## A Fully Abstract Presheaf Semantics of SCCS with Finite Delay (1999)

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Venue: | Department of Computer Science, University of Aarhus |

Citations: | 19 - 3 self |

### BibTeX

@INPROCEEDINGS{Hildebrandt99afully,

author = {Thomas Troels Hildebrandt},

title = {A Fully Abstract Presheaf Semantics of SCCS with Finite Delay},

booktitle = {Department of Computer Science, University of Aarhus},

year = {1999},

pages = {2--9},

publisher = {Elsevier}

}

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### Abstract

We present a presheaf model for the observation of infinite as well as finite computations. We apply it to give a denotational semantics of SCCS with finite delay, in which the meanings of recursion are given by final coalgebras and meanings of finite delay by initial algebras of the process equations for delay. This can be viewed as a first step in representing fairness in presheaf semantics. We give a concrete representation of the presheaf model as a category of generalised synchronisation trees and show that it is coreflective in a category of generalised transition systems, which are a special case of the general transition systems of Hennessy and Stirling. The open map bisimulation is shown to coincide with the extended bisimulation of Hennessy and Stirling. Finally we formulate Milners operational semantics of SCCS with finite delay in terms of generalised transition systems and prove that the presheaf semantics is fully abstract with respect to extended bisimulation

### Citations

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A Calculus of Communicating Systems
- Milner
- 1980
(Show Context)
Citation Context ... agents and so when two agents behave equally, i.e. they denote the same process. The approach used for CCS and SCCS, taking two agents to be equivalent if their derivation trees are strong bisimilar =-=[1-=-6], will identify agents that only differ on whether some infinite computations are admissible or not, in particularst is identified with t for any term t. Moreover, (by definition) both t and t shoul... |

924 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...nstruct final co-algebras for all relevant endofunctors as limits of ! op -chains. The definition is then extended to processes with more than one variable in the usual way as a limit with parameters =-=[13]-=-. From the explicit definitions given in Eq. (9)-(13) we can show that all basic operators preserve ! op -limits. From the general fact that limits commute with limits [13] we get that recursion prese... |

395 |
Category theory for computing science
- Barr, Wells
- 1990
(Show Context)
Citation Context ... ! Y is surjective open in b P then F (m) : F (X) ! F (Y ) is surjective open in b Q. 1.2 Initial Algebras and Final Coalgebras Below we recall the categorical analogues of pre- and post-fixed points =-=[2]-=-. Definition 1.5 Let F : P ! P be an endofunctor on a category P. A coalgebrasfor F is a pair (p; m) of an object and a morphism of P such that m : p ! F (p). Dually, an algebra for F is a pair (p; m)... |

313 |
Calculi for synchrony and asynchrony
- Milner
- 1983
(Show Context)
Citation Context ...he denotational semantics directly to the operational semantics given in Sec. 5 within the category GTS. First of all we will restrict attention to terms with only guarded recursion. Recall from e.g. =-=[18]-=- that a recursion rec x:t is guarded, if all free occurences of x in t is guarded, that is, within a subterm a : t 0 of t for some action a 2 Act. Let T g refer to the set of all closed, possibly anno... |

189 |
Sheaves in geometry and logic: A first introduction to topos theory
- Lane, Moerdijk
- 1992
(Show Context)
Citation Context ...l infinite sequences of actions to the category Fin. With the help of a simple Grothendieck topology we get indeed a suitable model for infinite computations from the category of seperated presheaves =-=[14] over Inf.-=- A careful generalisation of the models of synchronisation trees and transition systems lifts the relationship between the "standard" finitary models to the infinitary models and gives a con... |

116 | Bisimulation from open maps - Joyal, Nielsen, et al. - 1996 |

72 | Elementary transition systems - Nielsen, Rozenberg, et al. - 1992 |

45 | Presheaf models for concurrency
- Cattani, Winskel
- 1996
(Show Context)
Citation Context ...k on presenting models for concurrency categorically as initiated by Winskel and Nielsen [23] and developed further in the work on bisimulation from open maps [12] and presheaf models for concurrency =-=[3, 6, 9, 22]-=-. Our goal is twofold: We want to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to models for infinite computations and we want to g... |

27 | A relational model of non-deterministic dataflow
- Hildebrandt, Panangaden, et al.
- 2004
(Show Context)
Citation Context ...the more traditional domain theoretical approach to fairness and countable nondeterminism as in e.g. [20]. Finally, we hope to be able to extend the presheaf 27 model for (finitary) dataflow given in =-=[9]-=- to infinite computations along the lines of the present paper, giving a model of dataflow in which fairness, maybe even fair merge [19], can be expressed. Acknowledgements: Thanks to Glynn Winskel, M... |

20 |
A finite delay operator in Synchronous CCS
- MILNER
- 1992
(Show Context)
Citation Context ...ive an operational semantics of SCCS with finite delay in the generalised transition systems capturing exactly the definition of inadmissible computations given in terms of waiting subcomputations in =-=[17]-=-. We then give a denotational semantics in the presheaf model which we prove to be equationally fully abstract with respect to extended bisimulation. In all of the steps above we greatly benefit from ... |

19 | A powerdomain for countable non-determinism (extended abstract
- Plotkin
(Show Context)
Citation Context .... It would be interesting to explore if there is any relationship between the present approach and the more traditional domain theoretical approach to fairness and countable nondeterminism as in e.g. =-=[20]-=-. Finally, we hope to be able to extend the presheaf 27 model for (finitary) dataflow given in [9] to infinite computations along the lines of the present paper, giving a model of dataflow in which fa... |

18 | A categorical axiomatics for bisimulation - Cattani, Power, et al. - 1998 |

18 | The power of the future perfect in program logics
- Hennessy, Stirling
- 1985
(Show Context)
Citation Context ... in which it cannot perform an a-action. Aczel [1] proposes a final-coalgebra semantics, which gives a bisimulation closely related to the extended bisimulation introduced by Hennessy and Stirling in =-=[8]-=- for general transition systems. This bisimulation indeed distinguishes the two agents given above. The background of the present paper is the work on presenting models for concurrency categorically a... |

16 |
Axiomatising finite delay operators
- Hennessy
- 1984
(Show Context)
Citation Context ...t paper and in [1], more intuitive, equivalence than the one obtained from the fortification preorder in [17], which except for [1] has been the basis for previous semantics of SCCS with finite delay =-=[7, 11, 10]-=-. Benefitting from the categorical presentation, our semantics appears to give a conceptually simpler treatment of infinite computations than the one in [1]. A number of questions remains to be explor... |

12 | Presheaf models for the pi-calculus
- Cattani, Stark, et al.
- 1997
(Show Context)
Citation Context ...equivalence by choosing a path category, 3 i.e. a subcategory of the model at issue identifying the observable computations. The open maps approach increased its worth through the further development =-=[3, 6, 5, 22]-=- of the presheaf models for concurrency proposed in [12]. Here one starts with a path category P and then takes the category b P of presheaves over P as model, justified categorically by being the fre... |

11 | A linear metalanguage for concurrency
- Winskel
- 1998
(Show Context)
Citation Context ...k on presenting models for concurrency categorically as initiated by Winskel and Nielsen [23] and developed further in the work on bisimulation from open maps [12] and presheaf models for concurrency =-=[3, 6, 9, 22]-=-. Our goal is twofold: We want to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to models for infinite computations and we want to g... |

7 | Finite but unbounded delay in synchronous CCS
- Huth, Kwiatkowska
- 1996
(Show Context)
Citation Context ...and the general theory of bisimulation from open maps reduce the problem of relating the two semantics to finding an open map within the category of generalised transition systems. A number of papers =-=[1, 11, 7, 8, 21]-=- have already proposed denotational semantics for SCCS with finite delay and models for non-deterministic processes with infinite computations. As mentioned above, the approach we take is closely rela... |

6 | The expressive power of indeterminate primitives in asynchronous computations - Panangaden - 1995 |

5 |
Handbook of Logic in Computer Science, volume 4: Semantic Modelling, chapter 1: Models for concurrency
- Winskel
- 1995
(Show Context)
Citation Context ...is bisimulation indeed distinguishes the two agents given above. The background of the present paper is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen =-=[23]-=- and developed further in the work on bisimulation from open maps [12] and presheaf models for concurrency [3, 6, 9, 22]. Our goal is twofold: We want to extend the categorical approach (in which the ... |

4 |
A semantic universe for fairness
- Aczel
- 1996
(Show Context)
Citation Context ...congruence with respect to all the operators of the language (under an assumption of guarded recursion). This approach works reasonably, but is not completely satisfactory. As pointed out by Aczel in =-=[1]-=-, the fortification equivalence makes some non desirable identifications of agents due to the fact that infinite computations are treated totally separately from finite computations. For example, the ... |

3 |
The semantics of fair recursion with divergence. Submitted
- Huth, Kwiatkowska
(Show Context)
Citation Context ...t paper and in [1], more intuitive, equivalence than the one obtained from the fortification preorder in [17], which except for [1] has been the basis for previous semantics of SCCS with finite delay =-=[7, 11, 10]-=-. Benefitting from the categorical presentation, our semantics appears to give a conceptually simpler treatment of infinite computations than the one in [1]. A number of questions remains to be explor... |

3 | Introduction to Mathematical Logic. Undergraduate Texts in Mathematics - Malitz - 1979 |

3 |
Generalised synchronisation trees. Handwritten notes
- Winskel
- 1983
(Show Context)
Citation Context ...and the general theory of bisimulation from open maps reduce the problem of relating the two semantics to finding an open map within the category of generalised transition systems. A number of papers =-=[1, 11, 7, 8, 21]-=- have already proposed denotational semantics for SCCS with finite delay and models for non-deterministic processes with infinite computations. As mentioned above, the approach we take is closely rela... |

2 |
A Grothendieck topology for a partial order Here we give the definitions from [14] of a Grothendieck topology for a category P and the sup topology, specialised to the case where P is a partial order. Let P be a partial order and p ∈ P .Definep↓={p ′ ∈P|p
- Wyler
- 1991
(Show Context)
Citation Context ... Fin is equivalent to the category Sh( c Inf), of sheaves over Inf with respect to the sup topology. Sheaves, separated presheaves and presheaves are known to be closely related and rich in structure =-=[14, 25]-=-. We will especially make use of the fact, that they are related by a sequence of reflections, i.e. the inclusions Sh( c Inf) ,! Sp( c Inf) and Sp( c Inf) ,! c Inf both have left adjoints (reflectors)... |

1 |
Bisimulation from open maps. LICS '93
- Joyal, Nielsen, et al.
- 1996
(Show Context)
Citation Context ...ackground of the present paper is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen [23] and developed further in the work on bisimulation from open maps =-=[12]-=- and presheaf models for concurrency [3, 6, 9, 22]. Our goal is twofold: We want to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to... |