## A Theory of Metric Labelled Transition Systems (1995)

Venue: | Papers on General Topology and Applications: 11th Summer Conference at the University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Breugel95atheory,

author = {Franck Van Breugel},

title = {A Theory of Metric Labelled Transition Systems},

booktitle = {Papers on General Topology and Applications: 11th Summer Conference at the University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences},

year = {1995},

pages = {67--87}

}

### OpenURL

### Abstract

Labelled transition systems are useful for giving semantics to programming languages. Kok and Rutten have developed some theory to prove semantic models defined by means of labelled transition systems to be equal to other semantic models. Metric labelled transition systems are labelled transition systems with the configurations and actions endowed with metrics. The additional metric structure allows us to generalize the theory developed by Kok and Rutten. Introduction The classical result due to Banach [Ban22] that a contractive function from a nonempty complete metric space to itself has a unique fixed point plays an important role in the theory of metric semantics for programming languages. Metric spaces and Banach's theorem were first employed by Nivat [Niv79] to give semantics to recursive program schemes. Inspired by the work of Nivat, De Bakker and Zucker [BZ82] gave semantics to concurrent languages by means of metric spaces. The metric spaces they used were defined as solutio...

### Citations

1286 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...ace to itself with the semantic models as fixed point is needed. Kok and Rutten developed some theory to prove operational semantic models defined by means of labelled transition systems a la Plotkin =-=[Plo81]-=- equal to other semantic models---in particular denotational semantic models---by uniqueness of fixed point. For numerous applications of their theory we refer the reader to De Bakker and De Vink's te... |

75 |
les opérations dans les ensembles abstraits et leur application aux équations intégrales
- Banach, “Sur
- 1922
(Show Context)
Citation Context ...th the configurations and actions endowed with metrics. The additional metric structure allows us to generalize the theory developed by Kok and Rutten. Introduction The classical result due to Banach =-=[Ban22]-=- that a contractive function from a nonempty complete metric space to itself has a unique fixed point plays an important role in the theory of metric semantics for programming languages. Metric spaces... |

68 |
Solving reflexive domain equations in a category of complete metric spaces
- America, Rutten
- 1989
(Show Context)
Citation Context ... gave semantics to concurrent languages by means of metric spaces. The metric spaces they used were defined as solutions of recursive domain equations. By means of Banach's theorem America and Rutten =-=[AR89]-=- proved that a particular class of domain equations has unique solutions. Banach's theorem has also been used to prove semantic models to be equal. Kok and Rutten [KR90] applied a proof principle whic... |

52 |
Grundzüge der Mengenlehre
- Hausdorff
- 1914
(Show Context)
Citation Context ...d by A \Theta 1 2 \Delta C. As we will see below the introduction of the 1 2 \Delta gives rise to a less restrictive condition. The (compact) sets of these pairs are endowed with the Hausdorff metric =-=[Hau14]-=-. Definition 1.6 A compactly branching metric labelled transition system (C; A; !) is called nonexpansive if the function CB : C ! P c (A \Theta 1 2 \Delta C) defined by CB (c) = f (a; c 0 ) j c a \Ga... |

46 |
Topologies on spaces of subsets
- Michael
- 1951
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Citation Context ... : : : ; an ; c n+1 ) j c n an \Gamma\Gamma! c n+1 g fi fi fi (a 1 ; c 2 ; a 2 ; : : : ; an\Gamma1 ; c n ) 2 CB n\Gamma1 (c) o is a compact set of compact sets. From Michael's theorem (Theorem 2.5 of =-=[Mic51]-=-), the compactness of the set [ n f (a 1 ; c 2 ; a 2 ; : : : ; an ; c n+1 ) j c n an \Gamma\Gamma! c n+1 g fi fi fi (a 1 ; c 2 ; a 2 ; : : : ; an\Gamma1 ; c n ) 2 CB n\Gamma1 (c) o i.e. CB n (c), can ... |

37 |
Elements of generalized ultrametric domain theory
- Rutten
- 1995
(Show Context)
Citation Context ...abelled transition systems we have to restrict ourselves to nonexpansive and continuous semantics (for generalized metric spaces nonexpansiveness does not imply continuity as has been shown in, e.g., =-=[Rut95]-=-) rather than nonexpansive semantics, and instead of using the hyperspace of nonempty and compact sets endowed with the Hausdorff metric we have to employ the generalized convex powerdomain as defined... |

27 |
Control Flow Semantics. Foundations of Computing Series
- Bakker, Vink
- 1996
(Show Context)
Citation Context ...to other semantic models---in particular denotational semantic models---by uniqueness of fixed point. For numerous applications of their theory we refer the reader to De Bakker and De Vink's textbook =-=[BV95]-=-. Their results are only applicable to operational semantic models induced by finitely branching labelled transition systems. Although most programming languages can be modelled operationally by means... |

16 |
Infinite words, infinite trees, infinite computations
- Nivat
- 1979
(Show Context)
Citation Context ...te metric space to itself has a unique fixed point plays an important role in the theory of metric semantics for programming languages. Metric spaces and Banach's theorem were first employed by Nivat =-=[Niv79]-=- to give semantics to recursive program schemes. Inspired by the work of Nivat, De Bakker and Zucker [BZ82] gave semantics to concurrent languages by means of metric spaces. The metric spaces they use... |

15 |
The metric closure powerspace construction
- Kent
- 1987
(Show Context)
Citation Context ...opological machines was developed by Brauer [Bra70]. Our metric labelled transition systems are a special instance of his topological machines. Another instance are Kent's metrical transition systems =-=[Ken87]-=-. A metrical transition system is a labelled transition system with the configurations endowed with a (generalized) ultrametric (the labels are not provided with any additional structure). Neither Bra... |

12 |
Sur une methode de metrisation complete des certains espaces d'ensembles compacts, Fundamentae Mathematicae 43
- Kuratowski
- 1956
(Show Context)
Citation Context ... ) n of nonexpansive functions converging to O is introduced. Because the set of nonexpansive functions C ! 1 P nc (A 1 ) is closed (a consequence of the completeness of A and Lemma 3 of Kuratowski's =-=[Kur56]-=-), we can conclude that O is nonexpansive. The function On : C ! P n (A 1 ) is defined by On (c) = f a 1 a 2 \Delta \Delta \Delta a k\Gamma1 j c = c 1 a1 \Gamma\Gamma! c 2 a2 \Gamma\Gamma! \Delta \Del... |

11 |
An introduction to mathematical machine theory
- Ginsburg
- 1962
(Show Context)
Citation Context ...pact. Conclusion Already in the early sixties the problem what structure to add to an abstract machine---like a labelled transition system---to obtain a topological machine was formulated by Ginsburg =-=[Gin62]-=-. Shreider [Shr64] introduced a particular topological machine---a compact automaton---to study dynamic programming. A general and detailed study of topological machines was developed by Brauer [Bra70... |

9 |
Topological Models in Comparative Semantics
- Breugel
- 1994
(Show Context)
Citation Context ...e topological machines to give (operational) semantics as we have done in this paper. This paper only describes part of a theory of metric labelled transition systems developed in the author's thesis =-=[Bre94b]-=-. We have shown how to generalize finitely branching to compactly branching and nonexpansive. Similarly image finite can be generalized to image compact and binonexpansive. The semantic models we have... |

7 | Topological Models for Higher Order Control Flow
- Bakker, Breugel
- 1993
(Show Context)
Citation Context ...proved to be equal by uniqueness of fixed point exploiting the theory in [Bre91]. 2. The theory has also been used to relate an operational and a denotational semantics for a higher-order language in =-=[BB93]-=-. 3. Metric labelled transition systems have turned out to be very convenient to define abstraction operators as has been shown in [Bre95]. We are interested to see whether a similar theory can be dev... |

6 |
Sur la représentation des fonctions discontinues (deuxiéme partie
- Baire
- 1909
(Show Context)
Citation Context ... where joe i j denotes the length of the sequence oe i and oe i (n) denotes the n-th element of oe i . In case we endow the action set A with the discrete metric we obtain the usual Baire-like metric =-=[Bai09]-=-: dA 1 (oe 1 ; oe 2 ) = ae 0 if oe 1 = oe 2 2 \Gamman otherwise where n is the length of the longest common prefix of oe 1 and oe 2 . Second, we endow the (nonempty) sets of action sequences with the ... |

3 | From Branching to Linear Metric Domains (and back
- Breugel
- 1995
(Show Context)
Citation Context ...al and a denotational semantics for a higher-order language in [BB93]. 3. Metric labelled transition systems have turned out to be very convenient to define abstraction operators as has been shown in =-=[Bre95]-=-. We are interested to see whether a similar theory can be developed if we replace the metric spaces of a metric labelled transition system by algebraic complete generalized metric spaces. To develop ... |

3 |
Automata and the Problem of Dynamic Programming
- Shreider
- 1961
(Show Context)
Citation Context ...lready in the early sixties the problem what structure to add to an abstract machine---like a labelled transition system---to obtain a topological machine was formulated by Ginsburg [Gin62]. Shreider =-=[Shr64]-=- introduced a particular topological machine---a compact automaton---to study dynamic programming. A general and detailed study of topological machines was developed by Brauer [Bra70]. Our metric labe... |

2 |
Zu den Grundlagen einer Theorie Topologischer Sequentieller Systeme und Automaten, volume 31 of Berichte der Gesellschaft fur Mathematik und Datenverarbeitung. Gesellschaft fur Mathematik und Datenverarbeitung
- Brauer
- 1970
(Show Context)
Citation Context ...Gin62]. Shreider [Shr64] introduced a particular topological machine---a compact automaton---to study dynamic programming. A general and detailed study of topological machines was developed by Brauer =-=[Bra70]-=-. Our metric labelled transition systems are a special instance of his topological machines. Another instance are Kent's metrical transition systems [Ken87]. A metrical transition system is a labelled... |

2 | Comparative Semantics for a Real-Time Programming Language with Integration
- Breugel
- 1991
(Show Context)
Citation Context ...n three applications. 1. An operational and a denotational semantics for a fragment of the real-time language ACPrae have been proved to be equal by uniqueness of fixed point exploiting the theory in =-=[Bre91]-=-. 2. The theory has also been used to relate an operational and a denotational semantics for a higher-order language in [BB93]. 3. Metric labelled transition systems have turned out to be very conveni... |

2 | Generalizing Finiteness Conditions of Labelled Transition Systems
- Breugel
- 1994
(Show Context)
Citation Context ...so define branching operational semantic models. These 12 semantic models assign to each configuration a tree-like object. The details are provided in [Bre94b]. For an overview we refer the reader to =-=[Bre94a]-=-. If we replace nonexpansiveness by continuity (in Definition 1.6) all results still hold. In that case the induced operational semantic models are continuous (and not nonexpansive in general). In sev... |