Petri nets step transitions and proofs in partially commutative linear logic (original) (raw)
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A description of the non-sequential execution of Petri nets in partially commutative linear logic
Logic Colloquium, 2004
Un modèle de l'exécution non séquentielle des réseaux de Petri en logique linéaire partiellement commutative Résumé : Nous décrivons l'exécution d'un réseau de Petri dans la logique linéaire partiellement commutative, une logique intuitionniste introduite par Ph. de Groote qui contient et des connecteurs commutatifs et des connecteurs non commutatifs. Nous sommes ainsi capable de décrire fidèlement l'exécution en parallèle d'un réseau de Petri, du moins tant que celle-ci reste un ordre série-parallèle. Ce codage s'inspire de la description des langages algébriques par les grammaires de Lambek.
Process Semantics of Petri Nets over Partial Algebra
Lecture Notes in Computer Science, 2000
Petri nets are monoids" is the title and the central idea of the paper . It provides an algebraic approach to define both nets and their processes as terms. A crucial assumption for this concept is that arbitrary concurrent composition of processes is defined, which holds true for place/transition Petri nets where places can hold arbitrarily many tokens. This paper defines a similar concept for elementary Petri nets, which are elementary net systems with arbitrary initial marking. Since markings of elementary nets cannot be added arbitrarily, some operators are only defined partially; hence we employ concepts of partial algebra. The main result of the paper states that the semantics based on process terms agrees with the classical partial-order process semantics for elementary net systems. More precisely, we provide a syntactic equivalence notion for process terms and a bijection from according equivalence classes of process terms to isomorphism classes of partially ordered processes.
Pomset logic: a non-commutative extension of classical linear logic
Typed Lambda Calculi and Applications, 1997
We extend the multiplicative fragment of linear logic with a non-commutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae. We rstly examine coherence semantics, where we introduce the before connective, and ordered products of formulae. Secondly we extend the syntax of multiplicative proof nets to these new operations. We then prove strong normalisation, and con uence. Coming back to the denotational semantics that we started with, we establish in an unusual way the soundness of this calculus with respect to the semantics. The converse, i.e. a kind of completeness result, is simply stated: we refer to a report for its lengthy proof. We conclude by mentioning more results, including a sequent calculus which is interpreted by both the semantics and the proof net syntax, although we are not sure that it takes all proof nets into account. The relevance of this calculus to computational linguistics, process calculi, and semantics of imperative programming is brie y explained in the introduction.
2007
This paper provides a natural deduction system for Partially Commutative Intuitionistic Multiplicative Linear Logic (PCIMLL) and establishes its normalisation and subformula property. Such a system involves both commutative and non commutative connectives and deals with context that are series-parallel multisets of formulae. This calculus is the extension of the one introduced by de Groote presented by the second order for modelling Petri net execution, with a full entropy which allow order to be relaxed into any suborder -as opposed to the Non Commutative Logic of Abrusci and Ruet. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Up to now PCIMLL with full entropy had no natural deduction. In particular for linguistic applications, such a syntax is much welcome to construct semantic representations from syntactic analyses.
Petri nets with generalized algebra: a comparison
Electronic Notes in Theoretical Computer Science, 1999
In the last decade we can see substantial e ort to develop an abstract and uniform constructions for Petri nets. Most of such abstractions are based on algebraic characterizations of Petri nets. They work mostly over commutative monoids and their various subclasses, namely cancellative commutative monoids or cones of Abelian groups. In the paper we study relationships between Petri nets with generalized underlying algebra. More precisely, w e study Petri nets over commutative monoids, cancellative commutative monoids, cones of Abelian groups, and fully ordered cones of Abelian groups. As the main result, we show that classes of reachability graphs of Petri nets over cancellative commutative monoids and cones of Abelian groups coincide (up to isomorphism). In other words, partial order on used cancellative commutative monoid plays no role in expressive power of Petri nets. However, as shows the fact that the class of reachability graphs of nets over fully ordered cones is a proper subclass of the class of reachability graphs of nets over cancellative commutative monoids, the total order on used monoids plays an important r o l e i n expressive p o wer of Petri nets.
Focusing and Proof-Nets in Linear and Non-commutative Logic
Lecture Notes in Computer Science, 1999
Linesir Logic [4] has radsed a lot of interest in computer research, especially because of its resource sensitive nature. One hne of research studies proof construction procedures and their interpretation as computation£il models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Pocusing", has been described in [2]. Pocusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Pocusing Ccin also be interpreted in the proof-net formalism, where it appecirs, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the CEise of the Non-Commutative logic of [1], and all the computational exploitation of Pocusing which has been performed in the commutative case can thus be revised and adapted to the non commutative case. * This work was performed while the second author was visiting XRCE; this visit was supported by the European TMR [Training and Mobility for Researchers) Network "Linear Logic in Computer Science" (esp. the Rome and Marseille sites, XRCE being attEiched to the latter).
Propositional dynamic logic for Petri Nets
Logic Journal of IGPL, 2014
Propositional Dynamic Logic (PDL) is a multi-modal logic used for specifying and reasoning on sequential programs. Petri Net is a widely used formalism to specify and to analyse concurrent programs with a very nice graphical representation. In this work, we propose a PDL to reasoning about Petri Nets. First we define a compositional encoding of Petri Nets from basic nets as terms. Second, we use these terms as PDL programs and provide a compositional semantics to PDL Formulas. Finally, we present an axiomatization and prove completeness w.r.t. our semantics. The advantage of our approach is that we can do reasoning about Petri Nets using our dynamic logic and we do not need to to translate it to other formalisms. Moreover our approach is compositional allowing for construction of complex nets using basic ones.