N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1:66-101, 1991. Revised version of paper in LNCS 389.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a transition are not symmetric: the former is modeled as iterated linear implications while the latter as an asynchronous formula inside a monad. It would be tempting to represent both using the monadic encoding, which is akin to the way Petri nets are traditionally rendered in linear logic [Cer95, MOM91, GG90]. For example, transition t a would be represented as t a : b tok b tok ag f1 While this is not incorrect, the behavior of this declaration is not what we would expect: it is applicable not only in a linear context containing two declarations of type tok b and one of type tok a, but ....

N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1:66-101, 1991. Revised version of paper in LNCS 389.

....of duplicates. The second main di erence is that the formalism has a basic mechanism for choosing new symbols. This is important for modeling protocols that choose a new nonce or generate encryption keys. Our formalism can also be viewed as a Horn fragment of linear logic [Gir87b, Asp87, MOM89, GG90a, Kan94, Cer95] A similar fragment of linear logic is used in [KOS98] to represent real time nite state systems. Two other e orts using linear logic to model the state transition aspect of protocols (but not existential quanti cation for nonces) are [CD98, DMT98] The multi set ....

N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313-340, Manchester, United Kingdom, September 1989.

....The expressive power is so rich that one can construct a counter machine within the propositional fragment of linear logic. Some computational models of concurrency are applications of LL [26, 2, 20] In particular, the relation between linear logic and Petri nets [11, 36] has been well studied [16, 35, 21, 4, 23]. According to [25] the algebraic point of view of Petri nets seems to be related to the algebraic semantics for linear logic [28, 34] LL has a modal storage operator which means an infinite resource. Using this operator, one can distinguish the treatment of a reusable resource from the ....

N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313--340, Manchester, United Kingdom, September 1989.

....logic. Consumption of the data and a dynamic change in process environments can be expressed by linear logic. Some computational models of concurrency are applications of it [1] 7] 17] In particular, the relation between linear logic and Petri nets has been well studied [3] 5] 9] 12] [14], 16] The expressive power is so rich that one can construct a counter machine within the propositional fragment of linear logic. However, it is not enough to treat a dynamic change in process environments with the passage of time. Although [11] is a Graduate School of Science and Technology, ....

N.Mart-Oliet and J.Meseguer, "From Petri nets to linear logic," Lecture Notes in Computer Science, vol.389, pp.313-- 340, Sept. 1989.

....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 ....

N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313-340, Manchester, United Kingdom, September 1989.

....1 Introduction Since its introduction in [Gir87] linear logic has been understood as a framework to reason about concurrent computations. Several researcher have in fact observed the existence of natural connections between linear logic and the theory of Petri Nets, see e.g. Cer95,Kan94,MM91] In this work we will investigate this connection focusing on the relations between algorithmic techniques used for the analysis of Petri Nets and provability in fragments of linear logic. The fragment we consider in this paper is called LO [AP90] LO was originally introduced as a theoretical ....

....As we will show next, LO programs enjoy a simple operational reading that makes clear the connection between provability in linear logic and veri cation methods in Petri Nets. Let us illustrate these ideas with the help of some examples. Petri Nets in Propositional Linear Logic. Following [MM91] a Petri Net can be represented as a multiset rewriting system over a nite alphabet p; q; r; of place names. Among several possible ways, multiset rewrite rules can be expressed in linear logic using the connectives . ....

N. Mart-Oliet, J. Meseguer. From Petri Nets to Linear Logic. Mathematical Structures in Computer Science 1(1):69-101, 1991.

.... 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 ....

N. Mart-Oliet and J. Meseguer. From Petri Nets to Linear Logic. Technical report, SRI International, Computer Science Laboratory, Stanford, 1989.

....the presumption that the object nets structure remains unchanged in all processes. This paper sheds some light on possible extensions of high level Petri nets to incorporate the dynamic evolution of Petri net structures. The exposition is based on the Linear Logic encoding of Petri nets ( Bro89] [MOM89]) and coloured Petri nets ( Far98b] It provides a basic semantics for modifying net structures which can be employed in a framework of nets within nets, i.e. situations where Petri nets (so called token nets) themselves are used as tokens in an underlying environment net. 1 Introduction The ....

....are not consumed upon occurring. Any P T net can be represented by a formula from the (#, horn fragment of Linear Logic where the reachability of a certain marking in the net corresponds to the derivability of the associated sequent in a fragment of the Linear Logic sequent calculus ( Bro89] [MOM89]) These results extend also to a class of coloured Petri nets as illustrated in [Far98c] and [Far99] 2.3 Combining Petri Nets and Linear Logic In this paper we follow the definitions given in [Far98a] using Linear Logic formulae both as inscriptions of Petri Nets i.e. as arc inscriptions and ....

N. Mart-Oliet and J. Meseguer. From Petri Nets to Linear Logic. Technical report, SRI International, Computer Science Laboratory, Stanford, 1989.

....of linear logic [16] a re nement of modal logic with an intrinsic and natural accounting of process states and events. The choice of linear logic is natural because of the very close connection between multiset rewriting and simple fragments of linear logic, which has been studied extensively [3, 30, 15, 19, 5]. We extend this standard correspondence to include rst order parameters and existentially quanti ed variables. On the other hand, we also formally represent strand constructions as relatively simple formulas in rst order linear logic. This encoding is also shown to be sound and complete. As in ....

....intruder s knowledge I(m) In particular the initial state is just ; I0 , where I0 contains the information (e.g. keys) initially known to the intruder. 9 6. 2 Mapping to Linear Logic The close anity between multiset rewriting and simple fragments of linear logic has been known for a long time [3, 30, 15, 19, 5]. We extend this standard correspondence to take parameters and existentially quanti ed variables into consideration. A generic multiset M is mapped to the tensor product N M of its constituents, or 1 if M is empty. A multiset rewrite rule r : F ( x) 9 n: G( x; n) is translated into the ....

[Article contains additional citation context not shown here]

Mart -Oliet, N., and Meseguer, J. From Petri nets to linear logic. In Conference on Category Theory and Computer Science (1989), Springer-Verlag LNCS 389, pp. 313-337.

.... Girard himself has proposed quite different kinds of models such as coherent spaces [9] lattice theoretic models such as his phase semantics [9] and Hilbert space models [10] Although attempts to systematize the model theory of linear logic using category theory have been made (see for example [32,22,21]) particular models may not quite fit even those general notions of model. However, lattice theoretic models, including quantales, are in a sense the simplest, and are sufficient for completeness arguments. In hindsight we can view the process of endowing linear logic with such lattice theoretic ....

....are Sigma G terms on the constants S, for 1 i n and 1 j m. A map of signatures is just a function f : S S 0 . It induces a Sigma G homomorphism T Sigma G (f) T Sigma G (S) T Sigma G (S 0 ) and therefore an obvious function Sen LL (f ) Sen LL (S) Sen LL (S 0 ) 11 As in [32,22] we adopt a two sided sequent presentation. However, we could have adopted instead a one sided presentation as in [9] 32 Let LL denote the entailment system of linear logic as defined by its rules of deduction in any of its formulations, and let CE denote (unsorted) conditional equational ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1:69--101, 1991.

....of linear logic [16] a refinement of modal logic with an intrinsic and natural accounting of process states and events. The choice of linear logic is natural because of the very close connection between multiset rewriting and simple fragments of linear logic, which has been studied extensively [3, 30, 15, 19, 5]. We extend this standard correspondence to include first order parameters and existentially quantified variables. On the other hand, we also formally represent strand constructions as relatively simple formulas in first order linear logic. This encoding is also shown to be sound and complete. As ....

....knowledge I(m) In particular the initial state is just Pi; I0 , where I0 contains the information (e.g. keys) initially known to the intruder. 9 6. 2 Mapping to Linear Logic The close affinity between multiset rewriting and simple fragments of linear logic has been known for a long time [3, 30, 15, 19, 5]. We extend this standard correspondence to take parameters and existentially quantified variables into consideration. A generic multiset M is mapped to the tensor product N M of its constituents, or 1 if M is empty. A multiset rewrite rule r : F ( x) Gamma 9 n: G( x; n) is translated into ....

[Article contains additional citation context not shown here]

Mart' i-Oliet, N., and Meseguer, J. From Petri nets to linear logic. In Conference on Category Theory and Computer Science (1989), Springer-Verlag LNCS 389, pp. 313--337.

....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 ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poign'e, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313--340, Manchester, United Kingdom, September 1989.

....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 ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poign'e, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313--340, Manchester, United Kingdom, September 1989.

....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 ....

N. Mart ' i-Oliet and J. Meseguer, From Petri Nets to Linear Logic, In Category Theory in Computer Science, Lecture Notes in Computer Science 389, 1989.

.... ( a Omega c) Gammaffi b) b Omega d Omega d) Gammaffi (c Omega d) a; c; d; d c; d This sequent is provable in linear logic if and only if there is a sequence of Petri net rule applications that transform the token set fa; c; d; dg to fc; dg. This connection has been well studied [5, 14, 30, 6, 9], and extended to cover other models of concurrency [22, 2, 35] Linear logic has also been applied to several other areas of computer science. One key application of the resource sensitive aspect of the logic was the development of a functional programming language implementation in which garbage ....

.... work is progressing to exploit the unique features of linear logic for use as a type system to study computational complexity [13] and compiler optimization techniques [40, 8, 29, 41, 34, 25] as well as uses in logic programming [16, 3, 4] natural language processing [24, 38] and concurrency [5, 30, 35]. These recent contributions are developing linear logic from a theoretical curiosity into a tool that already has practical use within mainstream computer science. ....

N. Mart ' i-Oliet and J. Meseguer. From Petri nets to linear logic. In: Springer LNCS 389, ed. by D.H. Pitt et al., 1989. 313-340.

....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]

N. Mart'i-Oliet, J. Meseguer: "From Petri Nets to Linear Logic", in Proceedings of the Third Conference on Category Theory and Computer Science, Manchester, UK, 1989, pp. 313--337, LNCS 389, Springer-Verlag, 1989.

.... resource sensitive aspect of the logic was the development of a functional programming language implementation in which garbage collection was replaced by explicit duplication operations based on linear logic Introduction 5 [Laf88] Further studies have demonstrated connections with Petri nets [Asp87, GG89, MOM89, AFG90, GG90] and other models of concurrency [Laf90, AV90] With regard to concurrency, there is a similarity between proof nets, the inherent model of computation associated with cut elimination in multiplicative linear logic (cf [Gir87a, Gir87b, DR89, Laf90] and connection graphs, which were designed to ....

....these lemmas to obtain our main result in Theorem 3.7. 3.2 Linear Logic Augmented With Theories Essentially, a theory is a set of nonlogical axioms (sequents) that may occur as leaves of a proof tree. The use of theories described here is an extension of earlier work on multiplicative theories [GG89, MOM89]. We define a positive literal to be one of the given propositional symbols p i . A negative literal is one of the p i symbols. An atomic formula is any positive or negative literal. For the theories of interest here, an axiom may be any linear logic sequent of the form C; p i 1 ; p ....

N. Mart ' i-Oliet and J. Meseguer. From Petri nets to linear logic. In: Springer LNCS 389, ed. by D.H. Pitt et al., 1989. 313-340.

....unlike Propositional CL. This is mainly due to the exponentials and is proven by encoding the behavior of certain simple counting machines [112, 113] ffl Multiplicative exponential LL (no additives and quantifiers) is EXPSPACE hard, which follows from an encoding of Petri nets in that fragment [120, 122] (see Section 4.1) ffl First order multiplicative additive LL (no exponentials) is NEXPTIME hard [114] ffl Multiplicative additive LL (no exponentials and quantifiers) is PSPACE complete [112, 113] ffl Multiplicative LL and Horn LL are NP complete, which is proven by reduction to the ....

....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 ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. Mathemathical Structures in Computer Science, 1:66--101, 1991. Revised version of paper in LNCS 389.

....unlike Propositional CL. This is mainly due to the exponentials and is proven by encoding the behavior of certain simple counting machines [112, 113] ffl Multiplicative exponential LL (no additives and quantifiers) is EXPSPACE hard, which follows from an encoding of Petri nets in that fragment [120, 122] (see Section 4.1) ffl First order multiplicative additive LL (no exponentials) is NEXPTIME hard [114] ffl Multiplicative additive LL (no exponentials and quantifiers) is PSPACE complete [112, 113] ffl Multiplicative LL and Horn LL are NP complete, which is proven by reduction to the ....

....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 ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. In D. Pitt et al., editors, Category Theory and Computer Science, number 389 in LNCS, pages 313--340, Manchester, Sept. 1989.

....object ; that is, the functor ( C op Gamma C defined by A = A Gammaffi is an involution (viz. the canonical morphism A Gamma ( A Gammaffi ) Gammaffi ) is an isomorphism) In addition various coherence conditions must hold a good account of these may be found in [M OM89]. Coherence theorems may be found in [BCST, Bl91, Bl92] An equivalent characterization of autonomous categories is given in [CS91] based on the notion of weakly distributive categories. That characterization is useful in contexts where it is easier to see how to model the tensor Omega , the ....

Mart'i-Oliet, N. and J. Meseguer "From Petri nets to linear logic", in D.H. Pitt et al., eds. Category Theory and Computer Science, Manchester 1989, Lecture Notes in Computer Science 389, Springer-Verlag, Berlin, Heidelberg, New York, 1989.

....proof. Thus, we have logical bases for the design of proof strategies in CLL fragments and then we can propose sketches for their design. 1 Introduction Linear Logic is a powerful and expressive logic with connections to various topics in computer science as logic programming [9, 10] concurrency [12] or typed concurrent functional programming [1] For these applications in various fragments of Classical Linear Logic (CLL) it is important to propose eOEcient proof search procedures knowing that theorem proving is signi cantly more diOEcult for CLL than for classical logic since there is no ....

N. Mart#-Oliet and J. Meseguer. From Petri nets to linear logic. In proceedings of Category Theory and Computer Science, Manchester, sept. 1989, volume 389 of Lecture Notes in Computer Science, pages 313340. Springer Verlag, 1989.

.... having an apple and a quarter and a quarter, which is a strictly better situation than having an apple and a quarter (non idempotence of Omega ) Several researchers realized independently that this ACI operation on multisets corresponds to the conjunctive connective Omega (tensor) in linear logic [4, 9, 24, 32, 33]. This complementary point of view sees a net as a theory in this fragment of linear logic. Then, it is possible to establish a precise connection between reachability in Petri nets and provability in tensor logic. For example, in order to get the tensor theory corresponding to our Petri net ....

....reduce to the rules of tensor logic, the fragment of propositional linear logic consisting only of the connective Omega and its neutral element 1 (see [43, Sections 5.3. 1 2] Therefore, the following triple equivalence (including the previous equivalence in Theorem 3) is immediate: Theorem 4 [32, 43] Let N be a Petri net with set of places S and set of transitions T , and M;M 0 be markings on S. Then, the marking M 0 is reachable from M if and only if the sequent M M 0 is provable in tensor logic from the axioms corresponding to T , if and only if there is an N ] rewrite [M ] ....

N. Mart'i-Oliet and J. Meseguer, From Petri Nets to Linear Logic, Mathematical Structures in Computer Science 1, 1991, pages 69--101.

....consider the representation of a class of Petri nets, P T nets, in each of these languages. Keywords: Petri nets, multiset rewriting systems, linear logic, logic programming. 1 Introduction The relationship between linear logic [4] and Petri Nets [8] has been a major object of investigation [2, 3, 6]. It was soon noticed that the monoidal structure of multisets can serve as an abstract link between Petri nets (where markings are multisets and transition are rewrite rules on multisets) and the multiplicative fragment of linear logic (where either Omega or O is the operation of the monoid and 1 ....

N. Mart'i-Oliet, J. Meseguer: "From Petri Nets to Linear Logic", in Conf. on Category Theory and Computer Science, pp. 313--337, LNCS 389, Springer-Verlag, 1989.

....machines to show undecidability, since zero test has no natural counterpart in linear logic, but there is a natural counterpart of Fork: the additive conjunction . The remaining ACM instructions may be encoded using techniques very similar to the well studied Petri net reachability encodings [8, 18, 38, 9, 16]. The full proof of undecidability is presented in [32] 2.2 Propositional Multiplicative Additive Linear Logic The multiplicative additive fragment of linear logic (MALL) excludes the reusable modals ; Thus, every formula is used at most once in any branch of any cut free MALL proof. Also, ....

....There are many related problems of interest. A few representative nice fragments and some other interesting cases are sketched here. 5.1 Multiplicative Exponential Linear Logic The multiplicative exponential (MELL) fragment is currently of unknown complexity. By Petri net reachability encodings [8, 18, 38, 9, 16], it must be at least expspace hard. Although Petri net reachability is decidable, there is no known encoding of MELL formulas in Petri nets. A proof of decidability of MELL may therefore lead to a new proof of the decidability of Petri net reachability, and therefore be of independent interest. ....

N. Mart ' i-Oliet and J. Meseguer. From Petri nets to linear logic. In: Springer LNCS 389, ed. by D.H. Pitt et al., 1989. 313-340.

....of classical LL can be interpreted as extending the internal language of the category with a construct for reasoning about control as data. To this end, we will consider the very natural categorical characterization of classicality in terms of dualizing objects, due to Mart i Oliet and Meseguer [MOM90, MOM91]. Despite its apparent simplicity, it is actually slightly stronger than the interpretation based on autonomous categories [Bar79] outlined by Seely [See89] For any object C in an SMCC, there exists a natural (in A) transformation A (A ( C) C; we can think of it as the function a A : k ....

Narciso Mart'i-Oliet and Jos'e Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1, 1991.

....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 ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1:66--101, 1991. Revised version of paper in Springer LNCS 389.

.... an apple and a quarter and a quarter, which is a strictly better situation than having an apple and a quarter (non idempotence of Omega ) Several researchers realized independently that this ACI operation on multisets corresponds to the conjunctive connective Omega (tensor) in linear logic [7, 161, 68, 103, 104]. This complementary point of view sees a net as a theory in this fragment of linear logic. Then, it is possible to establish a precise connection between reachability in Petri nets and provability in tensor logic. For example, in order to get the tensor theory corresponding to our Petri net ....

....to the rules of tensor logic, the fragment of propositional linear logic consisting only of the connective Omega and its neutral element 1 (see [115, Sections 5.3.1 2] Therefore, the following triple equivalence (including the previous equivalence in Theorem 1) is immediate: Theorem 2. [103, 115] Let N be a Petri net with set of places S and set of transitions T , and M;M 0 be markings on S. Then, the marking M 0 is reachable from M if and only if the sequent M M 0 is provable in tensor logic from the axioms corresponding to T , if and only if there is an N ] rewrite [M ] ....

N. Mart'i-Oliet and J. Meseguer. From Petri nets to linear logic. In D. P. et al., editor, Category Theory and Computer Science, pages 313--340. Springer LNCS 389,

.... 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 ....

.... 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 ....

N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1:69-101, 1991.

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N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. Mathematical Structures in Computer Science, 1:66-101, 1991. Revised version of paper in LNCS 389.

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N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313-340, Manchester, United Kingdom, September 1989.

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N. Mart ' i-Oliet and J. Meseguer. From Petri nets to linear logic. In: Springer LNCS 389, ed. by D.H. Pitt et al., 1989. 313-340.

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N. Mart ' i-Oliet and J. Meseguer. From Petri nets to linear logic. In: Springer LNCS 389, ed. by D.H. Pitt et al., 1989. 313-340.

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N. Mart#-Oliet and J. Meseguer. From Petri nets to linear logic. Technical Report 89-4R2, SRI International, Computer Science Laboratory, Menlo Park, CA, USA, December 1989. In: Pitt, D.H. et al.: Lecture Notes in Computer Science, Vol. 389.

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N. Mart-Oliet and J. Meseguer. From Petri nets to linear logic. In P. Dybjer, A. M. Pitts, D. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Proceedings of the Conference on Category Theory and Computer Science, Springer-Verlag LNCS 389, pages 313-340, Manchester, United Kingdom, September 1989.

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Martí-Oliet, N., and Meseguer, J. "From Petri nets to linear logic". Math. Struct. in Comp. Sci. 1,1 (Mar. 1991).

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