Linear logic on Petri nets

  • Uffe Engberg
  • Glynn Winskel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 803)


This article shows how individual Petri nets form models of Girard's intuitionistic linear logic. It explores questions of expressiveness and completeness of linear logic with respect to this interpretation. An aim is to use Petri nets to give an understanding of linear logic and give some appraisal of the value of linear logic as a specification logic for Petri nets. This article might serve as a tutorial, providing one in-road into Girard's linear logic via Petri nets. With this in mind we have added several exercises and their solutions. We have made no attempt to be exhaustive in our treatment, dedicating our treatment to one semantics of intuitionistic linear logic.

Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. The strongest logic considered is intuitionistic linear logic, with ⊗, ⊸, &, ⊕ and the exponential ! (“of course”), and forms of quantification. This logic is shown sound and complete with respect to atomic nets (these include nets in which every transition leads to a nonempty multiset of places). The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. A start is made on decidability issues.


Linear logic Petri nets 


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  1. [AD93]
    G. T. Allwein and J. M. Dunn Dunn. Kripke models of linear logic. The Journal of Symbolic Logic, 58(2):514–545, 1993.Google Scholar
  2. [AJ92]
    Samson Abramsky and R. Jagadeesan. Games and full completeness for multiplicative linear logic. In Rudrapatna Shyamasundar, editor, FST and TCS 12, Foundations of Software Technology and Theoretical Computer Science, New Delhi, India, December 18–20, pages 291–301. Springer-Verlag (LNCS, 652), 1992.Google Scholar
  3. [Asp87]
    Andrea Asperti. A Logic for Concurrency. manuscript, November 1987.Google Scholar
  4. [AV88]
    Samson Abramsky and Steve Vickers. Linear Process Logic. Notes by Steve Vickers, 1988.Google Scholar
  5. [Ber78]
    G. Berry. Stable models of typed λ-calculi. In Fifth International Colloquium on Automata, Languages and Programs, pages 72–89. Springer-Verlag (LNCS 62), 1978.Google Scholar
  6. [Bla92]
    A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183–220, 192.Google Scholar
  7. [Bro89]
    Carolyn Brown. Relating Petri Nets to Formulae of Linear Logic. Technical Report ECS LFCS 89-87, University of Edinburgh, 1989.Google Scholar
  8. [Cat]
    Gian Luca Cattani. An existence predicate for a linear logic of Petri net.Google Scholar
  9. [CGW89]
    T. Coquand, C. Gunter, and G. Winskel. Domain theoretic models of polymorphism. Information and Computation, 81(2), 1989.Google Scholar
  10. [EW90]
    Uffe Henrik Engberg and Glynn Winskel. Petri Nets as Models of Linear Logic. In CAAP '90, Coll. on Trees in Algebra and Programming Copenhagen, Denmark, May 15–18, pages 147–161. Springer-Verlag (LNCS 431), 1990. Appears as Technical Report, DAIMI PB-301.Google Scholar
  11. [EW93]
    Uffe Henrik Engberg and Glynn Winskel. Completeness Results for Linear Logic on Petri Nets (Extended Abstract). In MFCS '93, Mathematical Foundations of Computer Science, Gdańsk, Poland, August 30–September 3. Springer-Verlag (LNCS 711), 1993. Appears as Technical Report, DAIMI PB-435.Google Scholar
  12. [GG89a]
    Carl Gunter and Vijay Gehlot. A Proof-Theoretic Operational Semantics for True Concurrency. Preliminary Report, 1989.Google Scholar
  13. [GG89b]
    Carl Gunter and Vijay Gehlot. Nets as Tensor Theories. Technical Report MS-CIS-89-68, University of Pennsylvania, October 1989.Google Scholar
  14. [Gir86]
    Jean-Yves Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45, 1986.Google Scholar
  15. [Gir87]
    Jean-Yves Girard. Linear Logic. Theoretical Computer Science, 50(1):1–102, 1987.Google Scholar
  16. [GL87]
    Jean-Yves Girard and Yves Lafont. Linear Logic and Lazy Computation. In Proc. TAPSOFT 87 (Pisa), vol. 2, pages 52–66. Springer-Verlag (LNCS 250), 1987.Google Scholar
  17. [Hac76]
    M. H. T. Hack. Decidability questions for Petri nets. PhD thesis, MIT, 1976.Google Scholar
  18. [Laf88]
    Yves Lafont. The Linear Abstract Machine. Theoretical Computer Science, 59:157–180, 1988.Google Scholar
  19. [LMSN90]
    P. Lincoln, J. Mitchell, A. Scedrov, and Shankar N.. Decision problems for propositional linear logic. In Foundations of Computer Science (FOCS'90), volume II, pages 662–671, St. Louis, MO, October 1990.Google Scholar
  20. [May84]
    E. W. Mayr. An algorithm for the general Petri net reachability problem. SIAM Journal of Computing, 13(3):441–459, 1984.Google Scholar
  21. [MOM89]
    Narciso Martí-Oliet and José Meseguer. From Petri Nets to Linear Logic. In Category Theory and Computer Science, Manchester, UK. Springer-Verlag (LNCS 389), 1989.Google Scholar
  22. [MOM91]
    Narciso Martí-Oliet and José Meseguer. From Petri Nets to Linear Logic: a Survey. International Journal of Foundations of Computer Science, 2(4):297–399, 1991.Google Scholar
  23. [Rei85]
    Wolfgang Reisig. Petri Nets, An Introduction, volume 4 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1985.Google Scholar
  24. [Ros89]
    Kimmo I. Rosenthal. A Note on Girard Quantales. To appear in: Cah. de Top. et G. D., 1989.Google Scholar
  25. [Sam]
    G. Sambin. Manuscript reported to us by Per Martin-Löf.Google Scholar
  26. [Yet]
    D. Yetter. Quantales and Non-Commutative Linear Logic. (preprint).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Uffe Engberg
    • 1
  • Glynn Winskel
    • 1
  1. 1.BRICS, Centre of the Danish National Research Foundation Computer Science DepartmentAarhus UniversityAarhus CDenmark

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