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....of the signal. Although this extension is expressive and similar in many ways to the level ruled Petri net, Vanbekbergen does not present any type of analysis algorithms for it. The event structure introduced by Winskel is a signal model that is similar to the Petri net and its many variants [39]. Burns adds fixed delays to the event structure and develops algorithms to compute the average case performance of control implementations in [40, 41] Myers added minimum and maximum bounds to the event rule structure in [42] Belluomini extends the event rule structure to timed event level ....

G. Winskel, "An introduction to event structures," in Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. Noordwijkerhout, Norway, June 1988.

....of Plotkin s method of Structured Operational Semantics [14] have been mostly developed with respect to Milner s Calculus of Communicating Processes [11] and other similar algebraic process languages. On the other hand, true concurrent models (including Petri nets [13] Winskel s event structures [12, 17] and Mazurkiewicz s traces [1] were primarily developed in relation to Petri nets. Meseguer s concurrent term rewriting logic [9] is another true concurrent model: it aims at describing a wide variety of concurrent systems. While there is some work done on concurrent semantics for process calculi ....

....complexity of detail involved in event diagrams (together with pending events) distract from the fundamental issues. Traces do not represent finite computations. Not having limit points, traces do not form domains. However, we can add limit points by taking a completion by ideals, as proposed in [17]. A most strikingly difference between [6, 4] and trace semantics concerns activation. 19 In the former works activation forms an irreflexive partial order on events such that each event has at most one immediate predecessor. This means we have to single out one only event as the direct cause of ....

G. Winskel. An introduction to event structures. In Linear time, Branching Time and Partial Orders in Logics and Models for Concurrency, Lecture Notes in Computer Science 354, pages 364--397, Springer-Verlag, May 1988. 22

....omniscience of the agents, which is frequently regarded as a drawback, especially if the agents are modeled to take decisions in real time. An alternative proposal to the classical notion of knowledge [19, 17, 18] is based on partial order structures of local states, called event structures [20, 16]. Knowledge is acquired by the agents via communication with local interactions. Each agent has (causal) knowledge about the most recent events of the other agents. This approach captures the changes in state due to actions, which is crucial for successful modeling of knowledge, but quite rarely ....

....E = i=1 E i a set of events, where E i is a set of events agent i participates in, for 1 i N , and agent(e) fi 2 N j e 2 E i g. Event e is called joint if belongs to at least two different sets E i . Event structures have been successfully applied in the theory of distributed systems [20] and several temporal logics have adopted them as frames [9, 16, 18] Next, we present a formal definition of event structure. Definition1. A labelled prime event structure (lpes, for short) is a 5 tuple ES = E; A; #; l) where 1. E is a finite set, called a set of events or action ....

Winskel, G., An Introduction to Event Structures, LNCS 354, Springer - Verlag, pp. 364-397, 1989.

....in particular events laid out along a time line. Programmers who would like to leave a useful legacy to their heirs need to spend more time reflecting on the nature of concurrency, learning what it feels like and how to control it. My own view of concurrency is that event structures [NPW81,Win80,Win88] o#er a good balance of abstractness and comprehensiveness in modeling concurrency. The extension of event structures to Chu spaces, or couples as I have started calling them [Pra95] simultaneously enriches the comprehensiveness while cleaning up the model to the point where it matches up to to ....

G. Winskel. An introduction to event structures. In Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, REX'88, volume 354 of Lecture Notes in Computer Science, Noordwijkerhout, June 1988. Springer-Verlag. All trademarks appearing in this article are the property of their respective owners.

....That this is indeed the case is easily shown on the basis of lemmas 6.3 and 6.7 and the form of the protocol rules. In fact [Fig. 12] with time flowing down the page, is an elementary event structure for a successful communication according to our protocol [Nielsen et al. 1981) Winskel (1986) Winskel (1988)] For obvious reasons, the commit events (d) are regarded as pinpointing the position of a communication within an execution. Thus even if the event sequences for two communications overlap, they are still regarded as taking place in the order of their commit events. Remark 7.16.1 We recall the ....

Winskel G., An Introduction to Event Structures. in: Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. de Bakker, de Roever, Rozenberg (eds.), LNCS 354, 364-397, (1988). 398

.... global action dependencies) The basis for the operational semantics, on the other hand, will be a very rich event based model; and the constructions defined for the operators of L are variations on known constructions on, for instance, prime event structures [52, 25] stable event structures [79] and families of posets [66, 69] We then give a mapping from the denotational to the operational 18 model showing the consistency of the two (up to bisimulation) this shows that the poor model is yet rich enough to be compositional. Unfortunately, due to the action dependencies and the ....

G. Winskel. An introduction to event structures. In de Bakker et al. [22], pages 364-397.

....respect to the new semantics. 1. Introduction The notion of nondeterminism arises naturally in describing concurrent systems. Various approaches to the theory and specification of such systems, for instance, CCS [Milner 1980] CSP [Hoare 1985] process algebras [Bergstra 1986] event structures [Winskel 1988], include the phenomenon of nondeterminism. But nondeterminism is also a natural concept in describing sequential programs, either as a means of indicating a don t care attitude as to which among a number of computational paths will actually be utilized in a particular computation (e.g. ....

Winskel, G., "An introduction to event structures", LNCS, vol. 354, Springer, 1988.

....spaces and linear functions. The stable functions are obtained via the co Kleisli construction associated with the model. Adding a partial order, we obtain event structures (E; which represent coherent prime algebraic domains [8] and, with a finiteness axiom added, coherent dI domains [2, 11, 12]. We can consider the corresponding categories of relations, again representing the linear stable functions; we must now add the condition: a ff b b ) 9a a: a ff b However, this only yields a model of intuitionistic linear logic [11, 13] By passing to bistructures we add enough ....

....are subsets x E which are ffl consistent: 8e 1 ; e 2 2 x. e 1 ffl secured: 8e; e e 2 x ) e 2 x. Ordered by inclusion, the configurations ( Gamma(E) form a coherent prime algebraic domain [8] such domains are precisely the infinitely distributive, coherent Scott domains [12]. An instance of the causal dependency ordering e e when e and e are distinct, is understood as meaning that the event e causally depends on the event e , that the event e can only after e has ocurred. Given this understanding it is reasonable to impose a finiteness axiom, expressing ....

Winskel, G., An introduction to event structures. In the lecture notes for the REX summerschool in temporal logic, May 88, in Springer Lecture Notes in C.S., vol.354, 1989.

....are subsets x E which are ffl consistent: 8e 1 ; e 2 2 x. e 1 #e 2 , ffl secured: 8e; e 2 x. 6 Ordered by inclusion, the configurations ( Gamma(E) form a coherent prime algebraic domain [13] such domains are precisely the infinitely distributive, coherent Scott domains [19]. An instance of the causal dependency ordering e e when e and e are distinct, is usually understood as meaning that the event e causally depends on the event e , that the event e can only occur after e has ocurred. Usually we impose a finiteness axiom: fe eg is finite, for ....

Winskel, G., An introduction to event structures. In the lecture notes for the REX summerschool in temporal logic, May 88, in Springer Lecture Notes in C.S., vol.354,

....of such an event structure are those subsets x E which are: ffl left closed: 8 e; e e 2 x ) e 2 x. Ordered by inclusion, the configurations form a coherent prime algebraic domain ( Gamma(E) 21] such domains are precisely the infinitely distributive, coherent Scott domains [27]. An instance of the causal dependency ordering e when e and e are distinct, is understood as meaning that the event e causally depends on the event e , in that the event e can only occur after e has occurred. Given this understanding it is reasonable to impose a finiteness axiom, ....

....states that a compact element dominates finitely many elements. Axiom I follows from the finiteness of compacts, proved in Proposition 2. Distributivity is then equivalent to prime algebraicity, i.e. the property that any element is the lub of the complete primes that it dominates. We refer to [31, 27] for a proof. Primealgebraicity is an immediate consequence of (2) 2) Consider a configuration [e] x . We show it is a complete prime. If Y is bounded above and [e] x v Y = Y , then e 2 y for some y in Y . Since [e] x y, we infer that [e] x y, by Lemma 5. Conversely, every ....

Winskel, G., An introduction to event structures, in REX School/Workshop on Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, (eds. de Bakker, J.W., de Roever, W.P. and Rozenberg, G.), LNCS, Vol. 354, pp. 364--397, Springer-Verlag, 1989.

....has joins got simply via unions so it is certainly a cpo, with least element ; and we can think of the union operation as being a form of nondeterministic sum. It s worth remarking that the domains obtained in this way are precisely the in nitely distributive algebraic lattices (see e.g. [18, 19]) and that these are just the same as the prime algebraic lattices of [17] and free join completions of partial orders. There are several choices about what to take as maps between nondeterministic domains. If we eschew fairness , the most generous we seem to have call for is that of all ....

G. Winskel. An introduction to event structures. In Proc. of REX summerschool in temporal logic, 'May 88, LNCS 354, 1988.

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: Winskel, G., An Introduction to Event Structures, LNCS 354, pp. 364-387, 1988.

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G. Winskel (1989): An Introduction to Event Structures. LNCS 354, Springer - Verlag, pp. 364-397.

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G. Winskel (1989): An introduction to event structures. In J.W. de Bakker, W.P. de Roever & G. Rozenberg, editors: REX School/Workshop on Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, Noordwijkerhout, The Netherlands, May/June 1988, LNCS 354, Springer, pp. 364-397.

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Winskel, G., An Introduction to Event Structures, LNCS 354, Springer - Verlag, pp. 364-397, 1989.

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G. Winskel, An Introduction to Event Structures, LNCS 354, Springer - Verlag, pp. 364-397, 1989.

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G. Winskel. An introduction to event structures. In J.W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, School/Workshop, Noodwijkerhout, The Netherlands, May/June 1988, pages 364--397. Springer-Verlag, LNCS 354, 1988.

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G. Winskel. An introduction to event structures. In J. W. de Bakker et al., editors, Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, number 345 in Lecture Notes in Computer Science, pages 364--397. Springer, 1989.

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Winskel, G., An Introduction to Event Structures, LNCS 354, Springer - Verlag, pp. 364--397, 1989.

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G. Winskel (1988), An Introduction to Event Structures, in Linear time, branching time, and partial order in logics and models for concurrency, J.W. de Bakker et al. (Eds.), Lecture Notes in Computer Science 354, 365--397, Springer-Verlag.

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G. Winskel. An introduction to event structures. In Proc. REX summer school 1988, LNCS 354.

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G. Winskel. An introduction to event structures. In J. W. de Bakker, editor, Linear Time, Branching Time, and Partial Order in Logics and Models for Concurrency, volume 354 of Lecture Notes in Computer Science, pages 364--399. Springer Verlag, 1988.

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G. Winskel. An introduction to event structures. In Proc. REX summer school 1988, LNCS 354.

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G. Winskel. Introduction to Event Structures. In G. Rozenberg J.W. de Bakker, W.- P. de Roever, editor, Linear Time, Branching Time, and Partial Order in Logics and Models of Concurrency, pages 364 - 397. LNCS 354, 1988.

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G. WINSKEL (1989): An introduction to event structures. In J.W. de Bakker, W.P. de Roever & G. Rozenberg, editors: REX School and Workshop on Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency, Noordwijkerhout, The Netherlands, May/June 1988, LNCS 354, Springer, pp. 364--397.

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