Engberg, U. and G. Winskel, Petri nets as models of linear logic, in: A. Arnold, editor, 15th Colloquium on Trees in Algebra and Programming (1990), pp. 147--161.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and notation to labeled multisets in which case a secondary ordering over labels orders di erent occurrences of the same elements. 5. 2 Petri Nets A Petri net N is a directed bi partite multigraph (P; T ; E) whose two types of nodes, P and T , are called places and transitions, respectively [Pet62, Rei85, EW90]. The multiedges E link places to transitions and transitions to place; each edge carries a multiplicity. It is convenient to take a transition centric view of edges and de ne them as a function E : T N N that associates each transition in t 2 T with a pair of multisets of places that ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of the 15th Annual Colloquium on Trees in Algebra and Programming, pages 147-161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 431.

....Pareschi and Castagnetti de ne an improved top down strategy for propositional LO based on the Karp Miller s covering graph of Petri Nets, i.e. a forward exploration with accelerations. The relation between Rewriting, Petri Nets and Linear Logic has been investigated in previous works like [11, 18, 36, 34]. Our point of view is based on the proof as computation metaphor proposed in [4, 3, 38] our connection with models for concurrency is inspired to works in this eld like [11, 13, 27, 37, 38] As an example, in [11] Cervesato shows how to encode Petri Nets in di erent fragments of linear logic ....

Engberg, U., & Winskel, G. (1990). Petri nets as models of linear logic. Pages 147-161 of: Arnold, A. (ed), Proceedings of colloquium on trees in algebra and programming. LNCS, vol. 389. Copenhagen, Denmark: SpringerVerlag.

....and notation to labeled multisets in which case a secondary ordering over labels orders di#erent occurrences of the same elements. 5. 2 Petri Nets A Petri net N is a directed bi partite multigraph (P, T , E) whose two types of nodes, P and T , are called places and transitions, respectively [Pet62, Rei85, EW90]. The multiedges E link places to transitions and transitions to place; each edge carries a multiplicity. It is convenient to take a transition centric view of edges and define them as a function E : T that associates each transition in t T with a pair of multisets of places that we ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of the 15th Annual Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 431.

....little discussion about which algorithms to use for proving LL sequents. It is intuitively clear that for a smaller set of logic connectives, operators and rules simpler proving techniques are applicable. Thus it makes sense to look for new proving methods. 4 Petri nets and LL It has been shown [7, 3] that Petri nets can be presented in the form of LL sequents. Thus at least a part of a set of LL sequents can be translated into Petri nets. One of the rst surveys on Petri nets is [28] where an overview of basic concepts and extensions and subclasses of Petri nets may be found. 4.1 Petri ....

U. Engberg, G. Winskel. Petri nets as models of Linear Logic. In: A. Arnold (ed). Proceedings of Colloquium on Trees in Algebra and Programming (CAAP'90), Copenhagen, Denmark, May 15-18, 1990.

....proof. Anyway, to enjoy the full power of Petri net analysis methods and tools, several conversions have to be applied to original LL programs , because not all our language constructs are directly representable as Petri net fragments. 35 3. 1 Mapping LL sequents to Petri nets It has been shown [22, 23, 12] that Petri nets can be presented in the form of LL sequents. Thus at least a part of a set of LL sequents can be translated into Petri nets. One of the rst surveys on Petri nets is [82] where an overview of basic concepts and extensions and subclasses of Petri nets may be found. 3.1.1 Petri ....

U. Engberg, G. Winskel. Petri nets as models of Linear Logic. In A. Arnold (ed). Proceedings of Colloquium on Trees in Algebra and Programming (CAAP'90), Copenhagen, Denmark, May 15-18, 1990.

.... 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 ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, CAAP'90, volume 431 of Lecture Notes in Computer Science, pages 147{

....techniques are applicable. Thus it makes sense to look for new proving methods. 6 Peep K ungas A B C Figure 1: The Petri net representation of LL sequent A B C. C A B Figure 2: The Petri net representation of LL sequent C A B. 4 Mapping LL sequents to Petri nets It has been shown [6, 3] that Petri nets can be presented in the form of LL sequents. Thus at least a part of a set of LL sequents can be translated into Petri nets. One of the rst surveys on Petri nets is [24] where an overview of basic concepts and extensions and subclasses of Petri nets may be found. Following ....

U. Engberg, G. Winskel. Petri nets as models of Linear Logic. In: A. Arnold (ed). Proceedings of Colloquium on Trees in Algebra and Programming (CAAP'90), Copenhagen, Denmark, May 15-18,

....even necessary. What one would like is to take the attractive connection between markings and tensor products, and include it in a richer surrounding logic. This is where the several works on linear logic and nets di er. Perhaps the most satisfying treatment has been given by Engberg and Winskel [16, 17] (and Brown s related [7] to which we refer for discussion of the early works on linear logic and nets. Our purpose in this paper is to re consider Engberg and Winskel s semantics, from the point of view of the logic BI of bunched implications [27] The semantics of [16, 17] is organized by ....

....Engberg and Winskel [16, 17] and Brown s related [7] to which we refer for discussion of the early works on linear logic and nets. Our purpose in this paper is to re consider Engberg and Winskel s semantics, from the point of view of the logic BI of bunched implications [27] The semantics of [16, 17] is organized by viewing a proposition as a set of markings closed under (backwards) reachability, with the idea that a proposition is a property consisting of the set of markings that can give rise to it. Engberg and Winskel de ne operations on these sets of markings that provide an ....

[Article contains additional citation context not shown here]

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of Colloquium on Trees in Algebra and Programming, pages 147-161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 389.

....another plan, which uses a previously canned plan P 1 . In this way an hierarchical plan representation is achieved and plan reuse is implemented. 5 Linear logic, planning and Petri nets One of the rst models of concurrency represented in linear logic were Petri nets [1] It has been shown [12] that Petri nets can be presented in the form of linear logic formulas. Thus at least a part of a set of linear logic formulas can be translated into Petri nets which in turn can be used for planning. One of the rst surveys on Petri nets is [31] where an overview of basic concepts and extensions ....

U. Engberg, G. Winskel. Petri nets as models of linear logic. In: A. Arnold (ed). Proceedings of Colloquium on Trees in Algebra and Programming (CAAP'90), Copenhagen, Denmark, May 15-18, 1990. Vol. 431 of Lecture Notes in Computer Science, Springer-Verlag, pp. 147-161, 1990.

....axioms (a tensor theory) and the dynamic behaviour of the net is described by the inference rules of the tensor fragment. Alongside the work on Petri nets and linear logic, came the realisation that models of linear logic had arisen in the form of quantales [9] Indeed, Winskel and Enberg in [7] point out a straightforward way in which a Petri net induces a quantale and so becomes a model for linear logic. Their paper provides evidence that linear logic with the right notion of satisfaction, can be a reasonably expressive specification logic for parallel processes. More explicitely, in ....

....point out a straightforward way in which a Petri net induces a quantale and so becomes a model for linear logic. Their paper provides evidence that linear logic with the right notion of satisfaction, can be a reasonably expressive specification logic for parallel processes. More explicitely, in [7] Engberg and Winskel associate to a given Petri net N aquantaleQ[N ] whose elements are the set of markings downwards closed with respect to the reachability relation. Each atomic proposition A is denoted by a set of reachable markings of the corresponding place A. General formulas of linear logic ....

[Article contains additional citation context not shown here]

U. Egberg, G. Winskel, Petri Nets as Models of Linear Logic, Lecture Notes in Computer Science Vol. 431, CAAP'90 , Copenhagen, Denmark, Springer-Verlag, 1990.

.... Linear Logic View of Object Systems Berndt Farwer farwer informatik.uni hamburg.de Abstract Linear Logic [Gir87] has been shown to incorporate a fragment suitable for representing P T nets and giving a semantics to the computations of such nets (e.g. Bro89] MOM89] [EW90]) This result is generalized to coloured nets. Furthermore a new kind of Petri nets is de ned: Linear Logic Petri Nets (LLPN) These nets are used as an intuitive semantics to well known and new high level net concepts. keywords: Linear Logic, Petri nets, object systems, concurrency 1 ....

U. Engberg and G. Winskel. Petri Nets as Models for Linear Logic. Technical report, Computer Science Department, Aarhus University, 1990.

.... by Lincoln and Mitchell [70] Chirimar et al. 33, 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 ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of CAAP '90. Lecture Notes in Computer Science vol. 431, Springer-Verlag, 1990.

....axioms (a tensor theory) and the dynamic behaviour of the net is described by the inference rules of the tensor fragment. Alongside the work on Petri nets and linear logic, came the realisation that models of linear logic had arisen in the form of quantales [9] Indeed, Winskel and Enberg in [7] point out a straightforward way in which a Petri net induces a quantale and so becomes a model for linear logic. Their paper provides evidence that linear logic with the right notion of satisfaction, can be a reasonably expressive speci cation logic for parallel processes. More explicitely, in ....

....[7] point out a straightforward way in which a Petri net induces a quantale and so becomes a model for linear logic. Their paper provides evidence that linear logic with the right notion of satisfaction, can be a reasonably expressive speci cation logic for parallel processes. More explicitely, in [7] Engberg and Winskel associate to a given Petri net N a quantale Q[N ] whose elements are the set of markings downwards closed with respect to the reachability relation. Each atomic proposition A is denoted by a set of reachable markings of the corresponding place A. General formulas of linear ....

[Article contains additional citation context not shown here]

U. Egberg, G. Winskel, Petri Nets as Models of Linear Logic, Lecture Notes in Computer Science Vol. 431, CAAP'90 , Copenhagen, Denmark, Springer-Verlag, 1990.

.... to build counter models as it is already done in Intuitionistic Logic (IL) 14] Here we focus on the relationships between ILL and the Petri nets that can naturally be made into models of ILL in such a way that many properties of nets (one might wish to state) become expressible in the logic [3]. Completeness for the logic with respect to nets as a model has been studied in [4] For instance, Petri nets form a sound and complete model for the Phi free fragment and in the case of the ( free fragment an extra axiom (of distributivity of over Phi) is necessary for the completeness. In ....

....bottom element. Indeed, the preceding construction succeeded because the underlying monoid was regular 3 . However, it seems that a slight modification would be necessary in case of non regularity to keep the lattice structure. Let us now compare this construction with the one for Petri nets in [3]. In this case, one used the downward closed subsets of the underlying (pre)order. In case of a flat order, these are all subsets of the monoid and one has a much bigger number of elements in the quantale. But there should exist another general method for such construction leading to smaller ....

[Article contains additional citation context not shown here]

U. Engberg and G. Winskel. Petri Nets as Models of Linear Logic. In CAAP 90, LNCS 431, pages 144--161, Copenhagen, Denmark, May 1990.

....rules for Classical sequent calculus which allow one does to arbitrarily duplicate and reduce formulas on both sides of sequents. The resulting logical framework and in particular its proof theoretical formalization revealed strong connections with many computational models such as the Petri nets [15, 21] and the calculus [4] Proof theoretic investigations yielded both Intuitionistic and Classical formulations of LL [43] with interesting results on cut elimination for sublogics based on fragments of connectives and on permutability of their rules [2, 18, 31] For people not acquainted with ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 389.

....even necessary. What one would like is to take the attractive connection between markings and tensor products, and include it in a richer surrounding logic. This is where the several works on linear logic and nets differ. Perhaps the most satisfying treatment has been given by Engberg and Winskel [15, 16] (and Brown s related [7] to which we refer for discussion of the early works on linear logic and nets. Our purpose in this paper is to re consider Engberg and Winskel s semantics, from the point of view of the logic BI of bunched implications [24] The semantics of [15, 16] is organized by ....

....Engberg and Winskel [15, 16] and Brown s related [7] to which we refer for discussion of the early works on linear logic and nets. Our purpose in this paper is to re consider Engberg and Winskel s semantics, from the point of view of the logic BI of bunched implications [24] The semantics of [15, 16] is organized by viewing a proposition as a set of markings closed under (backwards) reachability, with the idea that a proposition is a property consisting of the set of markings that can give rise to it. Engberg and Winskel define operations on these sets of markings that provide an ....

[Article contains additional citation context not shown here]

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 389.

....even necessary. What one would like is to take the attractive connection between markings and tensor products, and include it in a richer surrounding logic. This is where the several works on linear logic and nets differ. Perhaps the most satisfying treatment has been given by Engberg and Winskel [16, 17] (and Brown s related [7] to which we refer for discussion of the early works on linear logic and nets. Our purpose in this paper is to re consider Engberg and Winskel s semantics, from the point of view of the logic BI of bunched implications [27] The semantics of [16, 17] is organized by ....

....Engberg and Winskel [16, 17] and Brown s related [7] to which we refer for discussion of the early works on linear logic and nets. Our purpose in this paper is to re consider Engberg and Winskel s semantics, from the point of view of the logic BI of bunched implications [27] The semantics of [16, 17] is organized by viewing a proposition as a set of markings closed under (backwards) reachability, with the idea that a proposition is a property consisting of the set of markings that can give rise to it. Engberg and Winskel define operations on these sets of markings that provide an ....

[Article contains additional citation context not shown here]

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 389.

....the deduction process. This can be obtained eliminating the structural rules for Classical sequent calculus which allow one to arbitrarily duplicate and reduce formulas on both sides of sequents. The resulting logic revealed strong connections with many computational models such as Petri nets [16] and Milner s calculus [4] Proof theoretic investigations yielded both Intuitionistic and Classical formulations of LL [40] with interesting results on cut elimination for fragments of connectives and on permutability of rules [2, 19] For people not acquainted with these subjects, in the ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 389.

....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 ....

....Petri nets within linear logic. Thus the study of the connections between linear logic and Petri nets may benefit both areas of research. There are different ways in which one can tackle the problem. The current 5 approaches can be classified as proof theoretic [10, 39] model theoretic [9, 24] and category theoretic [11, 49] In the proof theoretic approach the idea is to try to relate computations in nets to proofs of linear logic formulae. Brown [10] equates a net to a formula of linear logic and proves that reachability corresponds to provability in linear logic. On the other hand ....

[Article contains additional citation context not shown here]

Engberg, U. and Winskel, G. Petri nets as models of linear logic. Technical Report DAIMI PB-301. Computer Science Department, Aarhus University, Denmark, 1990.

....sums depending on the non strictness and non totality of arguments, properties similar to those of the SCL category. It has been proved by Seely to correspond directly to Girard s linear logic [Laf88, Sce90] which in turn has been shown to fit hand in glove with computation using Petri nets [MOM89, EW90]. Since this relationship has strong implications for parallel computation, the Girard category is so far only mildly promising for categorizing control. Another farther fetched possibility is higher order categories such n categories. Seely has expressed the typed lambda calculus as a ....

U. Engberg and G. Winskel, Petri Nets as Models of Linear Logic, In Proc. CAAP, Lecture Notes in Computer Science 431, 1990.

....sequent calculus which does not contain weakening and contraction. To evaluate the impact of Girard s intuition on the theoretical research in Computer Science, it is enough to look at the connections between Linear Logic and other frameworks and computation models such as the Petri nets [25, 34], the Chemical Abstract machine [2] the calculus [13] functional paradigms [50] The Curry Howard isomorphism, where formulas are interpreted as types, proofs as terms (programs) and cut elimination as conversion (computation) has been extended to this new setting yielding definitions of more ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 389.

....infinite place capacities and finite markings; we define them in subsection 3.1. The main reason for choosing this model is the ease of interpreting these nets as linear theories (the same model was adopted also by the other authors who tackled the problem of relating Petri nets and linear logic [16, 10, 6, 8]) In section 3.2, we prove that these nets are equivalent to the more general class of place transition nets with infinite place capacities. In section 3.3, we show how the more traditional class of place transition nets can be cast into the previous model by means of the operation of ....

....has not received the same attention. The way transitions themselves are encoded is far less uniform. 10] encodes the preset and the postset of a transition as the left hand side and the right hand side of sequents and uses the cut rule as the means of applying a transition. Other authors [16, 8] make an indirect use of the multiplicative implication to express transitions as the hypothetical validity of their postset relatively to their preset. The most popular means of connecting Petri nets and linear logic is to use category theory [6, 16] In these papers, both Petri nets and linear ....

[Article contains additional citation context not shown here]

U. Engberg, G. Winskel: "Petri Nets as Models of Linear Logic", in Proceedings of the 15th Annual Colloquium on Trees in Algebra and Programming (A. Arnold Ed.), Copenhagen, pp. 147--161, LNCS 431, Springer-Verlag, 1990.

....based on Formulae as Types notion of CurryHoward Isomorphism[13] where computation is described in terms of proof normalization. The other is Formulae as States, Proofs as Computations approach[16] In this approach, connections between Petri Nets and linear logic have been investigated[16] 11][10][6] They relate Petri Nets to theories in linear logic using only Omega . Later, 6] extended this approach to the implication(0ffi) The latter approach, Proofs as Computations , is also investigated in a rather different way, in the context of logic programming[4] 5] 17] and concurrent ....

Engberg, U., and G. Winskel, "Petri Nets as Models of Linear Logic," in Proceedings of CAAP'90, vol. 431 of Lecture Notes in Computer Science, 1990.

....A good philosophical explanation of LL along these lines can be found in [142] A number of concurrency models have been expressed in LL. Some of them are described in the following subsections. 4. 1 LL and Petri Nets One of the first models of concurrency represented in LL were Petri nets [17, 21, 80, 61, 37, 38, 53, 54, 120, 122, 123]. Category theory is used widely in these works, for example in [107] to show that high level nets (whose markers are data structures) are also LL models. 39] relates two typical uses of category theory for Petri Nets, refinement (mapping a net to another one) and simulation (mapping possible ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Colloquium on Trees in Algebra and Programming (CAAP'90), number 431 in LNCS, pages 147--161, Copenhagen, Denmark, May 1990.

.... 18] An early application of the resource sensitive aspect of the logic was the implementation of a functional programming language in which garbage collection was replaced by explicit duplication operations based on linear logic [25] Further studies have demonstrated connections with Petri nets [3, 20, 30, 4, 12, 10] and other models of concurrency [26, 1] With regard to concurrency, there is a similarity between proof nets, the inherent model of computation associated with cut elimination in multiplicative linear logic (c.f. 14, 15, 9, 26] and connection graphs, which were designed to model connection ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of CAAP'90. Lecture Notes in Computer Science vol. 431, Springer, 1990.

....framework may be seen as a combination of local transitions and global, quantitative time correlations. In our framework transitions are instantaneous but events may have duration. 2 EXAMPLE: RAILROAD CROSSING CONTROLLER 4 Regarding the transitions, our framework is a refinement of the work in [11, 21, 18, 39, 15], which established a direct relationship between Petri nets and linear logic axiomatizations using conjunctive formulas. Here we consider only conjunctions of fixed finite length. In linear logic this restriction suffices for a faithful simulation of finite state transitions. We extend this ....

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, Proceedings of CAAP '90. Springer LNCS 431, 1990.

No context found.

Engberg, U. and G. Winskel, Petri nets as models of linear logic, in: A. Arnold, editor, 15th Colloquium on Trees in Algebra and Programming (1990), pp. 147--161.

No context found.

U. Engberg and G. Winskel. Petri nets as models of linear logic. In A. Arnold, editor, 15th Colloquium on Trees in Algebra and Programming, pages 147--161, Copenhagen, Denmark, 1990. Springer-Verlag LNCS 431.

No context found.

U. Engberg and G. Winskel. Petri nets as models of linear logic. In Proceedings of the 15th Colloquium on Trees in Algebra and Programming, volume 431 of Lecture Notes in Computer Science, pages 147161, Copenhagen, Denmark, 1990.

No context found.

Engberg, U. and Winskel, G. (1990) Petri nets as models of linear logic. In Proc. CAAP'90, LNCS 431, pp. 147-161. Springer-Verlag.

No context found.

U. Engberg and G. Winskel, "Petri nets as models of linear logic", in CAAP'90, Coll. on Trees in Algebra and Programming (LNCS 431, Springer-Verlag, 1990) pp. 147--161.

No context found.

Engberg, U.,Winskel, G. Petri Nets as Models of Linear Logic. Lambek, J. deductive systems and categories.I, Math. Systems Theory 2 (1968), 287-318.

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