S. Abramsky (1996), Retracing Some Paths in Process Algebra, in Proceedings of CONCUR 96, U. Montanari and V. Sassone (Eds.), Lecture Notes in Computer Science 1119, 1--17, Springer-Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the GoI construction. Furthermore, we show that the class of untyped # theories induced by wave style GoI models is richer than that induced by game models. Keywords: linear) graph model, traced monoidal category, weak linear category, categorical geometry of interaction. Introduction In [Abr96], Abramsky provides a categorical generalization of Girard s Geometry of Interaction (GoI) Gir89] embracing previous axiomatic approaches, such as that based on dynamic algebras [DR93,DR95] and the one in [AJ94] This generalization is based on traced monoidal categories, JSV96] and it ....

....the traced category together with some additional structure. Under these conditions on the GoI category is a weak linear category (WLC) i.e. a weakening of a linear category (see [BBPH92] Moreover, every reflexive object in a WLC gives rise to a linear combinatory algebra (LCA) Following [Abr96], there are two main variants of GoI. In the particlestyle GoI, the tensor on the underlying category is a coproduct and the strong monoidal functor is a countable copower. Girard s GoI is an instance of this. Composition in the GoI category can be intuitively understood by simulating the flow ....

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S.Abramsky. Retracing some paths in Process Algebra, Concur'96, U. Montanari and V. Sassone eds., 1996, 1--17.

.... strict maps) 12 7 Strict Maps in Int The Int construction, introduced in [9] turns a traced monoidal category into a compact closed category Int to which fully faithfully embeds (see Appendix E for a summary of the construction) its applications to computer science are studied, e.g. in [1,5]. It is natural to ask how the uniformity principles in and in Int are related. Unfortunately, the situation seems less clear than we first guess, and in this paper we can give only some elementary results and remarks. First, by a straightforward calculation, we have an obvious sort of ....

Abramsky, S., Retracing some paths in process algebra, in "Proc. Concurrency Theory (CONCUR'96)", Springer Lecture Notes in Computer Science 1119 (1996), pp. 1--17.

....category with a closed structure, with exponentials given as A B, making it into a is compact closed category. Viewed in process terms, it allows the interfaces of a process to be rearranged, by moving ports from one side of the interface to the other. It also allows the definition of a trace [JSV96, Abr96, Sel99] (see also Chapter 2) which may be used build cyclic process networks. One can also define coproducts and products, to model the additive connectives of linear logic. Thus one obtains a model of the fragment of classical linear logic [Gir87] without the exponential and its dual . ....

....in B. Morphism composition is parallel composition plus restriction, and a tensor product is given by parallel composition without interaction. The category has a feedback operation which enables one to connect input and output channels of a process. This feedback operation defines a trace [JSV96, Abr96] on the category. Note the similarity to interaction categories: they also have a trace, given by the compact closed structure. The processes in Buf are presented as labelled transition systems. The labels are divided into input actions, output actions, and the silent action t, as in CCS: for ....

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Samson Abramsky. Retracing some paths in process algebra. In Proc. CONCUR, 1996.

....causal constraints which plague dataflow semantics. The idea that non deterministic dataflow can be modelled by some kind of generalised relations fits with that of others, notably Stark in [38, 39] That dataflow should fit within a categorical account of feedback accords for instance with [23, 1]. But in presenting a semantics of dataflow as profunctors we obtain the benefits to be had from placing nondeterministic dataflow centrally within categories of models for concurrency, and in particular within presheaf models. One of our future aims is a dataflow semantics of the ....

....Kahn and SProf and constructs compact closed categories HKahn and HProf which then serve as the interpretations of higherorder Kahn processes and Port profunctors. In this section we give a summary of a categorical presentation of the geometry of interaction construction due to Abramsky [1] and also to Joyal, Street and Verity [20] We do not need the full generality of the latter presentation since we do not consider braiding or twists. Essentially, one obtain a higher order model by working with processes with bi directional input and output . For dataflow this can be ....

ABRAMSKY, S. Retracing some paths in process algebra. In CONCUR'96, vol. 1119 of LNCS, pp. 1--17.

....systems [54] which appear in Ch. 7 and the generalised synchronisation trees [139] Also the semantics of fair dataflow in [68] uses sets of finite and infinite traces representing completed observations. Global versus local interaction This distinction is essentially the one made by Abramsky in [3]. By global interaction we refer to the situation when independent agents of a concurrent system communicate via globally shared resources. A prime example are systems described in CCS like calculi, where agents communicate via shared names, usually referred to as ports. An important point is, ....

....interfaces which are local to two fixed agents, meaning that interaction can be modelled by composition. Dataflow networks [74, 33] as studied in Ch. 8 are classical examples of locally interacting systems. Local interaction has been advocated lately through the interaction semantics programme [6, 4, 5, 3] as the right setting for studying typed concurrency, opening up to a Curry Howard paradigm for concurrency. Others. The list above is far from complete. Among other distinctions and aspects of behaviour are: deterministic versus non deterministic (or probabilistic) systems, asynchronous 9 ....

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Samson Abramsky. Retracing some paths in process algebra. In CONCUR'96, volume 1119 of LNCS, pages 1--17, 1996.

....can also define a fixpoint operator from a trace on a symmetric monoidal category with a cartesian sub category of values , which can be used to model cyclic sharing in call by name languages. Traced monoidal categories have also been used to model feedback in models for concurrency, see e.g. [Abr96], Mil94] Sel99] HPW98] Theorem 6.2. The categories CGraph(S V ) CGraph(S V ,S C ) CGraph(S V ,S C ,S P ) form the initial partially traced strict focal premonoidal category over S = S V , S C , S P ) Proof. See [Jef98] There are some slight differences between our approach ....

Samson Abramsky. Retracing some paths in process algebra. In Proc. CONCUR '96, 1996.

.... composition of strategies to the parallel composition hiding in the process algebra CSP (because of the way our strategies are typed it is here possible to put these two constructions together without losing associativity; with the same idea one can also construct typed processes (cf. e.g. [Abr96], Ju99b] ffl In addition to associativity we also have (partial) neutral elements wrt. composition (given by the identity strategies as presented above) so in fact we can form a category (see below) As a special case of composition we get the application of strategies to their input: ....

S. Abramsky, Retracing some paths in process algebra, in:CONCUR '96: concurrency theory (Pisa), Springer Verlag, Berlin 1996, p. 1--17,

....with a closed structure, with exponentials given as A B, making it into a is compact closed category. Viewed in process terms, it allows the interfaces of a process to be rearranged, by moving ports from one side of the interface to the other. It also allows the definition of a trace [JSV96, Abr96, Sel99] (see also Chapter 2) which may be used build cyclic process networks. One can also define coproducts and products, to model the additive connectives, and a comonad satisfying the laws to model the exponential of linear logic. Thus one obtains a model of full linear logic. Furthermore, there is ....

....in B. Morphism composition is parallel composition plus restriction, and a tensor product is given by parallel composition without interaction. The category has a feedback operation which enables one to connect input and output channels of a process. This feedback operation defines a trace [JSV96, Abr96] on the category. Note the similarity to interaction categories: they also have a trace, given by the compact closed structure. The processes in Buf are presented as labelled transition systems. The labels are divided into input actions, output actions, and the silent action t, as in CCS: for sets ....

Samson Abramsky. Retracing some paths in process algebra. In Proc. CONCUR '96, 1996.

....Composition of transducers induces composition of resumptions, and resumption domains are the arrow sets of a category. The observation that process categories are nal coalgebraenriched was developed in an unpublished joint work of Abramsky and the third author; some instances are given in [Abr96b]. On the other hand, hyperfunction domains [A; B]H also form a category, as recently discovered in [LKS00] While double categories present both structural (Kleisli) and behavioral (coalgebra) aspect of transducers and functions with storage, by passing to nal coalgebras we extract the canonical ....

S. Abramsky. Retracing some paths in process algebra. In U. Montanari and V. Sassone, editors, CONCUR '96: Concurrency Theory, 7th International Conference, volume 1119 of Lecture Notes in Computer Science, pages 1-17, 1996.

....or application in undeclared ways. Overall, the myriad of formalisms has led to healthy diversity rather than fragmentation of the discipline: existing concurrency theories have proved suitable for a wide range of application domains, and comparative concurrency theory [Win87, Gla90, Old91, Abr96] has identified deep mathematical relationships between individual theories. Yet, potential users, such as designers of communication protocols and embedded systems, have been reluctant in applying concurrency theoretic methods. This reluctance is reinforced by the perceived need of having to ....

....a sufficiently broad framework could support the implementation of asynchronous protocols using synchronous circuits. The literature already contains a myriad of proposals for highly abstract, general purpose models of concurrency, e.g. GV92, BDH92, BH93, Mil93, Gla93, ABV94, WN95, MMP95, MR96, Abr96, AH96b] and the uniform framework we seek could very well evolve from this body of work. To make this discussion more concrete, and to illustrate the rich diversity among the proposed models, we briefly describe three of these efforts. A reactive module [AH96b] is a kind of state transition ....

S. Abramsky. Retracing some paths in process algebra. In Proceedings of CONCUR '96 -- Seventh Intl. Conf. on Concurrency Theory, volume 1119 of Lecture Notes in Computer Science, pages 1--17. Springer-Verlag, 1996.

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S. Abramsky, Retracing Some Paths in Process Algebra. In Proceedings of CONCUR `96, Springer Lecture Notes in Computer Science Vol. 1119, pp. 1-17. Springer-Verlag, 1996.

....logic. The construction of PER models from LCA s (ACA s) of [AL99,AL00] is quite simple and clear, and it yields models with extensionality properties. Many examples of linear combinatory algebras arise in the context of Abramsky s categorical version of Girard s Geometry of Interaction ([AJ94,Abr97,Abr96,AHS98]) In this paper, we de ne a fully complete PER model for the maximal theory on k . k is the simply typed calculus with nitely many ground constants in the ground type o. The theory equates two closed terms M;N of type T 1 : T n o if and only if, for all P 1 Q 1 of type ....

S.Abramsky. Retracing some paths in Process Algebra, Concur'96, U. Montanari and V. Sassone eds., 1996, 1-17.

....in which the term is used. This is a key point at which games models differ from other process models: the distinction between the actions of the system and those of its environment is made explicit from the very beginning. For a fuller discussion of the ramifications of this distinction, see [2]) In the games we shall consider, O always moves first the environment sets the system going and thereafter the two players make moves alternately. What these moves are, and when they may be played, are determined by the rules of each particular game. Since in a programming language a type ....

S. Abramsky. Retracing some paths in process algebra. In CONCUR 96: Concurrency Theory, Seventh International Conference, number 1119 in Lecture Notes in Computer Science, pages 1--17. Springer-Verlag, 1996.

....is that multiplicative proofs are modelled by permutations. Note that in any symmetric monoidal category, the symmetric group S(n) acts in a canonical way on the tensor power A = z In the original form of Geometry of Interaction [Gir89] dubbed particle style GoI in [Abr96]) the monoidal structure used for the representation is coproduct (disjoint union) In our setting, we can use products to exactly the same e ect. Thus in our example of the sequent the two proofs can be represented as maps on a product D D D D by (a; b; c; d) 7 (c; d; a; b) for the ....

....within our model. 7. 4 Connection with New Foundations It should be noted that mathematically, the concurrent games model is merely a rephrasing of the New Foundations for the Geometry of Interaction introduced by the present author and Radha Jagadeesan in [AJ92a] As demonstrated in [Abr96, AHS00], this and the original form of geometry of interaction in [Gir89] are indeed both instances of a single general scheme, as far as the interpretation of the multiplicatives and exponentials are concerned. However, as already emphasized in [AJ92a] the New Foundations setting has the ....

S. Abramsky. Retracing Some Paths in Process Algebra. In Proceedings of CONCUR 96, Lecture Notes in Computer Science Vol. 1119, pp. 1-17. Springer-Verlag 1996.

....that multiplicative proofs are modelled by permutations. Note that in any symmetric monoidal category, the symmetric group S(n) acts in a canonical way on the tensor power n O A = A : A z n : In the original form of Geometry of Interaction [Gir89] dubbed particle style GoI in [Abr96]) the monoidal structure used for the representation is coproduct (disjoint union) In our setting, we can use products to exactly the same e ect. Thus in our example of the sequent O ; the two proofs can be represented as maps on a product D D D D 21 by (a; b; c; d) 7 ....

....within our model. 7. 4 Connection with New Foundations It should be noted that mathematically, the concurrent games model is merely a rephrasing of the New Foundations for the Geometry of Interaction introduced by the present author and Radha Jagadeesan in [AJ92a] As demonstrated in [Abr96, AHS00], this and the original form of geometry of interaction in [Gir89] are indeed both instances of a single general scheme, as far as the interpretation of the multiplicatives and exponentials are concerned. However, as already emphasized in [AJ92a] the New Foundations setting has the ....

S. Abramsky. Retracing Some Paths in Process Algebra. In Proceedings of CONCUR 96, Lecture Notes in Computer Science Vol. 1119, pp. 1-17. Springer-Verlag 1996.

....model of computation. Our approach also has conceptual interest in that our constructions, while quite concrete, are based directly on ideas stemming from Linear Logic and Geometry of Interaction [18, 19, 20, 21] and developed in previous work by the present author and a number of colleagues [2, 3, 5, 6, 7, 9, 10]. Our work here can be seen as a (relatively) concrete manifestation of these more abstract and foundational developments. However, no knowledge of Linear Logic or Geometry of Interaction is required to read the present paper. The paper [16] contains some discussion of a reversible abstract ....

....exive, transitive closure) 8 4.2.3 Application f g = LApp(f; g) It can be veri ed that f g is a partial injective function. In fact, we have the following result. Theorem 4.1 (I; f AS ; f AK ) is a combinatory algebra. This theorem is a minor variation on the results established in [5, 6, 9, 7, 10]; see in particular [10] and the combinatory algebra of partial involutions studied in [7] The ideas on which this construction is based stem from Linear Logic [18, 21] and Geometry of Interaction [19, 20] in the form developed by the present author and a number of colleagues [2, 3, 5, 6, 9, 7, ....

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S. Abramsky, Retracing Some Paths in Process Algebra. In Proceedings of CONCUR `96, Springer Lecture Notes in Computer Science Vol. 1119, pp. 1-17. Springer-Verlag, 1996.

....third part of the paper, we build an example of a concrete fully complete model for ML types. This is built by linear realizability over a special linear combinatory algebra of partial involutions. This algebra arises in the context of Abramsky s generalization of Girard s Geometry of Interaction ([AJ94, Abr96, Abr96a, AHPS98]) The proof of full completeness consists in showing that this model satisfies the axioms in our axiomatization. In particular, proving that the model does not contain infinite typed Bohm trees is quite di#cult, and it requires the study of an intermediate model. This is the model generated by ....

..... T # X h( # X) For any morphism H : h 1 # h 2 in L(U m U) #m (H) H. 4. 2 Partial Involutions A#ne Combinatory Algebra Many examples of LCAs arise from Abramsky s categorical version of Girard s Geometry of Interaction (GoI) construction, based on traced symmetric monoidal categories ([Abr96, Abr96a, AHPS98]) A basic example of GoI LCA, introduced in [Abr96] can be defined on the space [N #N] of partial functions from natural numbers into natural numbers, applying the GoI construction to the the traced category Pfn of sets and partial functions. Here we briefly recall the definition of this LCA, ....

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S.Abramsky. Retracing some paths in Process Algebra, Concur'96 Conf. Proc., 1996.

....third part of the paper, we build an example of a concrete fully complete model for ML types. This is built by linear realizability over a special linear combinatory algebra of partial involutions. This algebra arises in the context of Abramsky s generalization of Girard s Geometry of Interaction ([AJ94, Abr96, Abr96a, AHPS98]) The proof of full completeness consists in showing that this model satisfies the axioms in our axiomatization. In particular, proving that the model does not contain infinite typed Bohm trees is quite difficult, and it requires the study of an intermediate model. This is the model generated by ....

.... X h( X) For any morphism H : h 1 h 2 in L(U m Theta U) 8m (H) H. 4. 2 Partial Involutions Affine Combinatory Algebra Many examples of LCAs arise from Abramsky s categorical version of Girard s Geometry of Interaction (GoI) construction, based on traced symmetric monoidal categories ([Abr96, Abr96a, AHPS98]) A basic example of GoI LCA, introduced in [Abr96] can be defined on the space [N N] of partial functions from natural numbers into natural numbers, applying the GoI construction to the the traced category Pfn of sets and partial functions. Here we briefly recall the definition of this LCA, ....

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S.Abramsky. Retracing some paths in Process Algebra, Concur'96 Conf. Proc., 1996.

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S. Abramsky (1996), Retracing Some Paths in Process Algebra, in Proceedings of CONCUR 96, U. Montanari and V. Sassone (Eds.), Lecture Notes in Computer Science 1119, 1--17, Springer-Verlag.

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S. Abramsky. Retracing some paths in process algebra. In Ugo Montanari and Vladimiro Sassone, editors, CONCUR '96: Concurrency Theory, 7th International Conference, volume 1119 of Lecture Notes in Computer Science, pages 1-17, Pisa, Italy, 26-29 August 1996. Springer-Verlag.

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S. Abramsky, Retracing some paths in process algebra, in:CONCUR '96: concurrency theory (Pisa), Springer Verlag, Berlin 1996, p. 1--17,

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S. Abramsky (1996), Retracing Some Paths in Process Algebra, in Proceedings of CONCUR 96, U. Montanari and V. Sassone (Eds.), Lecture Notes in Computer Science 1119, 1--17, Springer-Verlag.

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Abramsky, S. (1996) Retracing some paths in process algebra. Proceedings 7th International Conference on Concurrency Theory, LNCS 1119, 1--17.

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S.Abramsky. Retracing some paths in Process Algebra, Concur'96, U. Montanari and V. Sassone eds., 1996, 1-17.

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S.Abramsky. Retracing some paths in Process Algebra, Concur'96, U. Montanari and V. Sassone eds., 1996, 1-17.

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