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Petri nets are dioids: a new algebraic foundation for non-deterministic net theory

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Paolo Baldan at University of Padua
Paolo Baldan
Fabio Gadducci at University of Pisa
Fabio Gadducci
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Abstract and Figures

In a seminal paper Montanari and Meseguer have shown that an algebraic interpretation of Petri nets in terms of commutative monoids can be used to provide an elegant characterisation of the deterministic computations of a net, accounting for their sequential and parallel composition. A smoother and more complete theory for deterministic computations has been later developed by relying on the concept of pre-net, a variation of Petri nets with a non-commutative flavor. This paper shows that, along the same lines, by adding an (idempotent) operation and thus considering dioids (idempotent semirings) rather than just monoids, one can faithfully characterise the non-deterministic computations of a net.
a The P/T Petri net N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_0$$\end{document} and b one of its pre-net implementation R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}
a The P/T Petri net N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_0$$\end{document} and b one of its pre-net implementation R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}
… 
The set of rules generating ANP(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ANP}({R})$$\end{document}
The set of rules generating ANP(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ANP}({R})$$\end{document}
… 
The set of axioms quotienting ANP(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ANP}({R})$$\end{document}
The set of axioms quotienting ANP(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {ANP}({R})$$\end{document}
… 
The natural transformation ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}
The natural transformation ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}
… 
The process !a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$!_a$$\end{document}
+8
The process !a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$!_a$$\end{document}
… 
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Acta Informatica (2019) 56:61–92
https://doi.org/10.1007/s00236-018-0314-0
ORIGINAL ARTICLE
Petri nets are dioids: a new algebraic foundation
for non-deterministic net theory
Paolo Baldan1·Fabio Gadducci2
Received: 6 September 2016 / Accepted: 14 January 2018 / Published online: 24 January 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In a seminal paper Montanari and Meseguer have shown that an algebraic inter-
pretation of Petri nets in terms of commutative monoids can be used to provide an elegant
characterisation of the deterministic computations of a net, accounting for their sequential
and parallel composition. A smoother and more complete theory for deterministic computa-
tions has been later developed by relying on the concept of pre-net, a variation of Petri nets
with a non-commutative flavor. This paper shows that, along the same lines, by adding an
(idempotent) operation and thus considering dioids (idempotent semirings) rather than just
monoids, one can faithfully characterise the non-deterministic computations of a net.
1 Introduction
Petri nets [30] are one of the most studied and best known models for concurrent systems.
Due to the conceptual simplicity of the model and its intuitive graphical presentation, since
their introduction, which dates back to the Sixties [29], they have attracted the interest of
both theoreticians and practitioners.
The basic operational behaviour of Petri nets can be straightforwardly defined in terms of
sequences of transition firings, according to the “token game”. Concurrency in computations
can be made explicit by resorting to a semantics given in terms of (non-sequential) deter-
ministic processes à la Goltz and Reisig [15]. A process describes the transition firings in
a computation and their mutual dependency relations. Concretely, a deterministic processes
is an acyclic, deterministic net whose structure induces a partial order on transitions, which
can be seen as occurrences of transition firings in the original net. A deterministic process
BFabio Gadducci
fabio.gadducci@unipi.it
Paolo Baldan
baldan@math.unipd.it
1Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padua, Italy
2Dipartimento di Informatica, Università di Pisa, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Among the most recent ones we mention [8,9,10,11]. There the objective is typically the generation of all nets starting from a small number of components [9,10,11] or the preservation of the semantics [8] via composition. ...
... Among the most recent ones we mention [8,9,10,11]. There the objective is typically the generation of all nets starting from a small number of components [9,10,11] or the preservation of the semantics [8] via composition. The work in [11] 300 focuses on the algebraic structure of non-deterministic computations but is based on pre-nets [12] rather than Petri nets. ...
... There the objective is typically the generation of all nets starting from a small number of components [9,10,11] or the preservation of the semantics [8] via composition. The work in [11] 300 focuses on the algebraic structure of non-deterministic computations but is based on pre-nets [12] rather than Petri nets. The difference is that the preand post-sets of transitions are lists and not sets, so that the interfaces for composition are intrinsically ordered. ...
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