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

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Abstract

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.

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Notes

  1. Given functors \(F,G: {\mathscr {A}} \rightarrow {\mathscr {B}}\), a transformation\(\tau : F \Rightarrow G: {\mathscr {A}} \rightarrow {\mathscr {B}}\) is a family \(\tau = \{\tau _a: F(a) \rightarrow G(a) \mid a \in O_{\mathscr {A}}\}\) of arrows in \({\mathscr {B}}\) indexed by objects of \({\mathscr {A}}\). We say that \(\tau \) is natural if \(\tau _a ; G(f) = F(f) ; \tau _{b}\) for every arrow \(f: a \rightarrow b\) in \({\mathscr {A}}\) and an isomorphism if all its components \(\tau _a\)’s are so.

  2. Even if, despite the common usage, the terminology adopted in [19] is that of mega-graphs.

  3. In abstract data types terms, an order-sorted algebra with type \(\mathbf {AP}({R})\) included in type \(\mathbf {ANP}({R})\).

  4. Indeed, also for \(\rho \) and \(\nabla \) would suffice to consider only those instances associated to the objects of \(\mathbf {AP}({R})\), and consider the axioms involving them as definitions of derived operators.

References

  1. Baldan, P., Bonchi, F., Gadducci, F., Monreale, G.V.: Modular encoding of synchronous and asynchronous interactions using open Petri nets. Sci. Comput. Program. 109, 96–124 (2015)

    Article  MATH  Google Scholar 

  2. Baldan, P., Bruni, R., Montanari, U.: Pre-nets, read arcs and unfolding: a functorial presentation. In: Wirsing, M., Pattinson, D., Hennicker, R. (eds.) Algebraic Development Techniques (WADT 2002), Lecture Notes in Computer Science, vol. 2755, pp. 145–164. Springer (2003)

  3. Baldan, P., Gadducci, F.: Petri nets are dioids. In: Meseguer, J., Rosu, G. (eds.) Algebraic Methodologies and Software Technology (AMAST 2008), Lecture Notes in Computer Science, vol. 5140, pp. 51–66. Springer (2008)

  4. Bonchi, F., Gadducci, F., Kissinger, A., Sobocinski, P., Zanasi, F.: Rewriting modulo symmetric monoidal structure. In: Grohe, M., Koskinen, E., Shankar, N. (eds.) Logic in Computer Science (LICS 2016), pp. 710–719. ACM, New York (2016)

    Google Scholar 

  5. Bonchi, F., Sobocinski, P., Zanasi, F.: Full abstraction for signal flow graphs. In: Rajamani, S.K., Walker, D. (eds.) Principles of Programming Languages (POPL 2015), pp. 515–526. ACM, New York (2015)

    Google Scholar 

  6. Bruni, R., Gadducci, F., Montanari, U.: Normal forms for algebras of connections. Theor. Comput. Sci. 286(2), 247–292 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruni, R., Meseguer, J., Montanari, U., Sassone, V.: Functorial models for Petri nets. Inf. Comput. 170(2), 207–236 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Corradini, A., Gadducci, F.: An algebraic presentation of term graphs, via gs-monoidal categories. Appl. Categ. Struct. 7(4), 299–331 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Corradini, A., Gadducci, F.: A functorial semantics for multi-algebras and partial algebras, with applications to syntax. Theor. Comput. Sci. 286, 293–322 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Degano, P., Meseguer, J., Montanari, U.: Axiomatizing net computations and processes. In: Logic in Computer Science (LICS 1989), pp. 175–185. IEEE Computer Society (1989)

  11. Degano, P., Meseguer, J., Montanari, U.: Axiomatizing the algebra of net computations and processes. Acta Inform. 33(7), 641–667 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Engelfriet, J.: Branching processes of Petri nets. Acta Inform. 28(6), 575–591 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Esparza, J., Heljanko, K.: Unfoldings: a partial order approach to model checking. Springer, Berlin (2008)

    MATH  Google Scholar 

  14. Gadducci, F., Montanari, U.: Axioms for contextual net processes. In: Larsen, K., Skyum, S., Winskel, G. (eds.) Automata, Languages and Programming (ICALP 1998), Lecture Notes in Computer Science, vol. 1443, pp. 296–308. Springer (1998)

  15. Goltz, U., Reisig, W.: The non-sequential behaviour of Petri nets. Inf. Control 57(2/3), 125–147 (1983)

    Article  MATH  Google Scholar 

  16. Gorrieri, R.: Process algebras for Petri nets: the alphabetization of distributed systems. Springer, Berlin (2017)

    Book  MATH  Google Scholar 

  17. Green, A., Altenkirch, T.: From reversible to irreversible computations. In: Selinger, P. (ed.) Quantum Programming Languages (QPL 2006), Electronic Notes in Theoretical Computer Science, vol. 210, pp. 65–74. Elsevier, Amsterdam (2008)

    Google Scholar 

  18. Grosu, R., Lucanu, D., Stefanescu, G.: Mixed relations as enriched semiringal categories. Univers. Comput. Sci. 6(1), 112–129 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Hackney, P., Robertson, M.: On the category of PROPs. Appl. Categ. Struct. 23(4), 543–573 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Camb. Philos. Soc. 119(3), 447–468 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lack, S.: Composing PROPs. Theory Appl. Categ. 13(9), 147–163 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Laplaza, M.: Coherence for distributivity. In: Coherence in Categories, Lecture Notes in Mathematics, vol. 281, pp. 29–72. Springer (1972)

  23. Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  24. Martì-Oliet, N., Meseguer, J.: From Petri nets to linear logic through categories: a survey. Found. Comput. Sci. 2(4), 297–399 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. May, J.: The construction of \(\text{ E }_\infty \) ring spaces from bipermutative categories. Geom. Topol. Monogr. 16, 283–330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Meseguer, J., Montanari, U.: Petri nets are monoids. Inf. Comput. 88(2), 105–155 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Milner, R.: The Space and Motion of Communicating Agents. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  28. Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains, part 1. Theor. Comput. Sci. 13, 85–108 (1981)

    Article  MATH  Google Scholar 

  29. Petri, C.: Kommunikation mit automaten. Ph.D. Thesis, Institut für Instrumentelle Matematik, Bonn (1962)

  30. Reisig, W.: Petri nets: an introduction. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  31. Sangiorgi, D., Walker, D.: The Pi-Calculus: A Theory of Mobile Processes. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  32. Sassone, V.: An axiomatization of the category of Petri net computations. Math. Struct. Comput. Sci. 8(2), 117–151 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Selinger, P.: A survey of graphical languages for monoidal categories. Spring. Lect. Notes Phys. 13(813), 289–355 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Stefanescu, G.: Reaction and control I. Mixing additive and multiplicative network algebras. Log. J. IGPL 6(2), 348–369 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stefanescu, G.: Network Algebra. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  36. Winskel, G.: Event structures. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) Petri Nets: Applications and Relationships to Other Models of Concurrency, Lecture Notes in Computer Science, vol. 255, pp. 325–392. Springer (1987)

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Acknowledgements

We are indebted to Professor Peter May for the interaction and the fruitful discussions on bimonoidal categories, as well as to the reviewers for their remarks and pointers to the literature.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Paolo Baldan

  2. Dipartimento di Informatica, Università di Pisa, Largo Bruno Pontecorvo 3, 56127, Pisa, Italy

    Fabio Gadducci

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  1. Paolo Baldan
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  2. Fabio Gadducci
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Correspondence to Fabio Gadducci.

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Baldan, P., Gadducci, F. Petri nets are dioids: a new algebraic foundation for non-deterministic net theory. Acta Informatica 56, 61–92 (2019). https://doi.org/10.1007/s00236-018-0314-0

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