## Topological Models for Higher Order Control Flow (1993)

Venue: | PROCEEDINGS OF THE 9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, VOLUME 802 OF LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 7 - 3 self |

### BibTeX

@INPROCEEDINGS{Bakker93topologicalmodels,

author = {J. W. de Bakker and F. van Breugel},

title = {Topological Models for Higher Order Control Flow},

booktitle = {PROCEEDINGS OF THE 9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, VOLUME 802 OF LECTURE NOTES IN COMPUTER SCIENCE},

year = {1993},

pages = {122--142},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Semantic models are presented for two simple imperative languages with higher order constructs. In the first language the interesting notion is that of second order assignment x := s, for x a procedure variable and s a statement. The second language extends this idea by a form of higher order communication, with statements c ! s and c ? x, for c a channel. We develop operational and denotational models for both languages, and study their relationships. Both in the definitions and the comparisons of the semantic models, convenient use is made of some tools from (metric) topology. The operational models are based on (SOS-style) transition systems; the denotational definitions use domains specified as solutions of domain equations in a category of 1-bounded complete ultrametric spaces. In establishing the connection between the two kinds of models, fruitful use is made of Rutten's processes as terms technique. Another new tool consists in the use of metric transition systems, with a metric defined on the configurations of the system. In addition to higher order programming notions, we use higher order definitional techniques, e.g., in defining the semantic mappings as fixed points of (contractive) higher order operators. By Banach's theorem, such fixed points are unique, yielding another important proof principle for our paper.

### Citations

1286 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...rder to establish O = D, it suffices to show, e.g., that \Phi O (D) = D. The definition of each of the O's follows the customary pattern in that it is derived from some (SOS-style) transition system (=-=[Plo81]-=-). Mostly, these systems are finitely branching, a property on which the compactness of the resulting sets of meanings is based. However, in the comparative study of L co2 we need a generalisation to ... |

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...s has become a central topic in the field of semantics. Seminal in this development have been two schools of research, viz. that of (typed) -calculus in the area of functional programming (see, e.g., =-=[Bar92]-=- for a survey of the current situation) , and that of higher order processes in the theory of concurrency (see, e.g., [AR87, Tho90, MPW92]). ([LTLG92] can be used for a quick overview of much of the r... |

452 | The Chemical Abstract Machine - Berry, Boudol - 1990 |

313 | Functions as processes, in - Milner - 1990 |

258 | Expressing Mobility in Process Algebras: First-Order and Higher-Order Paradigms - Sangiorgi - 1992 |

219 | Barbed bisimulation, in - Milner, Sangiorgi - 1992 |

72 | The Pi Calculus - Milner - 1999 |

62 | An abstract framework for environment machines - Curien - 1991 |

60 | Towards a lambda-calculus for concurrent and communicating systems - Boudol - 1989 |

47 | On the Foundations of Final Semantics: Non-Standard Sets - Rutten, Turi - 1992 |

46 |
Topologies on spaces of subsets
- Michael
- 1951
(Show Context)
Citation Context ...d on channel c should be assigned to procedure variable x - denoted by (c; x). The transition system T 4 is presented in 6 Let (X; dX ) be a metric space. If X 2 Pco (Pco (X)) then S X 2 Pco (X) (cf. =-=[Mic51]-=-). 4. A parallel language with second order communication 13 Definition 4.3 The transition relation ! of T 4 is the smallest subset of (Res 4 \Theta SynState ext 4 ) \Theta (Res 4 \Theta SynState ext ... |

20 |
Processes as terms: Non-well-founded models for bisimulation
- Rutten
- 1992
(Show Context)
Citation Context ...delivering results in the denotational domain, is applied again. However, for L co 2 1. A sequential language with assignment 3 a complication arises, inducing the appearance of `processes as terms' (=-=[Rut92]-=-). Also, this is the point where, as signalled earlier, a compactly branching transition system appears, a notion which presupposes a metric on the configurations ([Bre94]). In the final stage of the ... |

17 |
Metric Semantics for Concurrency
- Meyer
- 1988
(Show Context)
Citation Context ...(oefff=vg))) = 1 2 \Delta d (O (r); D (r)) 1 2 \Delta sup f d (O (r 0 ); D (r 0 )) j r 0 2 Res 1 g: Consequently, for all r, d (O (r); D (r)) = 0. Hence O = D . ut The first proof follows [KR90] (cf. =-=[BM88]), bu-=-t with a substantial simplification thanks to our avoiding procedure environments. corollary 1.10 For all s, O (s) = D (s)(oe : f"g). 2. A sequential language with second order assignment The cen... |

12 |
Sur une methode de metrisation complete des certains espaces d'ensembles compacts, Fundamentae Mathematicae 43
- Kuratowski
- 1956
(Show Context)
Citation Context ...dX ) is a 1-bounded complete ultrametric space then the set of nonempty and compact subsets of X, Pnc (X), endowed with the Hausdorff metric based on dX is a 1-bounded complete ultrametric space (cf. =-=[Kur56]). 4 -=-Let (X; dX ) and (X 0 ; d X 0 ) be metric spaces. A function f : X ! X 0 is called nonexpansive if, for all x and x 0 , d X 0 (f (x); f (x 0 ))sdX (x; x 0 ): 6 Introduction D (e) = oe : f"g D (s ... |

10 | A Domain-theoretic Model for a Higher-order Process Calculus", LNCS
- Jagadeesan, Panangaden
- 1990
(Show Context)
Citation Context ... of further work. Other connections to explore include the relationships with the -calculus ([MPW92, Mil92]), the higher order -calculus ([San92, San93]), and the fl-calculus ([Bou89, BB92], cf. also =-=[JP90]-=-). In the -calculus, channel names are transmitted rather than processes, so an immediate correspondence is not to be expected. For another reason, the same holds for the fl-calculus: the notion of se... |

9 |
Topological Models in Comparative Semantics
- Breugel
- 1994
(Show Context)
Citation Context ...nce of `processes as terms' ([Rut92]). Also, this is the point where, as signalled earlier, a compactly branching transition system appears, a notion which presupposes a metric on the configurations (=-=[Bre94]-=-). In the final stage of the proof relating O and D, a lemma relating the transitions of both the original system (on which O for L co2 is based) and of the extended system (in which the configuration... |

4 | Three Metric Domains of Processes for Bisimulation - Breugel - 1993 |

3 |
CONcurrency and Functions: Evaluation and Reduction
- L'evy, Thomsen, et al.
- 1992
(Show Context)
Citation Context ... in the area of functional programming (see, e.g., [Bar92] for a survey of the current situation) , and that of higher order processes in the theory of concurrency (see, e.g., [AR87, Tho90, MPW92]). (=-=[LTLG92]-=- can be used for a quick overview of much of the relevant literature.) The aim of the present paper is to provide another perspective on this problem area by studying higher order notions embedded in ... |

1 | An Investigation into Functions as Processes. To appear - Sangiorgi - 1993 |