G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 5875, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....components. This opens the way towards a possibility of verifying certain properties of complex systems in a modular way, thus increasing eciency. Among existing approaches to concurrency based on berings, sheaf and presheaf theory we mention [MP86] Pfa91] Gog92] Lil93] Mal94] Win96] [CW96]. The starting point of our research is the work of Pfalzgraf [Pfa91] and the ideas of Goguen [Gog92] and Monteiro and Pereira [MP86] The idea of modeling states, actions and transitions by sheaves with respect to a topological space, and of using geometric logic for studying the link between ....

G.L. Cattani and G. Winskel. Presheaf models for concurrency. In D. van Dalen and M. Bezem, editors, Proceedings of Computer Science Logic '96, LNCS 1258, pages 58-75. Springer Verlag, Berlin, 1996.

....leads to familiar notions of bisimulation. One may then ask how one identifies a suitable category of paths in a model for concurrency; or even, if it is possible to construct a model for concurrency, given a category of computation paths. An answer is given by presheaf models for concurrency [CW96]. A presheaf over a category P is a contravariant functor into the category of sets, P Set. The category of presheaves, P, has presheaves as objects, and natural transformations as morphisms. A category P is a subcategory of the presheaf category P via the Yoneda embedding. In fact, P ....

....relabelling, and fibrations model inverse relabelling and restriction. Furthermore, because a category P is embedded in the presheaf category P, the presheaf category comes with a notion of bisimulation [JNW96] and weak bisimulation [FCW99] Presheaf models have been given to CCS like languages [CW96], a value passing version of CCS [Win96] and the p calculus [CSW97] An approach towards a theory of domains for concurrency based on presheaf models is presented in [CFW98] Presheaf models differ from the models presented before in that they do not present processes as morphisms, and types as ....

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Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Proc. CSL, 1996.

....have more structure than traditional traces, e.g. allowing path semantics to take nondeterministic branching into account in a limited way. For example, path equivalence is related to simulation equivalence in Sect. 3.5 below. The other path based approach is that of categories of presheaf models [14] in which processes denote mappings from computation paths to sets of realisers saying how each computation path may be realised. This extra structure allows the incorporation of complete branching information, and the corresponding notion of process equivalence is a form of bisimulation [26] ....

G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

....es for a typical path p the set X(p) of computation paths of shape p, and acts on e : p q to give a function X(e) saying how paths of shape q restrict to paths of shape p. In this way a presheaf can model the nondeterministic branching of a process. For more information of presheaf models, see [14, 4]. A presheaf category has all colimits and so in particular all sums (coproducts) for any set I , the sum i2I X i of presheaves X i over P has a contribution i2I X i (p) the disjoint union of sets, at p 2 P. The empty sum is the presheaf with empty contribution at each p. In process ....

....maps [13, 14] Open maps are a generalization of functional bisimulations, or zig zag morphisms, known from transition systems. Presheaves over P are openmap bisimilar i there is a span of surjective open maps between them. The maps of Lin and A preserve open maps and so open map bisimulation [7, 4], giving congruence results for free when a process language is interpreted in these categories. Interestingly, although the operational bisimulation of Park Milner is a congruence for the higher order language, maps of Cts need not preserve open maps. This suggests that one should look at other ....

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

....to unify models of concurrency and their notions of bisimulation into one framework, a domain theory based on presheaf models and open map bisimulation. The endeavor was initiated by Joyal, Nielsen and Winskel in [15] and pursued further in two PhD theses [3, 11] and a number of papers including [6, 20, 5, 4, 13, 21, 9, 12, 7]. Presheaf models are instantiations of the following situation: Let P be a small category in which objects are viewed as computation path shapes with morphisms saying how paths can be extended. The path category P embeds via Yoneda in the category = P , Set] of presheaves over P in which ....

....with arrows colimit preserving functors between the associated presheaf categories, and with natural transformations as 2 cells. The 2 category is equivalent to a bicategory Prof with the same objects, but with profunctors (which are often easier to work with) as arrows [7, 1, 2] It was shown in [6] that open map bisimulation is preserved along the arrows of Cocont, giving rise to congruence results for free when one interprets terms of a process language as such arrows. In fact, the arrows need only preserve connected colimits to preserve open map bisimulation, and this is exploited in ....

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, number 1258 in Lecture Notes in Computer Science, pages 58--75. Springer-Verlag, 1997.

....context of this work is an ongoing effort to unify models of concurrency and their notions of bisimulation into one framework, a domain theory based on presheaf models and open map bisimulation. The endeavor was initiated by Joyal, Nielsen and Winskel in [15] and pursued further in two PhD theses [3, 11] and a number of papers including [6, 20, 5, 4, 13, 21, 9, 12, 7] Presheaf models are instantiations of the following situation: Let P be a small category in which objects are viewed as computation path shapes with morphisms saying how paths can be extended. The path category P embeds via Yoneda ....

Gian Luca Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

....(co)monads on a biequivalent 2 category. At present this theory, even the definitions, are not sufficiently developed (though remedial work has started with Martin Hyland and John Power) Acknowledgements A good deal of the background for this work was developed with Gian Luca Cattani for his PhD [6]. Discussions with Martin Hyland and John Power have played a crucial role in the work on nonlinearity. ....

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

....allowing process passing, though there remains the problem of relating the notion of bisimulation obtained from open maps to a more traditional definition from the operational semantics. A tentative beginning is made of a domain theory supporting presheaf models. Introduction The papers [12, 4] explore presheaf models for concurrency. Here begins an investigation of the use of presheaves to model higher order features, most dramatic in the situation of process calculi where processes can be communicated as values. Something of higher order appears even in value passing process calculi ....

....path lifting property of h see [12] Say presheaves X;Y are bisimilar iff there is a span of surjective open maps between them, equivalently, iff there is R , X Theta Y such that the compositions with the projections R , X Theta Y X and R , X Theta Y are surjective open. In [12, 4] we defined bisimulation between rooted presheaves, presheaves X, over a category assumed to have an initial object I, for which X(I) is a 4 singleton. For rooted presheaves bisimulation is defined merely through the presence of a open maps (not requiring surjectivity) This is because open maps ....

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Cattani, G.L., and Winskel, G., Presheaf models for concurrency. Manuscript, 1996.

....This yields a form of domain theory but boosted to the level of using categories rather than partial orders as the appropriate domains. One argument for the definition of bisimulation based on open maps is the powerful preservation properties associated with it. Notable is the result of [8] that any colimit preserving functor between presheaf categories preserves bisimulation, Basic Research in Computer Science, Centre of the Danish National Research Foundation. which besides obvious uses in relating semantics in different models with different notions of bisimulation is, along ....

....of bisimulation is all well and good but it needs to be tested and its consequences understood for a range of process languages. Another argument in favour of the presheaf approach to bisimulation is that when tried against traditional process languages it yields persuasive results, as in [8, 40, 7]. But still these are just examples and it is hoped that a more satisfying and conclusive argument will come from an endeavour to ascertain the operational content of presheaf models more generally. One difficulty has been in understanding the operational significance of the bisimulation which ....

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CATTANI, G. L., AND WINSKEL, G. Presheaf models for concurrency. In CSL'96, vol. 1258 of LNCS, Springer, pp. 58--75. 17

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G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

....These categories are very rich in structure. They point towards more expressive languages than HOPLA and AL. In particular, the a#ne category accommodates the independence model of event structures to the extent of supporting the standard event structure semantics of CCS and related languages [12], as well as the trace of nondeterministic dataflow [23] In fact, AL can be given an event structure semantics which at first order provides a representation of the presheaf denotations of AL. Nevertheless, it is here we meet the limitations of AL, and HOPLA. They can be shown not to support ....

....of Lin. 28 There is a category analogous to A# based on presheaves rather than path sets; its maps preserve connected colimits in presheaf categories [39, 15] This a#ne category is host to the semantics of nondeterministic dataflow [23] event structure semantics of CCS and related languages [12] as well as a semantics for AL. It came as a recent surprise [38] that the presheaf denotations of first order processes in AL can be represented by event structures; the elements of definable presheaves can be understood as finite configurations of an event structure. In more detail, maps ....

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G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

....constructions arise in manipulating that structure. For concreteness, we restrict our attention for the bulk of the paper to the operations needed in modelling CCS processes by synchronisation trees. We occasionally refer to more involved examples that were treated using presheaf categories in [22, 6, 5]. We rst observe that, given a category of observations, P, the basic operations of CCS lead us to consider the free completion of the category Punder countable colimits: for our choice of P, that is equivalent to the category of countably branching trees, as we shall see in Section 2. The ....

....It also gives us a candidate for higher order structure (though outside Kl(T ) allowing a possible way to model a process passing extension of CCS. This analysis forms Sections 4 and 5. Finally, in Section 6, we compare, especially as far as higher order is concerned, this work with that of [22, 6, 5, 7] using presheaves and profunctors to model process calculi, and we suggest directions for future research. We do not address weak bisimulation at all here. In no way do we suggest that it is unimportant. But it is such a large issue that it requires a full paper devoted to it. There are delicate ....

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G. L. Cattani and G. Winskel. Presheaf Models for Concurrency. In Proceedings of CSL' 96, LNCS 1258, pages 58-75, 1997.

....observational congruence on presheaf models. Presheaf models have been shown to include traditional models like synchronisation trees and event structures [13] along with their notion of bisim ulation, to be related by powerful preservation properties associated with colimit preserving functors [9], and to form a domain theory for bisimulation [23, 7, 6] in which a wide range of, possibly higher order, process languages can receive a denotational semantics. In [17] an approach to an open map account of weak bisimulation on transition systems is based on weak morphisms , a reconstruction ....

....canonical choice for open map bisimulation in a presheaf category [13] Presheaf categories subsume more traditional categories of models. It was suggested in [13] that they deserve to be studied as categories of models for concurrency in their own right. Subsequent work on this line appeared in [9, 23, 26, 8, 7, 11, 6]. 5 Synchronisation trees. A simple example of a presheaf model is given by synchronisation trees. A synchronisation tree is a transition system whose transition graph is a tree with root the initial state. Definition 2.1 For a set of actions A, let STA be the full subcategory of TSA of ....

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G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 58--75. Springer-Verlag, 1997.

.... models have been shown to include traditional models like synchronisation trees and event structures [13] along with their notion of bisim ulation, to be related by powerful preservation properties associated with colimit preserving functors [9] and to form a domain theory for bisimulation [23, 7, 6] in which a wide range of, possibly higher order, process languages can receive a denotational semantics. In [17] an approach to an open map account of weak bisimulation on transition systems is based on weak morphisms , a reconstruction of the maps between transition systems to account for the ....

....s 0 t and f(t) t # . In other words, f not only preserves but also reflects reachable transitions. If f is a zig zag morphism, then its graph is a bisimulation between the two transition systems. In [13] the above situation was axiomatised in terms of the following path lifting property (see [13, 6] for more detailed explanations) Suppose that within a category of models, one can distinguish a subcategory of computation paths (or shapes) P. The objects of P account for runs (or history) of a process and their morphisms tell us how one path can be extended by another one. Given the ....

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G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999. Forthcoming.

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G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 5875, 1997.

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G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

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Gian Luca Cattani. Presheaf Models for Concurrency. PhD thesis, BRICS - University of Aarhus, 1999.

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Gian Luca Cattani. Presheaf Models for Concurrency. PhD thesis, BRICS - University of Aarhus, 1999.

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G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

No context found.

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

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G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

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G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

No context found.

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

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