Abstract
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.
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G. T. Allwein and J. M. Dunn Dunn. Kripke models of linear logic. The Journal of Symbolic Logic, 58(2):514–545, 1993.
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.
Andrea Asperti. A Logic for Concurrency. manuscript, November 1987.
Samson Abramsky and Steve Vickers. Linear Process Logic. Notes by Steve Vickers, 1988.
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.
A. Blass. A game semantics for linear logic. Annals of Pure and Applied Logic, 56:183–220, 192.
Carolyn Brown. Relating Petri Nets to Formulae of Linear Logic. Technical Report ECS LFCS 89-87, University of Edinburgh, 1989.
Gian Luca Cattani. An existence predicate for a linear logic of Petri net.
T. Coquand, C. Gunter, and G. Winskel. Domain theoretic models of polymorphism. Information and Computation, 81(2), 1989.
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.
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.
Carl Gunter and Vijay Gehlot. A Proof-Theoretic Operational Semantics for True Concurrency. Preliminary Report, 1989.
Carl Gunter and Vijay Gehlot. Nets as Tensor Theories. Technical Report MS-CIS-89-68, University of Pennsylvania, October 1989.
Jean-Yves Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45, 1986.
Jean-Yves Girard. Linear Logic. Theoretical Computer Science, 50(1):1–102, 1987.
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.
M. H. T. Hack. Decidability questions for Petri nets. PhD thesis, MIT, 1976.
Yves Lafont. The Linear Abstract Machine. Theoretical Computer Science, 59:157–180, 1988.
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.
E. W. Mayr. An algorithm for the general Petri net reachability problem. SIAM Journal of Computing, 13(3):441–459, 1984.
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.
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.
Wolfgang Reisig. Petri Nets, An Introduction, volume 4 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1985.
Kimmo I. Rosenthal. A Note on Girard Quantales. To appear in: Cah. de Top. et G. D., 1989.
G. Sambin. Manuscript reported to us by Per Martin-Löf.
D. Yetter. Quantales and Non-Commutative Linear Logic. (preprint).
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© 1994 Springer-Verlag Berlin Heidelberg
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Engberg, U., Winskel, G. (1994). Linear logic on Petri nets. In: de Bakker, J.W., de Roever, W.P., Rozenberg, G. (eds) A Decade of Concurrency Reflections and Perspectives. REX 1993. Lecture Notes in Computer Science, vol 803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58043-3_20
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DOI: https://doi.org/10.1007/3-540-58043-3_20
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