V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170(1-2):277--296, 1996.  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 ....

....Nets and Collective Conf. structures, CTS, Commutative processes T (N ) Nets and Individual Conc. Pomsets, Event Struct. Processes P(N ) Q(N ) Z(N ) MON SYM TABLE 1 From the logical viewpoint, it is not difficult to formulate a theory SYM of permutations and symmetries (cf. [19]) bridging the gap from strictly symmetric categories to categories symmetric only up to coherent isomorphism. On the other hand, the investigation of suitable algebraic models is still open, as our current best candidates, the symmetric strict monoidal categories P(N ) of concatenable processes ....

V. SASSONE (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170, 277--296, Elsevier.

....corresponding to linearizations of the partial ordering are considered equivalent. Most semantic and logic notions specifying the concurrent behavior of nets are based on the notion of process [23, 2] Processes play an important role in the Petri Nets are Monoids approach to net theory [19, 11, 26]. In this approach, a net N is analogous to a signature Sigma , and the symmetric monoidal category P(N ) associated to N is analogous to the cartesian category L( Sigma) of terms and substitutions freely generated by Sigma . As the (tuples of) terms in T Sigma (X) are the arrows of L( Sigma) ....

....by Sigma . As the (tuples of) terms in T Sigma (X) are the arrows of L( Sigma) the processes 1 of N are the arrows of P(N ) The construction of P(N ) provides a concise, algebraic description of the concurrent operational semantics of P T nets. Since P(N ) can be finitely axiomatized [26], this construction provides a finite axiomatization of non sequential processes. Moreover, the well understood setting of monoidal categories allows for an easy comparison with related models, like linear logic [17] The aim of this paper is to extend the above categorical approach to P T nets ....

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V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277--296, 1996.

....of De nition 2.2. This can be done by exploiting the de nition of G ; 0 and applying the de nition of parallel and sequential composition. The freeness result relies on previous characterization results for symmetric monoidal categories as suitable (strongly concatenable) Petri processes [37,36] and more recent analogous results for pre nets [6] to which our spaces are equivalent. First, let us give some terminology, to be used also in the sketched proofs of Propositions 4.5 and 4.8. For relational spaces we de ne the depth of a generator sentence x 1 : x n f ## ## y 1 : y m ....

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277-296, 1996.

....the coherence axioms of Def. 3. This can be easily done by exploiting the definition of G ; 0 and applying the definition of parallel and sequential composition. The initiality result relies on previous characterization results for symmetric monoidal categories as suitable Petri processes [31, 30], to which our spaces are equivalent. 2 4.2 GS Monoidal As illustrated in Section 2, gs monoidal categories are symmetric monoidal categories enriched with suitable transformations for copying and discharging information, which lack the naturality axiom. In our setting, this enrichment reflects ....

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277--296, 1996.

.... = 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 ....

....only up to a natural isomorphism. In fact, suitable auxiliary arrows called symmetries are present (see Table 4) that can model the possible reorganizations of minimal and maximal places of a process. We recall here the de nition of the category P(N) introduced in [21] and nitely axiomatized in [43]. De nition 1.10. Let N be a pt net. The category P(N) is the monoidal quotient of the free symmetric monoidal category F(N) generated by N , modulo the axioms: i) a;b = id a id b , if a; b 2 S, and a 6= b; and (ii) s; t; s 0 = t, if t 2 T 8 u; u 0 2 S u;u 0 : u u 0 u 0 ....

V. Sassone. An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170:277-296, 1996.

....of De nition 2.2. This can be done by exploiting the de nition of G ; 0 and applying the de nition of parallel and sequential composition. The freeness result relies on previous characterization results for symmetric monoidal categories as suitable (strongly concatenable) Petri processes [37,36] and more recent analogous results for pre nets [6] to which our spaces are equivalent. First, let us give some terminology, to be used also in the sketched proofs of Propositions 4.5 and 4.8. For relational spaces we de ne the depth of a generator sentence x 1 : x n f ## ## y 1 : y m ....

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277-296, 1996.

....2.2. This can be easily done by exploiting the definition of G ; 0 and applying the definition of parallel and sequential composition. The initiality result relies on previous characterization results for symmetric monoidal categories as suitable (strongly concatenable) Petri processes [40,39] and, more recently, to analogous results for pre nets [6] to which our spaces are equivalent. The proof is very long but standard and therefore we only sketch its four fundamental steps: ffl We first define the functor H mapping the arrows of the symmetric category freely generated by Sigma ....

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277--296, 1996.

....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 and transitive closure of relation ....

....of P T nets as a subcategory, and yields a coreflection corresponding exactly to the construction of the causal abstract net in Def. 2.20. 3.3. 1 Review of the Axiomatization of Concatenable Processes The algebraic structure of process is well exploited in (Degano et al. 1989; Degano et al. 1996; Sassone 1996). There it is shown how to associate a free symmetric strict monoidal category (see Appendix A) F [N ] to each net N in such a way that, under two suitable axioms, it characterizes the concatenable processes of N . This is due to the existence of a left adjoint functor F : Petri Gamma SSMC ....

[Article contains additional citation context not shown here]

Sassone, V. (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes, Theoretical Computer Science 170(1-2), 277--296.

....the coherence axioms of Def. 3. This can be easily done by exploiting the definition of G ; 0 and applying the definition of parallel and sequential composition. The initiality result relies on previous characterization results for symmetric monoidal categories as suitable Petri processes [31, 30], to which our spaces are equivalent. 2 4.2 GS Monoidal As illustrated in Section 2, gs monoidal categories are symmetric monoidal categories enriched with suitable transformations for copying and discharging information, which lack the naturality axiom. In our setting, this enrichment reflects ....

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170:277--296, 1996.

....3 if2b; cg. Review of Concatenable Processes. Causal firing sequences define 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 [5, 20] 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 deterministic occurrence net iff 8a 2 SK ; j ffl aj 1 ja ffl j 1 and F K is acyclic (F denotes the reflexive and transitive ....

....nets together with a function id : S Phi Gamma T , where T is a commutative monoid (T; Omega ; 0) and 0 , 1 and id are monoid homomorphisms) and its arrows are Petri net morphisms preserving identities and the monoidal structures. The algebraic structure of process is well captured in [20]. There it is shown how to associate a free symmetric strict monoidal category (see Appendix A) F [N ] to each net N in such a way that, under two suitable axioms, it characterizes the concatenable processes of N . This is due to the existence of a left adjoint functor F : Petri Gamma SSMC ....

V. Sassone. An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science, vol. 170, n.1--2, pp 277--296, 1996.

....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 ....

.... token interpretation, obvious candidates for suitable behavioural structures are event structures, concatenable pomsets and, especially, various kinds of concatenable processes [5, 20] From the logical viewpoint, it is not difficult to formulate a theory SYM of permutations and symmetries (cf. [19]) bridging the gap from strictly symmetric categories to categories symmetric only up to coherent isomorphism. On the other hand, the investigation of suitable algebraic models is still open, as our current best candidates, the symmetric strict monoidal categories P(N) of concatenable processes ....

V. SASSONE (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170, 277--296, Elsevier.

....a net. Our interest, instead, resides on abstract models that capture the mathematical essence of such spaces, possibly axiomatically, roughly in the same way as a prime algebraic domain (or, equivalently, a prime event structure) models the computations of a safe net [9] The research detailed in [6, 3, 4, 14, 7, 8, 16] identifies such structures as symmetric monoidal categories where objects are states, i.e. multisets of tokens, arrows are processes, and the tensor product and the arrow composition model, respectively, the operations of parallel and sequential composition of processes. At a higher level of ....

....composition of processes. At a higher level of abstraction, the next important question concerns the global structure of the collection of such spaces, i.e. the axiomatisation in the large of net computations. In other words, the space of the spaces of computations of Petri nets. Building on [3, 4, 16], the work presented in [15, 17] shows that the so called symmetric Petri categories, a class of symmetric strict monoidal categories with free (noncommutative) monoids of objects, provide one such an axiomatisation. In this paper, we retrace and illustrate the main results achieved so far along ....

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V. SASSONE (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes, Theoretical Computer Science n. 170, 277--296.

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V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170(1-2):277--296, 1996.

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V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170(1-2):277--296, 1996.

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V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoret. Comput. Sci., 170(1-2):277--296, 1996.

....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 ....

....and ITph lead to quite different concurrent semantics. For ordinary PT nets the algebraic approach has been pursued under both philosophies, characterizing different kinds of net processes, ranging from Best and Devillers commutative processes [3] that support the CTph) to concatenable processes [13,31] and strongly concatenable processes [32] that support the ITph) Note that the ITph relies on a tensor product which can be commutative only up to a monoidal natural isomorphism. Therefore, the algebraic approach requires some special mechanism in order to accommodate the lifting of the ....

[Article contains additional citation context not shown here]

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoretical Computer Science, 170:277--296, 1996.

....introduced by Goltz and Reisig in [7] several authors have shown that the semantics of nets in the ITph can still be understood in terms of symmetric monoidal categories. In particular, a simple variation of Goltz Reisig processes called concatenable processes is introduced in [5] see also [17]) which admits sequential composition and yields a symmetric monoidal category P(N) for each net N . Also several unfolding semantics (see e.g. 22,11] have been proposed that give a denotational interpretation of the interplay between concurrency, causality and nondeterminism. For contextual ....

....interpretation in terms of strongly concatenable contextual processes. Although we have worked only at the level of single nets, we believe that our approach can be extended to constructions between categories of nets and models, with restrictions analogous to those well known in the literature [17,18]. Acknowledgements. Thanks to Jos e Meseguer and the referees for helpful suggestions. ....

V. Sassone. An axiomatization of the algebra of Petri net concatenable processes. Theoretical Computer Science, 170:277-296, 1996.

....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 ....

....of minimal and maximal places of a process. It is worth remarking that, in concatenable processes, the ordering on minimal and maximal places is imposed on instances of the same place only. We recall here the definition of the category P(N) introduced in [11] and finitely axiomatized in [33]. Definition 1.11. Let N be a PT net. The category P(N) is the monoidal quotient of the free symmetric monoidal category F(N) generated by N , modulo the axioms a;b =id a id b if a; b 2 SN , and a 6= b s; t; s 0 =t if t 2 TN and s; s 0 are symmetries where ; id , and ; are, ....

[Article contains additional citation context not shown here]

V. Sassone (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170, 277--296, Elsevier.

....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 ....

....of minimal and maximal places of a process. Let us consider concatenable processes first, where the ordering on minimal and maximal places is imposed on instances of the same place only. We recall here the definition of the category P(N) introduced in [5] and finitely axiomatized in [18]. Definition 1.8 Let N be a PT net. The category P(N) is the monoidal quotient of the free symmetric strict monoidal category F(N) generated by N , modulo the axioms fl a;b = id a Omega id b if a; b 2 SN , and a 6= b s; t; s 0 = t if t 2 TN and s; s 0 are symmetries where fl, id, Omega ....

[Article contains additional citation context not shown here]

V. Sassone (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170, 277--296, Elsevier.

....and the laws that rule their composition, one can proceed trying to capture CP(N) by means of (categorical) universal constructions. This is the purpose of the following improved definition of the category P (N) whose equivalence with the original one in (Degano et al. 1996) has been proved in (Sassone 1996). Definition 1.5. The Category P (N) The category P (N) is the monoidal quotient (see Appendix A) of F (N) the free symmetric strict monoidal category generated by N, obtained by imposing the axioms c a;b = id a Phib if a;b 2 S N and a 6= b; s ; t ; s 0 = t if t 2 T N and s; s 0 are ....

V. SASSONE (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170, 277--296.

.... language, such as various forms of parallel and non deterministic composition (Winskel 1987; Meseguer and Montanari 1990; Brown et al. 1991; Mukund 1992) A unification of the process oriented and algebraic viewpoints has recently been proposed in (Degano et al. 1989; see also the related Sassone 1996; Sassone 1995) by showing that the commutative processes (Best and Devillers 1987) 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 Goltz Reisig processes (Goltz and Reisig 1983) on ....

.... on n categories as models of higher dimensional automata is introduced, but the modelling power obtained does not seem to be greater than that of ordinary PT nets, though the framework is highly elegantly linked to algebraic topology (Goubault and Jensen 1992; Goubault 1993; see also Cattani and Sassone 1996), Hoogers et al. 1992) in which the authors give semantics to PT nets in terms of generalized trace languages and discuss how using their work it could perhaps be possible to obtain a concept of unfolding for PT nets; and (Engelfriet 1991) where the unfolding of Petri nets is given in term of a ....

Sassone, V. (1996) An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science, to appear.

.... Consequently, we shall make no distinction between two processes : Theta N and 0 : Theta 0 N for which there exists an isomorphism : Theta Theta 0 such that 0 ffi = The equivalence of the following definition of P[N ] with the original one in [3] has been proved in [12]. The reader is referred to the cited works for a more explicit description of P[N ] a wider discussion, and for related examples. Definition 1.3 (The Category P[N ] The category P[N ] is the monoidal quotient of F(N ) the symmetric strict monoidal category whose monoid of objects is S Phi ....

V. Sassone. An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science, to appear.

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V. Sassone (1996), An Axiomatization of the Algebra of Petri Net Concatenable Processes. Theoretical Computer Science 170, 277--296, Elsevier.

No context found.

V. Sassone, An axiomatization of the algebra of Petri net concatenable processes, Theoretical Computer Science 170(1-2), 277--296 (1996).

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