Meseguer, J., Montanari, U., Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, Proc. LICS, 1988, pp.175-185  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... T ; pre; post; cxt) where P is a set of places, T such that P T = is a set of transitions, and pre; post; cxt : T P , where P is the free commutative monoid generated by the set P of places, are respectively a consumption function, a production function, and a context function (cf. MM 88] and [MR 95] For the purpose of this paper we assume that pre(t) cxt(t) 6= 0 and post(t) cxt(t) 6= 0 for all t 2 T , and we say that N is context free if cxt = 0, that is if cxt(t) 0 for all t 2 T . We denote by E the set P [ T , and use subscripts, EN , PN , TN , pre N , post N , cxt N ....

Meseguer, J., Montanari, U., Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, Proc. LICS, 1988, pp.175-185

....the other or concurrent 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 ....

J. Meseguer and U. Montanari. Petri nets are monoids: A new algebraic foundation for net theory. In Proc. LICS'89, 3rd Symposium on Logic in Computer Science, pp. 155--164. IEEE Computer Society Press, 1988.

.... post; cxt) where P is a set of places, T such that P T = is a set of transitions, and pre; post; cxt : T P Phi , where P Phi is the free commutative monoid generated by the set P of places, are respectively a consumption function, a production function, and a context function (cf. MM 88] and [MR 95] For the purpose of this paper we assume that pre(t) cxt(t) 6= 0 and post(t) cxt(t) 6= 0 for all t 2 T , and we say that N is context free if cxt = 0, that is if cxt(t) 0 for all t 2 T . We denote by E the set P [T , and use subscripts, EN , PN , TN , preN , post N , cxt N , when ....

Meseguer, J., Montanari, U., Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, Proc. LICS, 1988, pp.175-185

....## 1 # ) C) C : ## # ##, C # : # # #. Thus each CMeta[G] is traced. 7.4 Discussion Our attempts to define an algebra associated with SFC diagrams led us to consider various approaches to the algebraic theory of Petri nets. One influential such approach is due to Meseguer and Montanari [100]. There, a net is regarded as a graph N whose objects are the places of the net (more generally, multisets thereof) and whose arrows are the transitions. Every such graph can be freely completed to a tensor category T (N ) Although this was already close to our objective, it su#ers from a certain ....

Jose Meseguer and Ugo Montanari. Petri nets are monoids: A new algebraic foundation for net theory. In Proceedings of the Third Annual Symposium on Logic in Computer Science (LICS), pages 142--154. IEEE Computer Society Press, 1988.

....Petri net, process, sequential composition, parallel composition, interchange, partitioned matrix, multiplication, juxtaposition. 3 4 1 Problem Place Transition Petri nets, or briefly nets, are bipartite graphs representing concurrent systems (cf. GLT 80] R 85] GV 87] Wns 87] and [MM 88] for details) A graph of this type is shown in figure 1. Nodes depicted as circles represent places in which some resources called tokens may reside. Those depicted as boxes represent transitions which when executed consume tokens from places and produce tokens in places as indicated by ....

....we give a solution to the first of the two problems. The role of this solution lies in the fact that it allows one to represent processes of a net as partitioned matrices with special properties rather than as unfoldings, which is convenient for algebraic treatment. 2 Formalization According to [MM 88] a Place Transition Petri net (or briefly a net) can be defined as N = P l Phi ; T r; pre; post) where P l Phi is the free commutative monoid generated by a set P l of places, T r is a set of transitions such that P l T r = 6 ; and pre; post : T r P l Phi are respectively a ....

Meseguer, J., Montanari, U., Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, Proc. LICS, 1988, pp.175-185

.... post; cxt) where P is a set of places, T such that P T = is a set of transitions, and pre; post; cxt : T P Phi , where P Phi is the free commutative monoid generated by the set P of places, are respectively a consumption function, a production function, and a context function (cf. MM 88] and [MR 95] For the purpose of this paper we assume that pre(t) cxt(t) 6= 0 and post(t) cxt(t) 6= 0 for all t 2 T , and we say that N is context free if cxt = 0, that is if cxt(t) 0 for all t 2 T . We denote by E the set P [ T and we use subscripts, EN , PN , TN , preN , post N , cxt ....

Meseguer, J., Montanari, U., Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, Proc. LICS, 1988, pp.175-185

....6) making the decomposition unique (Lemma 5) We are going to use the above observation after moving in the next section to the category of free commutative monoids enriched with covers. 8 4 Free commutative monoids with covers Free commutative monoids are the building blocks of Petri nets, cf.[6]. In Petri net theory one is interested in a family of markings reachable from a given initial marking. Also, morphisms of Petri nets preserve the initial markings, and they map reachable markings in the source net, to reachable markings in the target net. This amounts to studying not just free ....

J. Meseguer and U. Montanari. Petri nets are monoids: a new algebraic foundation for net theory. In Proc. LICS 88, pp.:155--164, 1988.

....ffl This approach can be taken further using the concept of enrichment; Pratt s group in [ 6 ] gives several categories based on the reals with a monoidal structure. These are then used to enrich behavioural categories. Such techniques are comprehensively displayed in Kasangian and Labella s [ 15 ] . ffl Another starting point is idea that we can associate a set of predicates with a state of a system the things we know to be true of the behaviour by that point. This means that an observation is a functor F : B op Set, and all possible observations live in the topos [B op ; Set] ....

....[13] A. Jeffrey, Timed process algebra 6= time Theta process algebra, Technical Report 79, Programming Methodology Group, Chalmers University, 1991. 14] M. Joseph and A. Goswami, Relating computation and time, Technical Report RR 138, Department of Computer Science, University of Warwick, 1985. [15] S. Kasangian and A. Labella, On continuous real time agents, in Mathematical Foundations of Programming Semantics (I. Guessarian, Ed. Volume 469, Springer Verlag LNCS, 1991. 16] R. Koymans, Specifying real time properties with metric temporal logic, Real Time Systems, Volume 2 (1991) Pp. ....

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J. Meseguer and U. Montanari. Petri nets are monoids: A new algebraic foundation for net theory. Information and Computation, 88(2):105--155, 1990.

....as syntactic really have varying degrees of semantics. For example, transition systems and synchronisation trees have been used as semantics for CCS and CSP; also, CSP has a preferred model, based on failures and refusals [30] Petri nets have been used as models for linear logic (e.g. [35]) and set theoretic models have been given for Hewitt s actor approach [1] Moreover, CCS expressions have been used as models for temporal logic. One person s syntax is another person s semantics. Thanks also to the Programming Research Group, Oxford University. The research reported in this ....

....for example, to the security of real time distributed concurrent (possibly object oriented) databases. A definition of deadlock is also given, which again seems more general than anything in the literature. Category theory has by now been used in many studies of concurrency; for example, see [45, 35, 11]. But as far as I know, only Monteiro and Pereira [37] have previously studied concurrency using sheaves; however, their approach does not seem to be closely related to the present paper. Prerequisites and Notation Basic category theory and some intuition for concurrency are needed to read this ....

Jos'e Meseguer and Ugo Montanari. Petri nets are monoids: A new algebraic foundation for net theory. In Proceedings, Symposium on Logic in Computer Science. IEEE Computer Society, 1988. Full version in Report SRI-CSL-88-3, Computer Science Laboratory, SRI International, January 1988; submitted to Information and Computation.

....simple calculations. ffl Discovering and exploiting relations with other fields. Sufficiently abstract formulations can reveal surprising connections. For example, an analogy between Petri nets and the calculus might suggest looking for a closed category structure on the category of Petri nets [52]. ffl Dealing with abstraction and representation independence. In computing science, more abstract viewpoints are often more useful, because of the need to achieve independence from the often overwhelmingly complex details of how things are represented or implemented. A corollary of the first ....

....[26] 40] 56] among other places, and is mentioned in [61] 8.2 Monoidal Categories. There are many cases where a category has a natural notion of multiplication that is not the usual Cartesian product but nevertheless enjoys many of the same properties. The category of Petri nets studied in [52] has already been mentioned, and a variety of recent work suggests that monoidal categories may be broadly useful in understanding the relationships among the various theories of concurrency, e.g. see [12] 8.3 Indexed Categories. A strict indexed category is just a functor B op Cat. The ....

Jos'e Meseguer and Ugo Montanari. Petri nets are monoids: A new algebraic foundation for net theory. In Proceedings, Symposium on Logic in Computer Science. IEEE Computer Society, 1988. Full version in Report SRI-CSL-88-3, Computer Science Laboratory, SRI International, January 1988; submitted to Information and Computation.

....the ideas presented here generalise to nets in which events can fire and markings hold with multiplicities, as indicated in [99] though at present it is not known how to link up with other models via adjunctions. See also Meseguer and Montanari s study of several definitions of net morphisms [53]. Categories of Petri nets have been shown to form a model of Girard s linear logic, offering an interpretation of the logical operations of linear logic as operations on nets and of proofs as kinds of simulation morphisms like those here (see [17] Since the morphisms preserve behaviour, the ....

Meseguer, J., and Montanari, U., Petri nets are monoids: a new algebraic foundation for net theory, Proc. of LICS 88, pp. 155--164, 1988.

....computation. Tensor products (hence monoidal categories) arise naturally here when dealing with multiple inputs or outputs. Similarly, Degano, Meseguer and Montanari treat multisets as free modules over the semiring of the natural numbers to give an algebraic treatment of Petri nets [35, 14]. In this case arrows of the category represent transitions, and tensor product corresponds to running a number of transitions concurrently. In the system presented in this thesis the concurrence operation turns out to be categorical sum, not tensor product. Meseguer and Mart i Oliet [34] show ....

Jos'e Meseguer and Ugo Montanari. Petri nets are monoids: A new algebraic foundation for net theory. In Third Annual Symposium on Logic in Computer Science, pages 155-- 164, Edinburgh, New York, July 1988. The Computer Society's Technical Committee on Mathematical Foundations of Computing, Computer Society Press, Washington DC.

....if a coreflection exists between C and C 0 with the inclusion as left adjoint, then it is unique up to isomorphism. 3.1 Review of the Monoidal Structure of Petri Nets 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 ....

....it is unique up to isomorphism. 3.1 Review of the Monoidal Structure of Petri Nets 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 ....

Meseguer, J., and Montanari, U. (1988), Petri Nets are Monoids: A New Algebraic Foundations for Net Theory, in "Proceedings of the 3rd LICS Symposium", pp. 155--164, IEEE.

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Meseguer, J., Montanari, U., Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, Proc. LICS, 1988, pp.175-185

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