Petri nets with generalized algebra: a comparison (original) (raw)

Petri Algebras

2014

Abstract. The firing rule of Petri nets relies on a residuation operation for the commutative monoid of natural numbers. We identify a class of residuated commutative monoids, called Petri algebras, for which one can mimic the token game of Petri nets to define the behaviour of generalized Petri net whose flow relation and place contents are valued in such algebraic structures. We show that Petri algebras coincide with the positive cones of lattice-ordered commutative groups and constitute the subvariety of the (duals of) residuated lattices generated by the commutative monoid of natural numbers. We introduce a class of nets, termed lexicographic Petri nets, that are associated with the positive cones of the lexicographic powers of the additive group of real numbers. This class of nets is universal in the sense that any net associated with some Petri algebras can be simulated by a lexicographic Petri net. All the classical decidable properties of Petri nets however are undecidable o...

The essence of Petri nets and transition systems through Abelian groups

Electronic Notes in Theoretical Computer Science, 1998

In this paper we describe an abstract and (as we hope) a uniform frame for Petri net models, which enable to generalise algebra as well as enabling rule used in the dynamics of Petri nets. Our approach of such an abstract frame is based on using partial groupoids in Petri nets. Further, we study properties of Petri nets constructed in this manner through related labelled transition systems. In particular, we i n vestigate the relationships between properties of partial groupoids used in Petri nets and properties of labelled transition systems crucial for the existence of the state equation and linear algebraic techniques. We s h o w that partial groupoids embeddable to Abelian groups play an important role in preserving these properties.

General Morphisms of Petri Nets (Extended Abstract)

International Colloquium on Automata, Languages and Programming, 1999

A new notion of a general morphism of Petri nets is intro- duced. The new morphisms are shown to properly include the morphisms considered so far. The resulting category of Petri nets is shown to admit products. Potential applications of general morphisms are indicated.

Finite Completeness of Categories of Petri Nets

Fundamenta Informaticae, 2000

The problem of finite completeness of categories of Petri nets is studied. Since Petri nets have finite products, the problem reduces to the issue of the existence of equalizers. We show that the categories of Petri nets with general and Winskel morphisms do not admit equalizers, and hence are not finitely complete. An important class of multiplicative Petri net morphisms is also identified. The main positive result of the paper states that reachable Petri nets and multiplicative morphisms form a finitely complete category. As an application of this result, some well-known categories turn out to be finitely complete. Since, for instance, all morphisms between reachable safe Petri nets are multiplicative it follows that the category of reachable safe Petri nets nets is finitely complete.

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.

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.

An algebraic structure of petri nets

Lecture Notes in Computer Science, 1980

A relational model for non-deterministic programs is presented. Several predicate transformers are introduced and it is shown that one of them satisfies all the healthiness criteria indicated by Dijkstra for a useful total correctness predicate transformer.

Petri nets step transitions and proofs in partially commutative linear logic

2001

We encode the execution of Petri nets in Partially Commutative Linear Logic, an intuitionistic logic introduced by Ph. de Groote which contains both commutative and non commutative connectives. We are thus able to faithfully represent the concurrent firing of Petri nets as long as it can be depicted by a series-parallel order. This coding is inspired from the description of context-free languages by Lambek grammars. This report is an extended version (with complete proofs) of an article to appear in the proceedings of the Logic Colloquium 1999 (Utrecht).

Algebraic Conservative Petri Nets Based on Symmetric Groups

International Journal on Information Sciences and Computing, 2008

In this paper we define a new sub class of Petri nets called algebraic conservative Petri nets (ACPN) for a given symmetric group S. We prove that the resulting Petri net (ACPN) is a marked graph. In particular, we show that the algebraic n conservative Petri nets associated with S3 and S5 has decompositions respectively, for the sets of places such that each block is both siphon and trap and hence the underlying directed graphs of these algebraic conservative Petri nets are Eulerian. Also we show that each of the ACPN associated with these groups has a subset of places which are both siphon and trap such that the input transitions equal the output transitions and both of them equal to the set of all transitions of these algebraic conservative Petri nets and hence that the underlying directed graphs of these algebraic conservative Petri nets associated with S and S are Hamiltonian.

Functorial Models for Petri Nets

Information and Computation, 2001

We show that although the algebraic semantics of place/transition Petri nets under the collective token philosophy can be fully explained in terms of strictly symmetric monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory, because it lacks universality and also functoriality. We introduce the notion of pre-nets to overcome this, obtaining a fully satisfactory categorical treatment, where the operational semantics of nets yields an adjunction. This allows us to present a uniform logical description of net behaviors under both the collective and the individual token philosophies in terms of theories and theory morphisms in partial membership equational logic. Moreover, since the universal property of adjunctions guarantees that colimit constructions on nets are preserved in our algebraic models, the resulting semantic framework has good compositional properties.