S. Abramsky, S. J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In Deductive Program Design: Proceedings of the 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....two processes may communicate. Thus, one obtains an intuitive presentation of the communication abilities of processes in a network. A formal account of this graphical presentation would provide an intuitive way of reasoning about the connectivity of processes in a network. Interaction categories [AGN96] provide an account of the categorical structure necessary to model concurrent systems. In an interaction category, Objects are types. Morphisms A B are processes with the interface divided into two parts, A and B. Morphism composition is interaction. Identities and other structural ....

....the interface. As an example, there is a duality ( which takes a process p : A B to . Again, this has a nice graphical representation: B p Often, the duality is the identity on objects, so it can be seen as simply swapping the sides of the interface of a process. In [AGN96], interaction categories are introduced by presenting a canonical example, a category SProc of synchronous processes. Their objects are given by a set (of channel names) together with a safety specification . Morphisms A B are processes, presented as synchronisation trees (viewed up to ....

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Samson Abramsky, Simon Gay, and Rajagopal Nagarajan. Interaction categories and the foundations of typed concurrent programming. In Deductive Program Design: Proceedings of the 1994 Marktoberdorf Summer School. Springer-Verlag, 1996.

....x y , x y = y. Similarly, semilattices in the category of torsion commutative semigroups form a bireflective subcategory. One may also replace semigroups by monoids. Example 3. 6 The category Rel of sets and relations is bireflective in SProc, the interaction category of synchronous processes ([1], Proposition 3.0.4) Briefly, an object A of SProc is a pair ( Sigma A ; SA ) of sets with SA a nonempty prefix closed subset of Sigma A ; a morphism from A to B is a strong bisimilar class of Sigma A Theta Sigma B labeled transition systems whose traces are contained in SA Theta SB in ....

S. Abramsky, S. Gay, and R. Nagarajan. Interaction Categories and the Foundations of Typed Concurrent Programming. In Proceedings of NATO Advanced Study Institute: International Summer School, Marktoberdorf, 1994.

....presentation of ternis. They range from the fiowomial calculus of Stefanescu [32, 128] to the bicategories of processes of Walters et al..ii [80, 1] to the prc monoidal categories of Power and Robinson [122] to the actio structures of Milner [112] to the itcractio categories of Abramsky [1], and to the gs monoidal categories of Corradini and Gadducci [42, 41] to mention just a few (see also [31,125, 79, 2, 43, 69, 59] An element common to all these structures is the fact that they can be thought of as suitable enrichments of symmetric strict monoidal categories, which give the ....

S. Abramsky, S. Gay, and R. Nagarajan, Interaction categories and the foundations of typed concurrent programming, in: M. Broy, Ed., Deductive Program Designs: Proceedigs of the 1996.

....underway on a bicategory of port automata to close a gap in [36] This will provide further operational back up to the trace on port profunctors and help in the understanding of independence at higher order. The higher order models should be compared to the related, but clearly different, work in [2]. It remains to incorporate fairness into the profunctor model; it is hoped to exploit independence along the lines in [9] A compositional semantics of Verilog, or perhaps an interesting fragment, feels within reach. ....

ABRAMSKY, S., GAY, S., AND NAGARAJAN, R. Interaction categories and the foundations of typed concurrent programming. In Proc. of the 1994 Marktoberdorf summer school, Springer.

....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 ....

....is a bicategory, only satisfying the axioms of a TMC up to isomorphism. s ) will not be a category, e.g. composition is only associative up to isomorphism. One solution is consider the quotient of PProf s with respect to open map bisimulation, analogous to the definition of the category in [4, 43]. That is, instead of PProf s use the category with objects being path categories as usual, but taking arrows to be equivalence classes with respect to open map bisimulation. By the congruence result, this is indeed well defined and it is easy to check that we get a traced (strict) symmetric ....

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Samson Abramsky, Simon Gay, and Rajagopal Nagarajan. Interaction categories and the foundations of typed concurrent programming. In Proc. of the

....than being fixed by intuition. Such connectives have correspondents in different processes categories (e.g. arising from different families of functors) providing a remarkable level of genericity. From a methodological point of view this approach is in debt to Abramsky s interaction categories [1]. Our processes, however, are concretely defined over (also) concrete state spaces, leading to a rather different structure. In particular, by the presence of such state spaces, most diagrams, including the ones expressing identity and associativity of composition, only commute up to isomorphism. ....

S. Abramsky, S. Gay, and R. Nagarajan. Interaction categories and the foundation of typed concurrent programming. In M. Broy, editor, Deductive Program Design: Proc. of the

....distinct. We can use replication (or a single recursion) to achieve the same e#ect, but multiple recursion is preferable for well structured programs. This finishes the introduction of all language constructs we shall use in this paper. We give a simple example of a program. accept a(k) in k [1]; k (y) in P request a(k) in k (x) in k [x 1] inact. The first process receives a request for a new session via a, generates k, sends 1 and receives a return value via k, while the second requests the initiation of a session via a, receives the value via the generated channel, then returns the ....

....of the branches is selected, discarding the remaining ones. Note we do not allow reduction under various communication prefixes, as in standard process calculi. As an example, the simple program in 2. 1 has the following reduction (below and henceforth we omit trailing inactions) accept a(k) in k [1]; k (y) in P request a(k) in k (x) in k [x 1] # (#k) k [1] k (y) in P k (x) in k [x 1] # (#k) k (y) in P k [x 1] # P[2 y] Observe how interaction proceeds in a lock step fashion. This is due to the synchronous form of the present communication primitives. 3. Representing ....

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S. Abramsky, S. Gay, and R. Nagarajan, Interaction Categories and Foundations of Typed Concurrent Computing. Deductive Program Design, Springer-Verlag, 1995.

....[MW90] Furthermore, the relation of refinement between processes should have certain properties, the most crucial one is that it is compositional. This suggests to view refinement as given by 2 cells of a 2 category of processes since this would ensure a correct behavior of the refinement. In [AGN94] Abramski et.al. use a preorder given by strong simulation of synchronization trees to provide a 2 cell structure for the process category SP roc. However, this result is very specific for the case of SP roc and the strong simulation. This paper tries to adapt the idea of 2 cells as a notion ....

Samson Abramski, Simon J. Gay, and Rajagopal Nagarajan. Interaction categories and the foundation of typed concurrent programming. In M.Broy, editor, Deductive Program Design: Proceedings of the 1994 Marktoberdorf Summer School, NATO ASI Series F: Computer and System Sciences. Springer Verlag, 1994.

....tree like presentation of terms. They range from the flownomial calculus of Stefanescu [6, 34] to the bicategories of processes of Walters [18, 19] to the pre monoidal categories of Power and Robinson [28] to the action structures of Milner [24] to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs monoidal categories of Corradini and Gadducci [7, 8] just to mention a few (see also [9, 11, 15, 29] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a ....

....defined algebra of relations is new. e 4 ( e 4 ) b) e 2 ( e 2 ) b) 99 r r r r r r e 3 ( e 3 ) c) eeL L L L L L e 1 ( e 1 ) a) eeL L L L L L 99 r r r r r r Fig. 1. A partial order P = fe i g i=1: 4 with labels over the set A = fa; b; cg. x a y fflffl fflffl Y [1] OO OO X[1] X[1] Y [1] 7 ; w w w w w Upsilon ## ## G G G G G X[2] 0 B B B B B B B B B B B x 1 b y 1 fflffl fflffl Y [1] OO OO X[1] x 2 c y 2 fflffl fflffl Y [2] OO OO X[2] 1 C C C C C C C C C C C A ; Y [1] Upsilon ## ....

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S. Abramsky, S. Gay, and R. Nagarajan. Interaction categories and the foundations of typed concurrent programming. In Deductive Program Design, Nato ASI Series, pages 403--442. Springer Verlag, 1994.

....port names (not necessarily disjoint) An (A; B) process P is a set of sequences over the port alphabet Sigma A;B , i.e. P Sigma A;B , satisfying the following axioms. Assume x; y 2 Sigma A;B , a; a 0 2 A, b; b 0 2 B, v; v 0 2 V. 12 This is related to the work of Abramsky et al. in [AGN, Abr96]. The definition of a trace in a compact closed category seems closely connected to the canonical interpretation of the multi cut rule which can be imposed on linear logic. 13 This is a restriction of the original model, which was not required to be prefix closed, nor restricted to finite ....

....by including some infinite sequences, the fair ones, while excluding the unfair sequences. The problems of bisimulation between processes satisfying fairness conditions and of expressing fairness in profunctors relates to work of Aczel [Acz96] the specification structures of Abramsky et al. [AGN] and the focus on maximal traces in [Che96] Finally, it has recently come to our attention that there are problems in the semantics of hardware description languages which seem amenable to attack by dataflow models [Gor96, Gor97] It is my plan to investigate this possibility. 5 Conclusion The ....

Samson Abramsky, Simon Gay, and Rajagopal Nagarajan. Interaction categories and the foundations of typed concurrent programming. Tutorial notes on Samson Abramsky's interaction Categories.

....Abramsky s category SProc when applied to any of the standard models of interleaved concurrency. SProc is also obtained as a process category upon the category of sets in time (i.e. trees) and this sheds new light on the analogy between SProc and relations in time . 1 Introduction Abramsky [AGN94] has proposed interaction categories as a new semantic paradigm. These categories take concurrent processes as morphisms between interface specifications, with composition given by process interaction at a shared interface. This treatment of processes as morphisms provides a type discipline for ....

....processes as morphisms provides a type discipline for process construction. As in functional programming, this discipline may be used to facilitate correctness arguments for concurrent system implementations. A key example of an interaction category is the category SP roc of synchronous processes [AGN94]: its objects are trace specifications and its morphisms are strong bisimularity classes of transition systems whose traces lie within their interface specifications. One aspect of the original formulation of SP roc and of interaction categories in general is that morphisms are chosen to ....

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Samson Abramsky, Simon J. Gay, and Rajagopal Nagarajan. Interaction categories and the foundation of typed concurrent programming. In M.Broy, editor, Deductive Program Design: Proceedings of the 1994 Marktoberdorf Summer School, NATO ASI Series F: Computer and System Sciences. Springer Verlag, 1994.

.... construct par as the de Morgan dual of the tensor AOB = A Omega B ) The logical equations which hold in linear logic can be proved to be isomorphisms, for instance the Currying (A Omega B) C = A ( B ( C) Altogether the category of Chu spaces forms a autonomous category [IM93, Abr93b, AGN, Bie95, Pav94]. These structure has been identified as model for linear logic. Linear logic has been identified as type theory for concurrency by Abramsky [Abr93a, BS94, GN95, Gay95] Here we model concurrency in the frame of Chu spaces and use the logical connections as language constructs 5.2 Behavioral ....

Samson Abramsky, Simon Gay, and Rajagopal Nagarajan. Interaction categories and the foundations of typed concurent programming.

....needs to be added. interaction is modelled as composition in the category ( Thus interaction is aligned with the computation as cut elimination paradigm, and hence a unification of concurrency with other work in denotational semantics, type theory, categorical logic etc. becomes possible. See [AGN96a, Abr93, Abr95b] for a detailed discussion of this point. interaction is local. The dynamics of composition traces out information paths , which are closely related to the types of the processes which interact. There is no appeal to a global mechanism for matching names. As we will see, this is general enough to ....

....structures [AGN96b] Types can then carry strong correctness information, and the type inference rule for composition : Resumptions Geometry of Interaction 5.2 Modelling types and functions 5.3 State and concurrency becomes a compositional proof rule for process interaction. See [Abr93, Abr95b, AGN96a, AGN96b] for further discussion and applications. We shall mention some particular cases for the examples described above. In this case, we can get the structure of games as safety properties, and of winning strategies as liveness properties, as described in [AJ94a, Abr96b] In particular, the fact that ....

S. Abramsky, S. Gay, and R. Nagarajan. Interaction categories and the foundations of typed concurrent programming. In . Springer-Verlag, 1996. To appear.

....of (1.2) may not be typable when a name carries different types. Another difference is our explicit treatment of name transmission in a type, which we believe to be important for the kind of results we have obtained here. linear logic) Some ideas of Linear Logic [9] and Interaction Categories [1, 2, 8] are related (e.g. multi cuts in [8] Prasad [40] studies a term assignment to a generalisation of linear logic, where the distinction between linear and classical realms exists, which is also related. true concurrency) The causality of communication in process calculi has been studied ....

Abramsky, S., Gay, S. and Nagarajan, R., Interaction Categories and Foundations of Typed Concurrent Computing. Deductive Program Design, Springer-Verlag, 1995.

....In [GSB98a] we show how to combine these models to obtain a semantics for ROOM. This semantics is extended for the hierarchical specification of hybrid systems in [GSB98b] The calculus of interaction graphs is a simpler, more intuitive and more general formulation of interaction categories (see [AGN94] that uses ideas from [JSV96] It is not only closer to UML RT but, in our opinion, a better foundation for the theory of typed concurrent systems. This was only possible by having in mind the concrete implementation of UML RT. A semantic model for this calculus is given in [GBSR99] It is also ....

S. Abramsky, S. Gay, and R. Nagarajan. Interaction categories and the foundations of typed concurrent programming. To appear in Proc. Marktoberdorf Summer School, 1994.

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S. Abramsky, S. J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In Deductive Program Design: Proceedings of the 1994.

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S. Abramsky, S. J. Gay and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. Deductive Program Design: Proceedings of the 1994.

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S. Abramsky, S.J. Gay and R. Nagarajan, Interaction categories and the foundations of the typed concurrent programming. In: Deductive Program Design: Proceedings of the 1994.

....of subsystems becomes more complex. Interaction between components may take the form of a prolonged, dynamic pattern of communication rather than a simple procedure call, and both parties must agree about the expected nature of the dialogue. Our recent research on interaction categories [1, 2] has led to a framework for the design of sophisticated type systems which are able to specify communication protocols of this form. In this framework, type checking and verification are seen as different facets of a single activity and this opens up the possibility of developing a ....

....present paper. This includes the autonomous structure [5] corresponding to the multiplicative connectives Omega , O and ( Gamma) of linear logic [12] but not the additive structure (products and coproducts) or the delay operator. Complete definitions can be found in previous publications [1, 2, 3, 10]. Our use of ASProc in the present paper allows us to analyse asynchronous systems, in which different components are able to evolve at their own rate. This is in contrast to much of our previous work on interaction categories (for example, 11] which assumes universal synchronisation with ....

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S. Abramsky, S. J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In M. Broy, editor, Deductive Program Design: Proceedings of the 1994.

....including both compositional and non compositional approaches, begins to open up. We now turn to the specific applications of this framework which in fact originally suggested it, in the setting of the first author s interaction categories. 3 Interaction Categories Interaction Categories [1, 3, 4, 6] are a new paradigm for the semantics of sequential and concurrent computation. This term encompasses certain known categories (the category of concrete data structures and sequential algorithms [12] categories of games [7] geometry of interaction categories [8] as well as several new ....

....review the definition of SProc, the category of synchronous processes. Because the present paper mainly concerns the use of specification structures 6 for deadlock freedom, we omit the features of SProc which will not be needed in later sections. More complete definitions can be found elsewhere [1, 6, 18]. An object of SProc is a pair A = Sigma A ; SA ) in which Sigma A is an alphabet (sort) of actions (labels) and SA A is a safety specification, i.e. a non empty prefix closed subset of Sigma A . If A is an object of SProc, a process of type A is a process P with sort Sigma A such ....

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S. Abramsky, S. J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In M. Broy, editor, Deductive Program Design: Proceedings of the 1994.

....than in the sequential case, so too are the potential benefits: there is a wide range of concurrent program properties which one might hope to specify as types and verify by type checking. Examples are deadlock freedom, liveness and fairness. Abramsky s recent work on interaction categories [1 4] has established a semantic foundation for typed concurrency with the capability of constructing types which specify complex properties. It extends the Propositions as Types paradigm to concurrency, with classical linear logic [13] forming the core of the type system; the use of a classical linear ....

....correspondence to concurrency. The theory of interaction categories has been applied to several case studies, such as the cyclic scheduler [20] and the dining philosophers [15] in order to demonstrate the type theoretic approach to specification and verification. This work is reported elsewhere [4, 12]. Up to now, however, we have always constructed the systems being studied by describing their components as labelled transition systems and combining them by direct application of the categorical combinators. In the present paper, we have developed a formal syntax for the construction of typed ....

S. Abramsky, S. J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In M. Broy, editor, Deductive Program Design: Proceedings of the 1994.

....This work led to the development of a sort inference algorithm for the calculus [9] which continues to be cited in the literature on types for the calculus. Later my research moved in the direction of interaction categories, a semantic framework for typed concurrency proposed by Abramsky [1, 2, 4, 5]. My PhD thesis [11] dealt with the theory and applications of interaction categories. Its contributions included a new analysis of the semantics of synchronous data ow programs, in the interaction categories framework; the development of a synchronous process calculus with a type system based on ....

....view remains worthy of study in its own right, since it provides a useful abstraction. The research summarised above shows that types are being used to provide increasingly informative descriptions of system behaviour. The theory of interaction categories, developed by Abramsky, Gay and Nagarajan [1, 2, 4, 5] during the past several years, suggests a way of placing a range of type 4 systems within a single framework. Interaction categories are categories of typed concurrent processes, in which objects are types and morphisms are processes. The notion of speci cation structure allows the construction ....

S. Abramsky, S. J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In M. Broy, editor, Deductive Program Design: Proceedings of the

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ABRAMSKY, S., GAY, S. J., NAGARAJAN, R. Interaction categories and the foundations of typed concurrent programming. In Deductive Program Design: Proceedings of the 1994.

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Samson Abramsky, Simon J. Gay, and R. Nagarajan. Interaction categories and the foundations of typed concurrent programming. In Deductive Program Design: Proceedings of the 1994 Marktoberdorf Summer School, NATO ASI Series F. Springer-Verlag, 1994.

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S. Abramsky, S.J. Gay, and R. Nagarajan. Interaction categories and foundations of typed concurrent programming. In M. Broy, editor, Deductive Program Design: Proceedings of the 1994 Marktoberdorf International Summer School, NATO ASI Series F: Computer and System Sciences. Springer-Verlag, 1995.

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