P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proc. of LICS'89, pages 175-185, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this condition or its negation b. A process of a net is a model of a run or of a segment of a run of the represented system. Formally, such a process is defined as a particular net together with a particular mapping to the safe completion of the original net (cf. Petri 77] RT 86] BD 87] DMM 89] Eng 91] 2.3. Definition. A causal net is a net N = B; E; F ) such that (1) F , the reflexive and transitive closure of the flow relation F , is a partial order (2) for every b 2 B there exists at most one e 2 E satisfying eF b, and at most one e 2 E satisfying bF e . We say ....

Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in Proc. of 4th LICS Symposium, IEEE (1989) 175-185

....that the correspondence between processes and their characteristics is a homomorphism. Keywords: Contextual net, process, sequential composition, parallel composition, interchange, table, table multiplication, juxtaposition of tables, interchange of tables, homomorphism. 1. Introduction In [DMM 89] the authors defined concatenable processes of Place Transition Petri nets and operations of sequential and parallel composition of such processes. Under these operations, the processes of each net formed an algebra. In [Wnk 97] the author defined matrix like characteristics of concatenable ....

....97] the author defined matrix like characteristics of concatenable processes of the above nets and introduced operations on these characteristics that made the correspondence between processes and their characteristics a homomorphism. In the present paper we extend the definitions and results of [DMM 89] and [Wnk 97] to contextual nets similar to those in [MR 95] cf. also [BP 96] and [Ris 94] Contextual nets are models of concurrent systems with context dependent actions. A contextual net is shown in figure 1. Circles represent places in which some resources, called tokens, may reside. ....

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Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....can be observed in modelling real time systems, which implies a need of a representation of the lapse of time. To meet this need various solutions has been proposed known as timed Petri nets (cf. Ram 74] For the usual Petri nets there exist precise characterisations of behaviours (cf. DMM 89] for example) In the case of timed Petri nets the situation is less advanced since the existing semantics either do not reflect properly concurrency (cf. GMMP 89] for a review) or they oversimplify the representation of the lapse of time (as in [BG 92] In this paper we try to fill up this ....

....a review) or they oversimplify the representation of the lapse of time (as in [BG 92] In this paper we try to fill up this gap by representing the behaviour of a timed net by an algebra of structures called concatenable weighted pomsets. These structures correspond to concatenable processes of [DMM 89] with some extra information about the lapse of time. If the lapse of time is represented only in terms of delays between situations then we call such structures free time consuming processes. If also the time instants at which situations arise are given then we call them timed time consuming ....

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Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....the conditions m Phi (pre N (t) pre N 0 (m(t) for all t 2 TN , m Phi (post N (t) post N 0 (m(t) for all t 2 TN , and m Phi (cxt N (t) cxt N 0 (m(t) for all t 2 TN . Concatenable processes of a net N (called in the sequel processes) can be defined as follows (cf. DMM 89] and [MR 95] A process of the first type is an equivalence class of structures of the form A = X; Y; pred; suc; at; m; i; j) called contextual process instances, where: NA = X; Y; pred; suc; at) is a finite contextual net with the following properties (a contextual process net) ....

.... definitions it follows that the correspondence ff 7 table(ff) between processes of a net N and PN matrices over (T N ) Phi is a homomorphism from the algebra PROCESSES(N ) to PMATRICES(PN ; T N ) Phi ) cf. Wnk 95] 4 A characterization of matrices corresponding to processes In [DMM 89] it has been shown that each processes ff of a context free net N can be obtained by combining symmetries and one token and one event processes of N in the sense that it can be represented in a sequential form oe 1 ; ff 1 ; oe n ; ff n ; oe n 1 ; where oe 1 , oe n ,oe n 1 are ....

Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

.... commutative processes of N in the sense of Best and Devillers [11] This result, showing that the equational theory of true concurrency provided by rewriting logic agrees with more traditional notions of true concurrency in the case of Petri nets, has been proved by Degano, Meseguer and Montanari [28, 29] using an earlier categorical model of Petri net computations denoted T [N ] 80] that is in fact identical to TN . Since Petri nets are in some ways a very simple concurrency model, in practice it is often convenient to specify systems at a higher level, yet using the same basic properties of ....

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing net computations and processes. In Proc. LICS'89, pages 175--185. IEEE, 1989.

....when they could have happened in any order, because they affect independent subsystems. These features make net models suitable for representing in a satisfactory way concurrent and distributed systems in many interdisciplinary applications. Meseguer and Montanari in [22,23] and successively in [12,13,31,32,6,14] several authors) have recasted these facts in algebraic terms to unveil properties of net computations and, especially, of the intrinsic concurrency of the net model. The underlying idea of the so called Petri nets are monoids approach is to lift the algebraic structure of states to the level ....

....9. a; b;s) a;b) s a;id q =id p ; a = a (a;b) a 0 ; b 0 ) a a 0 ) b b 0 ) a (b s) a b) s a id =id a = a id p q =id p id q (a b) g q;q =g p;p ; b a) g p;q ; g q;p =id p id q g p;q r = g p;q id r ) id q g p;r ) Figure 10. Q (N) for PT nets [12,32] and, therefore, take a non commutative monoid of objects: it is commutative only on some objects, in particular on the markings. We use the symbol for the monoidal operation. Definition 11. For N = S;T; 0 ; 1 ; V) a contextual net, B(N) is the monoidal category with objects the bimolecules ....

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P. Degano, J. Meseguer, and U. Montanari. Axiomatizing net computations and processes. In Proc. LICS'89, 4th Symposium on Logic in Computer Science, pp. 175--185. IEEE Computer Society Press, 1989.

....such that the correspondence between processes and their characteristics is a homomorphism. Keywords: Contextual net, process, sequential composition, parallel composition, interchange, table, table multiplication, juxtaposition of tables, interchange of tables, homomorphism. 1. Introduction In [DMM 89] the authors defined concatenable processes of Place Transition Petri nets and operations of sequential and parallel composition of such processes. Under these operations, the processes of each net formed an algebra. In [Wnk 97] the author defined matrix like characteristics of concatenable ....

....97] the author defined matrix like characteristics of concatenable processes of the above nets and introduced operations on these characteristics that made the correspondence between processes and their characteristics a homomorphism. In the present paper we extend the definitions and results of [DMM 89] and [Wnk 97] to contextual nets similar to those in [MR 95] cf. also [BP 96] and [Ris 94] Contextual nets are models of concurrent systems with context dependent actions. A contextual net is shown in figure 1. Circles represent places in which some resources, called tokens, may reside. Boxes ....

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Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

.... an initial graph, or processes with an initial and with a resulting graph, may be equipped respectively with some canonical representation of the initial graph, or with some canonical representations of the initial and of the resulting graph, and then, by analogy with concatenable processes of [DMM 89] called respectively left concatenable or concatenable dynamic graphs. Such graphs may be concrete or abstract, that is considered up to isomorphism. An important feature of abstract left concatenable and concatenable dynamic graphs is that one can concatenate two such graphs if the resulting ....

Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in Proc. of 4th LICS Symposium, IEEE, 1989, 175-185

....timed process, actual timed process. 1. Motivation The idea of representing behaviours of concurrent systems with the aid of partial orders has appeared to be fruitful. From one side, it has allowed to develop an adequate theory of Petri nets (cf. P 77] Maz 77] Wi 80] Wi 82] GR 86] DMM 89] for example) From the other side, it has allowed to reduce dramatically the computational complexity of practical analysis of concurrent systems (cf. GW 91] GK 91] Pe 93] PP 95] for example) The core of this idea is that all possible processes of a concurrent system are represented ....

....[PP 95] for example) The core of this idea is that all possible processes of a concurrent system are represented as partially ordered sets of executions of actions, or occurrences of state components, or both, where the lack of order between elements represents their causal independence. In [DMM 89] such sets are equipped with some extra arrangements of their minimal and maximal elements, which allows to define a sort of concatenation. However, with processes thus represented only those features of concurrent systems can be reflected which can be expressed in terms of causality and choice. ....

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Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....represented by labelled partially ordered sets of a sort and operations of composing such processes sequentially and in parallel have been introduced. In [W 82] the algebras of processes thus obtained have been characterized as partially monoidal categories satisfying some specific axioms. In [DMM 89] and [DMM 91] similar but more advanced results have been presented for arbitrary Place Transition nets. In this paper we are interested in time consuming computations as those in timed Petri nets with tokens carrying information. We want to describe such computations by specifying the tokens ....

Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....of data rather than states, and processes caused by such markings should represent reactions to the respective streams. Solution We represent the behaviour of a timed net by an algebra of structures called concatenable weighted pomsets. These structures correspond to concatenable processes of [DMM 89] with some extra information about the lapse of time, and they can be combined with the aid of operations similar to those on concatenable processes (a sequential and a parallel composition and interchanges) The concatenable weighted pomsets represent processes of the considered net, where a ....

....of adequacy, economy, aggregability, and compositionality, and so we replace it by a representation in terms of free and timed processes. Our idea of defining free and timed processes of a timed net as combinations of executions of transitions and presences of tokens in places follows that in [DMM 89] where processes of an usual Petri net are defined as morphisms of a monoidal category which is freely generated by the set of transitions and the set of places. In the context of timed nets a similar idea has been exploited in [BG 92] where processes of a timed net are represented by ....

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Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....N 0 , m(T r N ) m(T r N 0 ) and the unique extension of m to a monoid homomorphism m Phi : P l Phi N P l Phi N 0 satisfies the conditions m Phi (pre N (t) pre N 0 (m(t) for all t 2 T r N and m Phi (post N (t) post N 0 (m(t) for all t 2 T r N . According to [DMM 89] the behaviour of a net N can be represented by the set of its concatenable processes (called in the sequel processes) a process being an equivalence class of process instances of the form A = P; T; pred; suc; m; i; j) such that (1) NA = P Phi ; T; pred; suc) is a finite net with the ....

.... over S, PMATRICES(V;S) From the respective definitions it follows that the correspondence ff 7 table(ff) between processes of a net N and P l N matrices over (T r N ) Phi is a homomorphism from PROCESSES(N ) to PMATRICES(P l N ; T r N ) Phi ) cf. Wnk 95] 4 Solution In [DMM 89] it has been shown that each processes ff of a net N can be obtained by combining symmetries and one place and one transition processes of N in the sense that it can be represented in a sequential form oe 1 ; ff 1 ; oe n ; ff n ; oe n 1 ; where oe 1 , oe n ,oe n 1 are symmetries and ff ....

Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....Definiuje sie operacje sk ladania proces ow tak, ze kazdy proces sieci kontekstowej mozna otrzyma c sk ladajac procesy odpowiadajace miejscom i tranzycjom tej sieci. Key words: Contextual net, process, sequential composition, parallel composition, interchange. 3 4 1 Introduction In [DMM 89] concatenable processes of Place Transition Petri nets and operations of composing such processes sequentially and in parallel have been defined such that processes of each net form an algebra. In the present paper we extend the definitions and results of [DMM 89] on processes of contextual nets ....

....3 4 1 Introduction In [DMM 89] concatenable processes of Place Transition Petri nets and operations of composing such processes sequentially and in parallel have been defined such that processes of each net form an algebra. In the present paper we extend the definitions and results of [DMM 89] on processes of contextual nets similar to those in [MR 95] Contextual nets are models of concurrent systems with context dependent actions. A contextual net is shown in figure 1. Circles represent places in which some resources called tokens may reside. Boxes represent actions called ....

[Article contains additional citation context not shown here]

Degano, P., Meseguer, J., Montanari, U., Axiomatizing Net Computations and Processes, in the Proceedings of 4th LICS Symposium, IEEE, 1989, pp.175-185

....in Computer Science, Centre of the Danish National Research Foundation. # Supported by EU Human Capital and Mobility grant ERBCHBGCT920005. ### Our first aim, therefore, is to define a set CM of combinators of Petri nets. Several works have focused on algebraic aspects of Petri nets, e.g. [16, 23, 11, 5, 7, 3, 2, 14, 21]. Here we shall consider a minimal set of combinators, focusing on (versions of) parallel composition, relabelling, restriction, and recursion, and allowing a rather general form of interaction. Similarly to [2] our approach is entirely based on a notion of interface for Petri nets, introduced in ....

P. Degano, J. Meseguer, and U. Montanari, "Axiomatizing Net Computations and Processes", in Proceedings of the #th LICS Symposium, pp. ###--###, IEEE Press, ####.

....hoc chosen, category of symmetries. In this paper we give a fully equational description of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets. Introduction C oncatenable processes of Petri nets have been introduced in [3] to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. ....

....to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. [14, 5, 2, 12, 3]. Generally speaking, Petri net processes whose standard version is given by the Goltz Reisig non sequential processes [5] are structures needed to account for the causal relationships which rule the occurrence of events in computations. Thus, ideally, processes are simply computations in ....

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P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proceedings of the #th LICS Symposium, pp. ###--###, IEEE, ####.

....operations at the level of a concurrent programming language, such as various forms of parallel and non deterministic composition [Win87, MM90] A unification of the process oriented and algebraic views has recently been proposed by two of the present authors in joint work with P. Degano [DMM89], by showing that the commutative processes [BD87] of a net N are isomorphic to the arrows of a symmetric monoidal category T [N ] Moreover, they introduced the concatenable processes of N a slight variation of GoltzReisig processes [GR83] on which sequential composition is defined and ....

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In proceedings of 4th LICS Symposium, pp. 175--185, IEEE, 1989.

....Since the transitions of the transition system are not just proofs but rather normalized proofs, their identity carries a lot of semantic information about the behaviour of the model, especially when concurrency is involved. For instance, recent work by Degano, Meseguer and Montanari [64] shows that in the case of Petri nets (corresponding to the tensor fragment of linear logic) the normalized proofs are Petri nondeterministic processes, a well studied model of (deterministic) concurrent behaviour. It will be interesting to study if similar characterizations hold also for ....

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing net computations and processes. In Proc. 4th Symp. on Logics in Computer Science, pages 175--185. IEEE, 1989.

....Approach to the Category of Net Computations Vladimiro Sassone ffi BRICS Computer Science Dept. University of Aarhus Abstract. We introduce the notion of strong concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a functor Q[ from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net N , the strong concatenable processes of N are isomorphic to the arrows of Q[N ] In addition, we ....

....to the understanding of nets, they fail, in our view, to give a comprehensive account of net behaviours. The idea of looking at nets as algebraic structures [10, 7, 13, 14, 2] has been given an original interpretation by considering monoidal categories as a suitable framework [6] In fact, in [6, 3] the authors have shown that the semantics of Petri nets can be understood in terms of symmetric monoidal categories where objects are states, arrows processes, and the tensor product and the arrow composition model, respectively, the operations of parallel and sequential composition of ....

[Article contains additional citation context not shown here]

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proceedings of the 4th LICS Symposium, pp. 175--185, IEEE, 1989.

....category of symmetries Sym N . In this paper we give a completely abstract characterization of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets. Introduction Concatenable processes of Petri nets have been introduced in [3] to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. ....

....to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. [14, 5, 2, 12, 3]. Generally speaking, Petri net processes whose standard version is given by the Goltz Reisig non sequential processes [5] are structures needed to account for the causal relationships which rule the occurrence of events in computations. Thus, ideally, processes are simply computations in ....

[Article contains additional citation context not shown here]

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proceedings of the 4th LICS Symposium, pp. 175--185, 1989.

....as a category, yielding in this way an axiomatization of its space of computations. One may call this use of categories in the small , as opposed to their use in the large to study the global properties of the entire class of nets as illustrated above. This idea has been exploited in [6], where it is shown that the commutative processes [2] of a net N are isomorphic to the arrows of a symmetric monoidal category T [N ] Moreover, 6] introduced the concatenable processes of N a slight variation of Goltz Reisig processes [8] and structured them as the arrows of the symmetric ....

....opposed to their use in the large to study the global properties of the entire class of nets as illustrated above. This idea has been exploited in [6] where it is shown that the commutative processes [2] of a net N are isomorphic to the arrows of a symmetric monoidal category T [N ] Moreover, [6] introduced the concatenable processes of N a slight variation of Goltz Reisig processes [8] and structured them as the arrows of the symmetric monoidal category P[N ] In particular, the distributivity of tensor product and arrow composition in monoidal categories is shown to capture the ....

[Article contains additional citation context not shown here]

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proceedings of the 4th LICS Symposium, pp. 175--185, IEEE, 1989.

....concurrency model for nets. Namely, when the finite concurrent computations of a net N are formalized as arrows in the category TN , they exactly correspond to the commutative processes of N in the sense of Best and Devillers [13] This result has been proved by Degano, Meseguer and Montanari [40, 41] using an earlier categorical model of Petri net computations denoted T [N ] 122] that is in fact identical to TN . Since Petri nets are in some ways a very simple concurrency model, in practice it is often convenient to specify systems at a higher level, yet using the same basic properties of ....

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing net computations and processes. In Proc. LICS'89, pages 175--185. IEEE, 1989.

.... of using an additional relation of weak causality to be able to fully express the causal structure of net computations in the presence of read or inhibitor arcs can be found already in [20, 21] 13 The papers [34, 17, 38] extend the theory of concatenable processes of traditional nets [15] to c nets, by showing that the concatenable processes of a c net N form the arrows of a symmetric monoidal category CP[N ] where objects are the elements of the free commutative monoid over the set of places (multisets of places) In particular, in [17] a purely algebraic characterization of ....

....a c net N , the marking coincides with the initial marking of N . 8. 2 Concatenable processes As in [17, 38] a notion of concatenable process for contextual nets, endowed with a notion of sequential (and parallel) composition, can be easily de ned, generalizing the concatenable processes of [15]. Obviously a meaningful operation of sequential composition can be de ned only on the unmarked processes of a c net. In order to properly de ne the operation of concatenation of processes, we need to impose a suitable ordering over the places in min( and max( for each process . Such ordering ....

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proc. 4th Annual Symposium on Logic in Computer Science, Asilomar, CA, USA, pages 175-185, 1989.

....cg. 2 Review of Concatenable Processes. Causal firing sequences establish a correspondence among the tokens produced and consumed via firings. This is due to the implicit orders which are imposed on the markings and is strictly related to a process view of computations. Concatenable processes (Degano et al. 1989; Degano et al. 1996; Sassone 1996) are obtained from processes by imposing a total ordering on the origins that are instances of the same place and, similarly, on the destinations. A net K is a causal net if 8a 2 SK ; j ffl aj 1 ja ffl j 1 and F K is acyclic (F denotes the reflexive ....

....Petri net theory can be profitably developed within category theory. Among the existing approaches we mention (Winskel 1987; Meseguer and Montanari 1988; Brown and Gurr 1990) We follow the approach initiated by Meseguer and Montanari (1988) other references are (Meseguer and Montanari 1990; Degano et al. 1989; Degano et al. 1996; Meseguer et al. 1996; Meseguer et al. 1998) which focuses on the monoidal structure of Petri nets, where the monoidal operation means parallel composition. The basic observation is that a (place transition) Petri net is just a graph (S Phi ; T; 0 ; 1 ) where the set ....

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Degano, P., Meseguer, J., and Montanari, U. (1989), Axiomatizing Net Computations and Processes, in "Proceedings 4th LICS Symposium", pp. 175-185, IEEE.

....of arrows corresponds to the concatenation of derivations. Models of computation of this kind (i.e. categories of computations, possibly equipped with some additional algebraic structure) have been proposed (with various names) for many formalisms, like Phrase Structure Grammars [47] Petri nets [48,49], CCS [50] Logic Programming [51,52] and others. Although in this section we focus on the DPO approach, it is obvious that models of computation can be defined easily for the SPO approach as well, and also for all other approaches to graph transformation. A concrete model of computation for a ....

P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proceedings 4th Annual Symposium on Logic in Computer Science, pages 175--185, 1989.

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P. Degano, J. Meseguer, and U. Montanari. Axiomatizing Net Computations and Processes. In Proc. of LICS'89, pages 175-185, 1989.

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