in the time-honoured theory of formal languages. Not surprisingly, the equational theory of shuffle has received considerable... |

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Citation Context ...well behaved in bisimulation semantics has led to general algorithms for the generation of (finite) equational axiomatizations for behavioural equivalences from their operational semantics—see, e.g., =-=[3, 6]-=- and the references in [4] for further details. Parallel composition appears as the shuffle operator in the time-honoured theory of formal languages. Not surprisingly, the equational theory of shuffle... |

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Citation Context ...on languages for reactive systems that arose from the pioneering work of figures like Bergstra, Hoare, Klop and Milner. Well known examples of such languages are ACP [7], CCS [26], CSP [23] and Meije =-=[5]-=-. These algebraic description languages for processes differ in the basic collection of operators that they offer for building new process descriptions from existing ones. However, since they are desi... |

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Citation Context ...uages incorporating some notion of parallel composition—see, e.g., the textbooks [7, 20, 26] and the classic papers [8, 22, 25] for general references. Early ω-complete axiomatizations are offered in =-=[19, 28]-=-. More recently, Fokkink and Luttik have shown in [17] that the process algebra PA [8] affords an ω-complete axiomatization that is finite if so is the underlying set of actions. An analysis of the re... |

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Citation Context ...sed by Pratt. In proving this result he essentially rediscovered the concept of pomset [33]—a model of concurrency based on partial orders whose algebraic aspects have been investigated by Gischer in =-=[18]-=-—, and proved that the equational theory of series-parallel pomsets coincides with that of languages over concatenation and shuffle. The argument adopted by Tschantz was based on the observation that ... |

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Citation Context ... language CCSH afford the same finite non-empty set of traces, and have therefore the same depth. Since the SOS rules defining the operational semantics of the language CCSH are in de Simone’s format =-=[34]-=-, we have that: 8sFact 2.1 Bisimulation equivalence is a congruence over the language CCSH. Bisimulation equivalence is extended to arbitrary CCSH terms thus: Definition 2.4 Let t, u be CCSH terms. Th... |

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Citation Context ...xiomatizations of various notions of behavioural equivalences or preorders over languages incorporating some notion of parallel composition—see, e.g., the textbooks [7, 20, 26] and the classic papers =-=[8, 22, 25]-=- for general references. Early ω-complete axiomatizations are offered in [19, 28]. More recently, Fokkink and Luttik have shown in [17] that the process algebra PA [8] affords an ω-complete axiomatiza... |

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Citation Context ...uages incorporating some notion of parallel composition—see, e.g., the textbooks [7, 20, 26] and the classic papers [8, 22, 25] for general references. Early ω-complete axiomatizations are offered in =-=[19, 28]-=-. More recently, Fokkink and Luttik have shown in [17] that the process algebra PA [8] affords an ω-complete axiomatization that is finite if so is the underlying set of actions. An analysis of the re... |

20 | Axiomatizing GSOS with termination
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Citation Context ...well behaved in bisimulation semantics has led to general algorithms for the generation of (finite) equational axiomatizations for behavioural equivalences from their operational semantics—see, e.g., =-=[3, 6]-=- and the references in [4] for further details. Parallel composition appears as the shuffle operator in the time-honoured theory of formal languages. Not surprisingly, the equational theory of shuffle... |

18 |
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Citation Context ...is essentially a combination of the left and communication merge operators as its behaviour is described by the first and the last rule in (1). Apart from having soundness problems (see the reference =-=[1]-=- for a general discussion of this problem, and corrected proofs of Hennessy’s results), the proposed axiomatization of observation congruence offered in op. cit. is infinite, as it used a variant of t... |

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Citation Context ...e, e.g., the textbooks [7, 20, 26] and the classic papers [8, 22, 25] for general references. Early ω-complete axiomatizations are offered in [19, 28]. More recently, Fokkink and Luttik have shown in =-=[17]-=- that the process algebra PA [8] affords an ω-complete axiomatization that is finite if so is the underlying set of actions. An analysis of the reasons why operators like the left merge and the commun... |

7 | Nonfinite axiomatizability of the equational theory of shuffle
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Citation Context ... of bisimulation equivalence over the recursion free fragment of CCS and PA without the left merge operator given in [29, 30] are paralleled in the world of formal language theory by those offered in =-=[9, 11, 15]-=-. In the first of those references, Bloom and Ésik proved that the valid inequations in the algebra of languages equipped with concatenation and shuffle have no finite basis. Ésik and Bertol showed in... |

6 |
Nonfinite axiomatizability of shuffle inequalities
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Citation Context ... of bisimulation equivalence over the recursion free fragment of CCS and PA without the left merge operator given in [29, 30] are paralleled in the world of formal language theory by those offered in =-=[9, 11, 15]-=-. In the first of those references, Bloom and Ésik proved that the valid inequations in the algebra of languages equipped with concatenation and shuffle have no finite basis. Ésik and Bertol showed in... |

6 |
Series and parallel operations on pomsets
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- 1999
(Show Context)
Citation Context ...the papers [10, 12, 13] to deal 4swith several other operators, infinite pomsets and infinitary languages, and sets of pomsets. The axiomatizations by Gischer and Tschantz have later been extended in =-=[13, 16]-=- to a two-sorted language with ω powers of the concatenation and parallel composition operators. The axiomatization of the algebra of pomsets resulting from the addition of these iteration operators i... |

4 |
Unique decomposition of processes (note
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Citation Context ... has no 0 factors cannot be omitted in statement 2. For instance, if t = 0 | / x and σ(x) = a0, then depth(σ(t)) < depth(σ(x)). In the proof of our main result, we shall make use of some notions from =-=[27, 28]-=-. These we now proceed to introduce for the sake of completeness and readability. Definition 4.1 A closed term p is irreducible if p ↔ q | r implies q ↔ 0 or r ↔ 0, for all closed terms q, r. We say t... |

4 | Shuffle binoids - Bloom, Ésik - 1998 |

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3 | Axiomatizing shuffle and concatenation in languages - Bloom, Ésik - 1997 |

2 |
shuffle algebras in language varieties, Theoret
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Citation Context ...ion operators on pomsets are modelled by the concatenation and shuffle operators on languages. Tschantz’s technique of coding pomsets with languages homomorphically was further extended in the papers =-=[10, 12, 13]-=- to deal 4swith several other operators, infinite pomsets and infinitary languages, and sets of pomsets. The axiomatizations by Gischer and Tschantz have later been extended in [13, 16] to a two-sorte... |

2 |
generated by languages with poset operations
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(Show Context)
Citation Context ...ion operators on pomsets are modelled by the concatenation and shuffle operators on languages. Tschantz’s technique of coding pomsets with languages homomorphically was further extended in the papers =-=[10, 12, 13]-=- to deal 4swith several other operators, infinite pomsets and infinitary languages, and sets of pomsets. The axiomatizations by Gischer and Tschantz have later been extended in [13, 16] to a two-sorte... |

2 |
Languages under concatenation and shuffling
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Citation Context ...ly, the equational theory of shuffle has received considerable attention in the literature. Here we limit ourselves to mentioning some results that have a special relationship with process theory. In =-=[35]-=-, Tschantz offered a finite equational axiomatization of the theory of languages over concatenation and shuffle, solving an open problem raised by Pratt. In proving this result he essentially rediscov... |

1 |
Axiomatizing the subsumption and subword preorders on finite and infinite partial words, Theoret
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Citation Context ...m the addition of these iteration operators is, however, necessarily infinite because, as shown in op. cit. no finite collection of equations can capture all the sound equalities involving them. (See =-=[14]-=- for closely related developments.) The results of Moller’s on the non-finite axiomatizability of bisimulation equivalence over the recursion free fragment of CCS and PA without the left merge operato... |

1 | Formal verification of parallel programs - Hoare - 1985 |