Brown, C. and D. Gurr, A categorical linear framework for Petri nets, in: Proc. LICS'90 (1990), pp. 208--218.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....system, consider H : O(H) H that maps each [ 2 SO(H) to its final state s 2 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 ....

C. BROWN AND D. GURR (1990), A Categorical Linear Framework for Petri Nets, in Proceedings of the 5th Symposium on Logics in Computer Science, 208-- 218, IEEE Press.

.... 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 explained in [48] both the categorical place transition net models of [53] and the linear logic representations of place transition nets inspired rewriting logic as a generalization of both formalisms. But, as shown in this paper, the extra algebraic ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proc. Fifth Annual IEEE Symposium on Logic in Computer Science, pages 208-218, June 1990.

....Winskel, and we argue that it provides a more natural logic for the net semantics. We then brie y consider a more expressive logic based on an extension of BI with classical and modal features. 1 Introduction A number of authors have explored the connection between Petri nets and linear logic [5, 8, 19, 9, 24, 25]. The basic idea is to encode markings as tensor products a 1 an of atomic formulae, and transitions as judgements M N with markings in antecedent and consequent positions. Given a judgement of this form as an axiom, for each transition in a net, one obtains that M N is derivable from ....

....as in branching time temporal logics, Hennessy Milner logic or temporal logic based on partial order semantics [15, 33, 4, 30, 31] but in a way compatible with multiplicative conjunction. There have, to date, been a number of suggestive uses of substructural logics in concurrency (e.g. [11, 3, 2, 22, 8, 19, 9, 24, 25]) While there has not yet been a de nitive or completely convincing account, the idea of using multiplicative conjunction to describe spatial distribution of resources or processes remains appealing. In this connection it is worth stressing, in conclusion, that there appears to be no compelling ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proceedings of the Sixth Annual Symposium on Logic in Computer Science, pages 208-219, Philadelphia, Pennsylvania, June 1990. IEEE Computer Society Press.

....N ) The symmetries of CP(N) are given by concatenable processes with empty set of transitions (token permutation is expressed by di erent orderings and ) 1. 1 Petri nets are monoids Several interesting aspects of net theory can be pro tably developed within category theory, see, e.g. [50,30,9,22,38,33]. We focus on the so called Petri 1 F denotes the re exive and transitive closure of relation F . 7 u 2 S idu : u u t: u v 2 T t: u v : u v; u 0 v 0 : u u 0 v v 0 : u v; v w ; u w Table 2. neutral: id = commutativity: ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proceedings 5th LICS Symposium, pages 208-218. IEEE Press, 1990.

.... 34] Wadler [90] Mackie [77] and Benton et al. 24] A strong relationship of the multiplicative fragment of linear logic to Petri nets has been demonstrated by Gehlot and Gunter [57, 44, 43] Asperti et al. 12, 14] Engberg and Winskel [41] Marti Oliet and Meseguer [79] and Brown and Gurr [30]. Interpretations of linear logic proofs in concurrent paradigms such as the chemical abstract machine or Milner s calculus are given by Abramsky [1] and by Bellin and P.J. Scott [23] With regard to concurrency, we also note a remarkable similarity between a programming paradigm based on proof ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proc. 5-th IEEE Symp. on Logic in Computer Science, Philadelphia, June 1990.

....Petri (respectively Petri ) to indicate the category of (respectively marked) PT nets and their morphisms, with the obvious componentwise composition of arrows. Several different proposal of net morphisms can be found in the literature that give rise to different categories of nets as, e.g. [5, 6, 26, 1]. Comparing these categories to Petri is not trivial, and we cannot address here the question as to whether and to what extent our results would lift to these choices. We limit ourselves to remark that our morphisms follow the two main algebraic approaches to nets namely Winskel s [35] and ....

....In the CTph, instead, the two firings are always considered to be concurrent, because 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 ....

C. Brown and D. Gurr (1990), A Categorical Linear Framework for Petri Nets, in Proceedings of the 5th Symposium on Logics in Computer Science, 208--218, IEEE Press.

....and we argue that it provides a more natural logic for the net semantics. We then briefly consider a more expressive logic based on an extension of BI with classical and modal features. 1 Introduction A number of authors have explored the connection between Petri nets and linear logic [5, 8, 17, 9, 21, 22]. The basic idea is to encode markings as tensor products a 1 Omega Delta Delta Delta Omega an of atomic formulae, and transitions as judgements M N with markings in antecedent and consequent positions. Given a judgement of this form as an axiom, for each transition in a net, one obtains ....

....as in branching time temporal logics, Hennessy Milner logic or temporal logic based on partial order semantics [14, 29, 4, 27, 28] but in a way compatible with multiplicative conjunction. There have, to date, been a number of suggestive uses of substructural logics in concurrency (e.g. [11, 3, 2, 20, 8, 17, 9, 21, 22]) While there has not yet been a definitive or completely convincing account, the idea of using multiplicative conjunction to describe spatial distribution of resources or processes remains appealing. In this connection it is worth stressing, in conclusion, that there appears to be no compelling ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proceedings of the Sixth Annual Symposium on Logic in Computer Science, pages 208--219, Philadelphia, Pennsylvania, June 1990. IEEE Computer Society Press.

....and we argue that it provides a more natural logic for the net semantics. We then briefly consider a more expressive logic based on an extension of BI with classical and modal features. 1 Introduction A number of authors have explored the connection between Petri nets and linear logic [5, 8, 19, 9, 24, 25]. The basic idea is to encode markings as tensor products a 1 Omega Delta Delta Delta Omega an of atomic formulae, and transitions as judgements M N with markings in antecedent and consequent positions. Given a judgement of this form as an axiom, for each transition in a net, one obtains ....

....as in branching time temporal logics, Hennessy Milner logic or temporal logic based on partial order semantics [15, 33, 4, 30, 31] but in a way compatible with multiplicative conjunction. There have, to date, been a number of suggestive uses of substructural logics in concurrency (e.g. [11, 3, 2, 22, 8, 19, 9, 24, 25]) While there has not yet been a definitive or completely convincing account, the idea of using multiplicative conjunction to describe spatial distribution of resources or processes remains appealing. In this connection it is worth stressing, in conclusion, that there appears to be no compelling ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proceedings of the Sixth Annual Symposium on Logic in Computer Science, pages 208--219, Philadelphia, Pennsylvania, June 1990. IEEE Computer Society Press.

.... 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 ....

C. Brown and D. Gurr, A categorical linear framework for Petri nets, in Proceedings 5th LICS Symposium, 208--218, IEEE Computer Society Press (1990).

....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 monoidal operation means ....

Brown, C., and Gurr, D. (1990), A Categorical Linear Framework for Petri Nets, in "Proceedings of the 5th LICS Symposium", pp. 208--218, IEEE.

....The transitions and places of a Petri net [18] hint at this duality by being respectively universal and existential: a firing transition involves every incident edge whereas a token passing through a place involves just one path through that place. But it was only relatively recently observed [6,7] that this could be made a true duality in the morphism reversing sense. True duality arises in many mathematical situations, the best known of which have important applications. The duality of locally compact abelian groups, which in the finite case reduces to the self duality of finite abelian ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In J. Mitchell, editor, Logic in Computer Science, pages 208--218. IEEE Computer Society, June 1990.

....processes. Finally we connect them with event structures, showing how any event structure can be represented as an eks. These structures have previously been studied by Barr[Bar79] as instances of Chu s construction on sets, and by Lafont and Streicher[LS91] as games over 2. Brown and Gurr[BG90] have used similar structures to study Petri nets. 1.1 Theory of an eks An eks (E; Q) is a set Q of subsets of 2 E . We can therefore write it out as a formula in Disjunctive Normal Form in propositional logic, with variables E and clauses Q, with each clause corresponding to a q 2 Q containing ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In J. Mitchell, editor, Logic in Computer Science, pages 208--218. IEEE Computer Society, June 1990.

.... the construction described by Po Hsiang Chu in the appendix of Barr s book on autonomous (i.e. self dual closed) categories [Bar79] Chu s construction takes a closed monoidal category V with pullbacks and completes it to a self dual category Chu(V; k) De Paiva [dP89a, dP89b] and Brown and Gurr [BG90, BGdP91] apply the Chu construction to respectively a version of Godel s Dialectica and Petri nets. Lafont and Streicher study Chu spaces over K, which they call games [LS91] the term Chu construction had been around previously, and the name Chu space for V = Set was suggested to us by Barr in email. ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In J. Mitchell, editor, Logic in Computer Science, pages 208--218. IEEE Computer Society, June 1990.

....and a net morphism consists of a pair of matrices that are interpreted as relations. Meseguer and Montanari [52] suggest monoids as the structure on the transitions and places. This means that a net morphism consists of a pair of monoid homomorphism. A recent idea by Carolyn Brown and Doug Gurr [11] is to turn the map on the places around and relax the commutativity requirement to a requirement of lax naturality [46] The flow relation is represented by a pair of subobjects of the cartesian product of the set of places with the set of transtions. Finally Husberg [43] defines a multiset as a ....

....conscious logic and has thus recently gained considerable interest from the computer science community. Applications include implementation models for functional languages [34, 47, 63] the study of process algebras [3, 2, 1] logic programming [4, 14, 40] planning in AI [50] and net theory [5, 9, 10, 11, 8, 24, 39, 49]. The list is by no means meant to be comprehensive because the field is advancing very rapidly. The applications relevant to our purposes are the applications to net theory. The aim in these studies is twofold. Linear logic is a logic with many new connectives that are difficult to understand ....

[Article contains additional citation context not shown here]

Brown, C. and Gurr, D. A categorical linear framework for Petri nets. 5th Symposium on Logic in Computer Science. IEEE Computer Press, 1990.

....to ZSCGraph) having the ordinary category Petri of P T nets as a subcategory, 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 ....

C. Brown and D. Gurr. A Categorical Linear Framework for Petri Nets. In Proceedings of the 5th LICS Symposium, 208--218, 1990.

.... with my student Vineet Gupta [Gup94] Rob van Glabbeek and Gordon Plotkin have separately been invaluable sounding boards and sources of ideas, and jointly of results [VGP95] Carolyn Brown, with Doug Gurr, was the first to explore the connection between the Chu construction and concurrency [BG90], and has been a useful source of insights. Dusko Pavlovic has developed a formal sense in which Chu spaces are universal for mathematics, more correctly their couniversality. I am also grateful to H. Andreka, I. Nemeti, I. Sain for the opportunity to present a one week course on Chu spaces to an ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In J. Mitchell, editor, Logic in Computer Science, pages 208--218. IEEE Computer Society, June 1990.

....as a means of producing constructive models of linear logic [Bar91] The subsequent history of Chu spaces has been one of successive weakenings of the enriching category V . Barr and Chu took V to be any symmetric monoidal closed category with pullbacks, de Paiva [dP89] and Brown and Gurr [BG90] restricted to order enrichment, and finally Lafont and Streicher banished enrichment altogether by taking V = Set [LS91] and calling the resulting objects games after von Neumann and Morgenstern. Chu spaces are indeed games, and moreover of the asynchronous kind, ideally suiting them as a model ....

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In J. Mitchell, editor, Logic in Computer Science, pages 208--218. IEEE Computer Society, June 1990.

....system, consider H : O(H) H that maps each [ Gamma 2 S O(H) to its final state s 2 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 ....

C. BROWN AND D. GURR (1990), A Categorical Linear Framework for Petri Nets, in Proceedings of the 5th Symposium on Logics in Computer Science, 208--218, IEEE Press.

....structure of nets and their computations. Since such a structure is relevant 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 (Reisig 1985; Nielsen et al. 1981; Winskel 1984; Winskel 1987; Brown and Gurr 1990; Brown et al. 1991) has been given an original interpretation by considering monoidal categories as a suitable framework (Meseguer and Montanari 1990) In fact, in (Meseguer and Montanari 1990; Degano et al. 1996) the authors have shown that the semantics of Petri nets can be understood in terms ....

C. BROWN AND D. GURR (1990), A Categorical Linear Framework for Petri Nets. In Proceedings of the 5th LICS Symposium 208--218, IEEE Press.

....the ordinary category Petri of P T nets as a subcategory, and yields a coreflection 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 ....

C. Brown and D. Gurr. A Categorical Linear Framework for Petri Nets. In Proceedings of the 5th LICS Symposium, 208--218, 1990.

....1 ) is unde ned then 1 S ( t 1 ) 1 S (t 1 ) v) 8t 1 2 T 1 : If T (t 1 ) t 2 then: 1 S ( t 1 ) t 2 and 1 S (t 1 ) t 2 . 8s 2 t 2 : W 1 ( S (s) t 1 ) W 2 (s; t 2 ) 8s 2 t 2 : W 1 (t 1 ; S (s) W 2 (t 2 ; s) Following [2], we have directly de ned the map on places as a partial function in the reverse direction, rather than as a relation in the forward direction whose inverse is a partial function (as in [14] m m m A A AU A A AU s 1 s 3 s 2 t 1 t 2 m m s 0 1 s 0 2 t ....

C. Brown, D. Gurr, \A Categorical Linear Framework for Petri Nets", Proc. 5th Ann. IEEE Symp. on Logic in Computer Science, Philadelphia, USA, 1990, pp. 208-218.

No context found.

Brown, C. and D. Gurr, A categorical linear framework for Petri nets, in: Proc. LICS'90 (1990), pp. 208--218.

No context found.

C. Brown and D. Gurr. A categorical linear framework for Petri nets. In Proc. LICS'90, pages 208--218, Philadelphia, PA, 1990. IEEE Computer Society Press.

No context found.

C. Brown and D. Gurr. A categorical linear framework for petri nets. In Logic in Comptuer Science (LICS'90), pages 208-219, Philadelphia, PA, June 1990. IEEE Computer Society Press.

No context found.

Algebra 59, 1--41. Brown, C. and Gurr, D. (1991) A categorical linear framework for Petri nets. In Proc. 5th Ann.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC