Pratt, V. (1986). Modelling concurrency with partial orders. Int. Journal of Parallel Programming, 15(1):33--71.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....coded into the alphabet. In earlier work [16] one of us worked on series parallel terms (sp terms) which are de ned by adding to the concatenation product a parallel composition operation. The corresponding notion of behaviour is that of N free (labelled, nite) posets also known as pomsets [21]. Under certain conditions (of bounded parallel width, de ned later) recognizable sp languages were shown to be equal to those de ned by series rational expressions [16] The paper also de ned a kind of branching automata such that the notion of regularity matched the previous two notions. ....

V. Pratt. Modelling concurrency with partial orders, IJPP 15(1) (1986) 33-71.

....are output in name only is why we consider this restriction to be merely cosmetic. We plan to extend our framework to remove these cosmetic restrictions. There are many other traits that distinguish models of computation, including: levels of abstraction, partial orders models (e.g. POMSETS [11] and Mazurkiewicz Traces [10] vs total orders, action based vs value based, different styles of computation, communication and coordination. For all these traits, our framework is unbiased. We have constructed different models with several different combinations of these traits, all of which ....

V. R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):33--71, Feb. 1986.

....Algorithms. 1. Introduction Partial order semantics of formalisms designed to describe concurrent systems have a long history (approximately 30 years) Concentrating on Petri nets, instances of those semantics are processes [2] Mazurkiewicz traces [14, 6] partially ordered multisets (pomsets) [18], prime) event structures [17] partial words [10] and semiwords [20, 26] or branching processes [7] the list is far from being complete) Especially a finite representation of a branching process of a 1 bounded Petri net, called the finite prefix of the maximal branching process of a Petri net ....

....Semantics of Petri Nets. Semiwords as partial order semantics of (general) Place Transition nets were introduced by Starke [20] The idea to use equivalence classes labelled partial orders (partial words) as semantics of concurrent systems goes back to Grabowski [10] Independent of Starke, Pratt [18] developed a more process algebraic theory of partial words (which are now called pomsets) The relation between partial words and processes of Petri nets was justified by Vogler [26] At the end of this section we will briefly discuss the relation of semiwords and Mazurkiewicz traces. A ....

V. PRATT, Modelling concurrency with partial orders, International Journal of Parallel Programming, 15 (1986), pp. 33--71.

....is said to be strict if it is a monotone function with respect to the partial order relation on events. Define ES A to be the subcategory of ESA with the same objects but only strict morphisms. In [13] open map bisimulation for event structures was studied with respect to the choice of pomsets [18] as objects of the path category. Definition 2.4 Define a pomset to be a triple (P, l) with P a finite set, P, #) a partial order and l : P A a labelling function. Any pomset gives rise to an event structure (see [13] The full subcategory of ESA of pomsets is a large category, though ....

V. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Processing, 15:33--71, 1986.

....[5] are a categorical generalisation of sets and relations. The objects of Prof are small categories and arrows are profunctors; profunctors are like receptive monotone relations but with the category 2 2 2 replaced by Set. The traces can also be viewed as a specific kind of pomsets [31] and the path categories as a subcategory of the category of pomsets given in [19] 8 Definition 10 Let P and Q be small categories. A profunctor X : P Q is a bifunctor X : P Theta Q Gamma Set. For p; q objects of respectively P and Q we will write X q for the application (Xp)q and ....

PRATT, V. Modelling concurrency with partial orders. International Journal of Parallel Programming, 1 (1986).

....equivalences results are known, see [6, 17] In [26] we compared standard language equivalence with two generalisations based on traditional approaches capturing non interleaving behaviour. The rst known as pomset equivalence was based on pomsets representing global causal dependency [22], and the second known as location equivalence on locality [4] representing spatial distribution of events. The two notions of non interleaving equivalences were shown to be decidable on BPP contrasting the result of Hirshfeld [10] that language equivalence is undecidable. Moreover, larger ....

....knowledge rst positive decidability result for a natural weak behavioural equivalence on the full class of BPP Many other non interleaving equivalences exist besides our chosen pomset and location equivalences, and which deserve to be explored. For instance, the augmentation closure of Pratt [22] is an obvious candidate. Also we would like to emphasise that we do not claim that our notion of location equivalence is the only natural capture of spatial distribution, other possibilities exist. We showed that pomset and location equivalence are decidable on BPP (BPP ) with renaming and ....

V.R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):33-71, 1986.

....s for the subcategory with strict morphisms. In event structures a configuration, to be thought of as a computation path, carries more structure than simply a string of actions. A configuration inherits the shape of a pomset from the causal dependency and labelling of the event structure. Pomsets [9] are partial orders of labelled events and so can be identified with special event structures where all finite subsets of events are consistent. Definition 2. Say a pomset is prime when it has a top event e with respect to the causal dependency relation # so its set of events is [e] 2 ....

....to the di#erent ways one pomset can extend another. Definition 3. Define Pom to be the full subcategory of event structures E with objects finite pomsets. Define Pom s to be the subcategory of Pom where all morphisms are strict morphisms. An epimorphism in Pom is called an augmentation (following [9], though note the switch of direction relative to loc. cit. It is clear that all isomorphisms in Pom are augmentations (and strict) and that restricting to augmentation morphisms also yields a subcategory of Pom. Proposition 4. In Pom, any morphism f : P Q factors uniquely to within ....

Pratt, V.R., Modelling concurrency with partial orders. International Journal of Parallel Programming, 15,1, p.33-71, Feb. 1986.

....sequence of consecutive transitions starting at the initial state. For the noninterleaving (or independence) model of event structures a computation path is reasonably taken to be a slight generalisation of this to a partial order of events, and as the events are labelled, this amounts to a pomset [9]. The morphisms described, say f : X Y , quite naturally have the feature that they preserve behaviour in the sense of sending computation paths of X to computation paths of Y . Roughly, open maps are required to satisfy the additional property that they preserve labels and, whenever a path of X ....

Pratt, V.R., Modelling concurrency with partial orders, International Journal of Parallel Programming, 15,1, pp. 33--71, 1986.

.... X to X is a function f : S S ffl f(i) i ffl for all transitions s Gamma s 1 in X, f(s) Gamma f (s ) in X ffl s Gamma s 1 I u Gamma u 1 in X implies f(s) Gamma f (s 1 ) I f(u) Gamma f (u 1 ) in X As observations it is naturally to take Pratt s pomsets [Pra86]. We identify the category Pom L of pomsets with its full and faithful embedding in TI L (for details see [JNW93] The category TI L has pullbacks, so Pom L bisimilarity is an equivalence relation in TI L . The following proposition characterizes Pom L open morphisms in TI L [JNW93] ....

V. R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1), 1986.

....those special synchronisation trees consisting of a finite single branch of nonzero length. Define L # to be the extension of L to all finite strings. To take account of the added independence structure of event structures, the shape of their computation paths is taken to be a finite pomset [25]. Definition 1.5 (Labelled pomsets) Let L be a set. The category Pom L is taken to be the subcategory of L , for a labelling set L, consisting of those finite non empty event structures in which all subsets of events are in the consistency relation. In other words the objects P of Pom L are ....

V. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Processing, 15:33--71, 1986.

....to identify strings L # with the equivalent subcategory of consisting of those special synchronisation trees consisting of a finite single branch. To take account of the added independence structure of event structures, the shape of their computation paths is taken to be a finite pomset [111]. Definition 2.2.2 (Labelled pomsets) Let L be a set. The category PomL is taken to be the subcategory of L , for a labelling set L, consisting of those finite event structures in which all subsets of events are in the consistency relation. In other words the objects P of PomL are triples P = ....

Vaughan Pratt. Modelling concurrency with partial orders. International Journal of Parallel Processing, 15:33--71, 1986.

....preserves the trace of Kahn. It must be possible to represent di#erent dependencies between input and output for a particular inputoutput pair in the relation. This is precisely what moving to the bicategory of profunctors does for us. The traces can also be viewed as a specific kind of pomsets [110] and the path categories as a subcategory of the category of pomsets given in [71] 113 8.3.1 Profunctors The (bi)category Prof of profunctors, or bimodules, or distributors [15] are a categorical generalisation of sets and relations. The objects of Prof are small categories and arrows are ....

Vaughan R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):33--71, february 1986.

....However, let us define a pomset to be an almost totally ordered multiset iff it is of one of the two simple forms considered in the proof of Proposition 16, i.e. allowing at most two (maximal) elements to be unordered. Note that in the range of subclasses of pomsets considered in the literature, [19], this class is as close to BranL as one can get Let us denote the full subcategory of PomL consisting of object of this simple form by AtomL . Corollary 21 (i) A morphism in NL is PomL open iff it is AtomL open. ii) Two nets are PomL bisimilar iff they are AtomL bisimilar. Proof: ....

Pratt, V.R., Modelling concurrency with partial orders, International Journal of Parallel Programming, 15,1, pp. 33--71, 1986.

....the set E L Gamma1 (P ) is totally ordered by P (process line ordering) In general, the visual order provides more ordering than intended by the designer. Therefore every chart has an associated causal structure providing the intended ordering [1] Causal structures are related to pomsets [9], event structures [8] and Mazurkiewicz traces [4] A causal structure is obtained from a chart by means of a given semantics, which depends on the system architecture. Formally, the causal structure of a chart M = E ; P ; L; T ) is given as tr(M) E ; OE; P ; L; T ) where the only ....

V. R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):33--71, 1986.

....line. This imposes a total order among all events of P , for every process P . In general, the visual order provides more ordering than intended by the designer. Therefore we associate with every chart a causal structure providing the intended ordering. Causal structures are related to pomsets [11], event structures [9] and traces [5] A causal structure is obtained from an MSC by means of a given semantics. Formally, the causal structure of an MSC M is a quintuple tr(M ) hE; OE; L; T; Pi, where the only component that differs from the definition of an MSC is the relation OE, called the ....

V. R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):33--71, 1986. 17

....it models that, after nishing one task, a system can start another one. Therefore, the natural mathematical model is that of a (free) monoid . To model not only the behavior of a sequential system, but also allow parallelism, labeled partially ordered sets or pomsets were suggested [Gra81,Pra86,Gis88] In this setting, there is not only one, but there are (at least) two natural operations: A parallel system can start a new job after nishing the rst one, or it can perform two jobs in parallel. These two operations are mathematically modeled by the sequential and the parallel product on ....

V. Pratt. Modelling concurrency with partial orders. Int. J. of Parallel Programming, 15:33{ 71, 1986.

....Foundation (DFG) 1 pomsets without auto concurrency: concurrent events must have different labels. These pomsets are called semi words in [Sta81, Die94] For studies how general pomsets can be used to represent parallel processes and how they can be composed, we refer the reader e.g. to [Pra86, Gis88]. There are several ways to describe the behaviors of a system. For instance, logic formulas are suited for specification purposes. Depending on the properties we have to express, we can use various logics such as temporal logics, first order logics or (monadic) second order logics. On the other ....

V. Pratt. Modelling concurrency with partial orders. Int. J. of Parallel Programming, 15:33--71, 1986.

....The edge relation can be chosen to be the covering relation and concurrent events must have di erent labels. These pomsets are called semi words in [Sta81,Die94] For studies how general pomsets can be used to represent parallel processes and how they can be composed, we refer the reader e.g. to [Pra86,Gis88]. Dependence graphs (known from the theory of Mazurkiewicz traces [DR95] are another example of dags that model concurrent behaviors: A dependence graph is a dag whose edges are dictated by a static dependence relation between the actions of the system. Asynchronous cellular automata (ACAs for ....

V. Pratt. Modelling concurrency with partial orders. Int. J. of Parallel Programming, 15:33-71, 1986.

....recognizes an event and assigns a domain dependent symbol to it. The event symbols together with their attributes are then inserted into the database. Example systems for symbolic processing in activity recognition are provided in [1] The use of partial orders to model concurrency is studied in [7], while [2, 6] are examples of the design of partially ordered databases. The main di#erence between our data model and that of others is in fixing of the temporal relation between eventss and identification of a specific class of partial orders, namely semiorders. With appropriate semantic ....

V. R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1), 1986.

....it models that, after nishing one task, a system can start another one. Therefore, the natural mathematical model is that of a (free) monoid . To model not only the behavior of a sequential system, but also allow parallelism, labeled partially ordered sets or pomsets were suggested [11, 18, 10]. In this setting, there is not only one, but there are (at least) two natural operations: A parallel system can start a new job after nishing the rst one, or it can perform two jobs in parallel. These two operations are mathematically modeled by the sequential and the parallel product on ....

V. Pratt. Modelling concurrency with partial orders. Int. J. of Parallel Programming, 15:33-71, 1986.

....t 0 u a u 0 t j u t t j u 0 TABLE 5. The operational semantics of pccs has shown CSP is completely expressive for de Simone format Can the partial order used to model concurrent prioritized actions be unified with the partial order structures investigated by MAZURKIEWICZ (1986) PRATT (1986), WINSKEL (1989) and many others The appendices contain detailed comparisons of PPA with three other approaches to priority. A Translating prioritized CCS into PPA CLEAVELAND and HENNESSY s (1988) prioritized variant of CCS has actions which are either low priority (written a) or high priority ....

PRATT, V. (1986). Modelling concurrency with partial orders. Internat. J. Parallel Programming, 15(1):33--71.

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Pratt, V. (1986). Modelling concurrency with partial orders. Int. Journal of Parallel Programming, 15(1):33--71.

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V. Pratt (1986), Modelling Concurrency with Partial Orders. International Journal of Parallel Programming 15, 33--71, Plenum.

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V. Pratt (1986), Modelling Concurrency with Partial Orders. International Journal of Parallel Programming 15, 33--71, Plenum. 17

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V. R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15#1#:33#71, Feb. 1986.

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