J. Meseguer, U. Montanari, and V. Sassone (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freksa et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....SH . It is easy to verify that this forms the counit of the adjunction. 1.4. Monoidal Categories. Several interesting aspects of Petri net theory can be profitably developed within category theory, see e.g. 21, 11, 2] Here we focus on the approach initiated in [11] other relevant references are [5, 13, 19, 15, 20]) which exposes the monoidal structure of Petri nets under the operation of parallel composition. In [11, 5] it is shown that the sets of transitions can be endowed with appropriate algebraic structures in order to capture some basic constructions on nets. In particular, the commutative processes ....

J. MESEGUER, U. MONTANARI, AND V. SASSONE (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freska et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.

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J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In Foundations of Computer Science: Potential - Theory - Cognition, vol. 1337 of Lect. Notes in Comput. Sci., pages 239--249. Springer, 1997.

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J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In Ch. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: Potential - Theory - Cognition, to Wilfried Brauer on the occasion of his sixtieth birthday, volume 1337 of Lect. Notes in Comput. Sci., pages 239--249. Springer Verlag, 1997.

....SH . It is easy to verify that this forms the counit of the adjunction. 1. 4 Monoidal Categories Several interesting aspects of Petri net theory can be profitably developed within category theory, see e.g. 21, 11, 2] Here we focus on the approach initiated in [11] other relevant references are [5, 13, 19, 15, 20]) which exposes the monoidal structure of Petri nets under the operation of parallel composition. In [11, 5] it is shown that the sets of transitions can be endowed with appropriate algebraic structures in order to capture some basic constructions on nets. In particular, the commutative processes ....

J. MESEGUER, U. MONTANARI, AND V. SASSONE (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freska et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.

.... body of the paper, this work, including the abovementioned functorial semantics and the semantic equivalences, generalizes in some ways, and complements in others, a substantial body of work initiated by the second author in joint work with Ugo Montanari under the motto Petri nets are monoids [53, 45, 46, 56, 21, 54, 55, 57, 12, 13], in which categorical models are naturally associated as semantic models to Petri nets, and are shown to be equivalent to well known true concurrency models. Our work is also related to linear logic representations of Petri nets [45, 46, 4, 11, 10, 26] All this is not surprising, since, as ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freska, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: Potential, Theory, Cognition, volume 1337 of Lecture Notes in Computer Science, pages 239-249. Springer-Verlag, 1997.

....4 . 4 Note that a Petri net closely resembles a hyper signature, modulo the commutativity of the multiset operator. In fact, considering net processes as terms of the (symmetric or strictly symmetric) monoidal theory associated to a net, lies at the heart of the Petri nets are monoids approach [26,59]. See also Section 3.3. 14 Proposition 3.1 (correspondence for firing semantics) Given a Petri net N and a marking M for SN , M ) M 0 holds if and only if the sequent [s M ] s M 0 ] 0 1 is entailed by RN . Alternatively, the multiset structure of markings could be recovered by ....

....of a (possibly commutative) monoid, freely generated from a set of basic rules, considered here as the founding elements of the monoid. Thus, transitions may recover information on the spatial distribution of a system, as shown by the correspondence results with the process semantics for nets [26,59]. Structured transition systems [18,22,29] represent an obvious generalization of the paradigm; states may enjoy now a rather complex structure, which is lifted to the level of transitions, still assuming the basic rules as the founding ingredients. Rewriting logic (in its unconditional version) ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: potential - theory - cognition, volume 1337 of Lect. Notes in Comp. Science, pages 239--249. Springer Verlag, 1997.

.... We conclude by de ning the inverse functor of H, i.e. proving H surjective (thanks to the acyclicity of our structure, this can be easily done by considering any total sorting of the ow relation) 2 The use of symmetric monoidal categories as an algebraic semantic framework for net processes [6,29] guarantees that symmetric spaces o er a natural structure for those net based systems where causality information plays a central role. Using spaces, Petri net computations nd an immediate representation in terms of places as sorts, tokens as variables, and transitions as operations. In ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: Potential - Theory - Cognition, to Wilfried Brauer on the occasion of his sixtieth birthday, volume 1337 of Lect. Notes in Comp. Science, pages 239-249. Springer Verlag, 1997.

....3.3. 4 Note that a Petri net resembles closely a hyper signature, modulo the commutativity of the multi set operator. In fact, considering net processes as terms of the (symmetric or strictly symmetric) monoidal theory associated to a net, lies at the heart of the Petri nets are monoids approach [26,59]. See also Section 3.3. 14 3.2.2 Encoding substitution: on the calculus As Petri nets represent a foundational paradigm for distributed computations, the untyped calculus is universally acknowledged as a canonical representative for functional languages. Roughly, each element t of the ....

....of a (possibly commutative) monoid, freely generated from a set of basic rules, considered here as the founding elements of the monoid. Thus, transitions may recover information on the spatial distribution of a system, as shown by the correspondence results with the process semantics for nets [26,59]. Structured transition systems [18,22,29] represent an obvious generalization of the paradigm; states may enjoy now a rather complex structure, which is lifted to the level of transitions, still assuming the basic rules as the founding ingredients. Rewriting logic (in its unconditional version) ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: potential - theory - cognition, volume 1337 of LNCS, pages 239--249. Springer Verlag, 1997.

.... = associativity: 0 = 0 ) 0 = 0 ) identities: id u=id v ; id u id v = id u v , functoriality: 0 ; 0 ) 0 ) 0 ) Table 3. nets are monoids approach initiated in [30] see also [21,31,43,32,44,12]) The idea is to extend (part of) the algebraic structure of states to the level of proof terms associated to the rules in Tables 1 in such a way to capture the basic laws of concurrent and causal computations. The proof terms we consider are inductively de ned in Table 2. In [30,21] it is shown ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science (dedicated to W. Brauer), volume 1337 of Lect. Notes in Comput. Sci., pages 239-249. Springer Verlag, 1997. 45

.... of a net can still be understood in terms of symmetric monoidal categories, but all the proposed constructions do not work properly in the large, i.e. they fail to preserve at the semantic level some ordinary simulation morphisms between nets given at the level of theories (cf. 11, 34] see [24] for an overview) As recalled above, the functoriality (and hence the lifting of simulation morphisms on nets to the level of computation models) is instead an essential property for guaranteeing the compositionality of the semantic framework. 2 Since we restrict our attention to strict ....

....the firing of t does not affect the enabling condition of t 0 . 1.1. Collective Token Semantics Several interesting aspects of Petri net theory can be profitably developed within category theory, see, e.g. 35, 21, 5] We focus on the approach initiated in [21] other relevant references are [11, 23, 33, 24, 34]) which reveals the monoidal structure of Petri nets under the operation of parallel composition. In [21, 11] it is shown that the sets of transitions can be endowed with appropriate algebraic structures in order to capture some basic constructions on nets. In particular, the commutative processes ....

J. Meseguer, U. Montanari, and V. Sassone (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freksa et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.

.... We conclude by de ning the inverse functor of H, i.e. proving H surjective (thanks to the acyclicity of our structure, this can be easily done by considering any total sorting of the ow relation) 2 The use of symmetric monoidal categories as an algebraic semantic framework for net processes [6,29] guarantees that symmetric spaces o er a natural structure for those net based systems where causality information plays a central role. Using spaces, Petri net computations nd an immediate representation in terms of places as sorts, tokens as variables, and transitions as operations. In ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: Potential - Theory - Cognition, to Wilfried Brauer on the occasion of his sixtieth birthday, volume 1337 of Lect. Notes in Comp. Science, pages 239-249. Springer Verlag, 1997.

.... the semantics of a net can still be understood in terms of symmetric monoidal categories, but their constructions do not work properly in the large, i.e. they fail to preserve at the semantic level some ordinary simulation morphisms between nets given at the level of theories (cf. 5,19] see [13] for an overview) More precisely, a simple variation of processes called concatenable processes is introduced in [5] which admits sequential composition and yields a symmetric strict monoidal category P(N) for each net N , but such construction is not functorial. Indeed, for N and N 0 two ....

....O , D are bijections O : O(P ) jO(P )j and D : D(P ) jD(P )j respectively. 1. 2 Categorical Semantics Several aspects of Petri net theory can be profitably developed within category theory, see e.g. 20,10] Here we focus on the approach initiated in [10] other relevant references are [5,12,18,13,19]) which exposes the monoidal structure of Petri nets under the operation of parallel composition. In [10,5] it is shown that the sets of transitions can be endowed with appropriate algebraic structures in order to capture some basic constructions on nets. For example, the commutative processes of ....

J. Meseguer, U. Montanari, and V. Sassone (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freksa et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.

....computations a pt net can perform, together with some information about the causal dependencies among the executed events. To this aim several different notions of processes have been proposed and classified accordingly to the concrete information provided needed by their underlying structure [2, 15, 12, 21]. The algebraic semantics for pt nets focus instead both on the algebraic structure of the computation space of a single net, and on the global structure of the class of all nets, providing useful operations that can be used to combine nets (e.g. parallel and nondeterministic compositions) ....

.... N [ Fnan Fnan oo Dom Pr[ Fnan Fnan oo Figure 2: From safe nets to coherent finitary prime algebraic domains. allow for an abstract formal framework where many notions can be straightforwardly formulated and many constructions find an adequate (universal) characterization [4, 32, 18, 12, 21, 20]. Categories can be used in the large, when referring to the category of nets and morphisms between them to characterize their global properties (e.g. composition as universal construction) or in the small, by noticing that the space of computation of each pt net possesses a category ....

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J. Meseguer, U. Montanari, and V. Sassone, Representation theorems for Petri nets, in: W. Brauer, C. Freksa, M. Jantzen, and R. Valk, Eds., Foundations of Computer Science, LNCS 1337, 239--249, SpringerVerlag (1997).

.... ffl We conclude by defining the inverse functor of H, i.e. proving H surjective (thanks to the acyclicity of our structure, this can be easily done by considering any sorting of the flow relation) 2 The use of symmetric monoidal categories as an algebraic semantic framework for net processes [6,30], together with the result of Theorem 4.2, guarantee that symmetric Sigma spaces offer a natural structure for those net based systems where causality information plays a central role. We recall that Petri nets, introduced by Petri in [34] see also [37] are one of the most widely used and ....

J. Meseguer, U. Montanari, and V. Sassone. Representation theorems for Petri nets. In C. Freksa, M. Jantzen, and R. Valk, editors, Foundations of Computer Science: potential - theory - cognition, volume 1337 of LNCS, pages 239--249. Springer Verlag, 1997.

....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 of nodes is the free commutative monoid S Phi over the set of ....

....net computations which is strictly related to a process understanding of the causal behaviour of a net, but which is not functorial. The main problem is that there exist reasonable morphisms of nets which cannot be extended to a monoidal functor. We report below a convincing example illustrated in Meseguer et al. 1998). Example 3.20 Consider the nets N and N 0 pictured in Fig. 9 and the net morphism f : N Gamma N 0 such that f(t i ) t 0 i , f(a i ) a 0 , f(b) b 0 and f(c) c 0 for i = 0; 1. Morphism f cannot be extended to a functor P[f ] P[N ] Gamma P[N 0 ] In fact, if such an ....

Meseguer, J., Montanari, U., and Sassone, V. (1998), Representation Theorems for Petri Nets, Festschrift in honor of Prof. Wilfried Brauer to appear.

....corresponding exactly to the construction of the abstract net in Def. 3.9. 4.1 Petri Nets are Monoids Petri net theory can be pro tably developed within category theory. Among the existing approaches we mention [20,12,3] We follow the approach initiated in [12] other references are [13,4,14,15]) This approach focuses on the monoidal structure of Petri nets, where the monoidal operation means parallel composition. The basic observation is that a Petri net is just a graph where the set of nodes is a commutative monoid freely generated by the set of places. De nition 4.1 [Graph] A graph ....

J. Meseguer, U. Montanari, and V. Sassone. Representation Theorems for Petri Nets. Festschrift in honor of Prof. Wilfried Brauer to appear.

....and yields a coreflection corresponding exactly to the construction of the causal abstract net in Def. 5. 4. 1 Review of Petri Nets are Monoids Petri net theory can be profitably developed within category theory [22, 13, 2] We follow the approach initiated in [13] other references are [14, 5, 15, 16]) 3 We say that a binary tree is complete if any internal node has exactly two children. A (place transition) Petri net is a graph (S Phi ; T; 0 ; 1 ) where the set of nodes is the free commutative monoid S Phi over the set of places S (functions 0 ; 1 : T Gamma V are called ....

....there exist reasonable morphisms of nets which cannot be extended to a monoidal functor. We illustrate 4 Symmetries cu;v for u; v 2 S Phi N denote any term obtained from c a;b for a; b 2 SN by applying recursive rules analogous to axioms (3) given in Th. 11. below the example presented in [16]. We will show that this kind of morphisms can be avoided in the category of ZS nets, our choice being justified by the necessity to preserve atomic behaviours through morphisms. Example 7. Consider the nets N and N 0 pictured below and the net morphism f : N Gamma N 0 s.t. f(t i ) t 0 ....

J. Meseguer, U. Montanari, and V. Sassone. Representation Theorems for Petri Nets. Festschrift in honor of Prof. Wilfried Brauer to appear.

....corresponding exactly to the construction of the abstract net in Def. 3.9. 4.1 Petri Nets are Monoids Petri net theory can be profitably developed within category theory. Among the existing approaches we mention [20,12,3] We follow the approach initiated in [12] other references are [13,4,14,15]) This approach focuses on the monoidal structure of Petri nets, where the monoidal operation means parallel composition. The basic observation is that a Petri net is just a graph where the set of nodes is a commutative monoid freely generated by the set of places. Definition 4.1 [Graph] A graph ....

J. Meseguer, U. Montanari, and V. Sassone. Representation Theorems for Petri Nets. Festschrift in honor of Prof. Wilfried Brauer to appear.

No context found.

J. Meseguer, U. Montanari, and V. Sassone (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freksa et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.

No context found.

J. Meseguer, U. Montanari, and V. Sassone (1997), Representation Theorems for Petri Nets, in Foundations of Computer Science, C. Freksa et al. (Eds.), Lecture Notes in Computer Science 1337, 239--249, Springer-Verlag.

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