Abstract
Linear logic has been recently introduced by Girard as a logic of actions that seems well suited for concurrent computation. In this paper, we establish a systematic correspondence between Petri nets, linear logic theories, and linear categories. Such a correspondence sheds new light on the relationships between linear logic and concurrency, and on how both areas are related to category theory. Categories are here viewed as concurrent systems whose objects are states, and whose morphisms are transitions. This is an instance of the Lambek-Lawvere correspondence between logic and category theory that cannot be expressed within the more restricted framework of the Curry-Howard correspondence.
Preview
Unable to display preview. Download preview PDF.
References
Andrea Asperti, A Logic for Concurrency, manuscript, November 1987.
Michael Barr, *Autonomous Categories, Volume 752 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979.
Michael Barr, Personal communication, December 1988.
Pierpaolo Degano, José Meseguer and Ugo Montanari, Axiomatizing Net Computations and Processes, in: Proc. Logic in Computer Science, Asilomar, 1989, pages 175–185.
H. J. Genrich and K. Lautenbach, System Modelling with High-Level Petri Nets, Theoretical Computer Science, 13, 1981, pages 109–136.
Jean-Yves Girard, Linear Logic, Theoretical Computer Science, 50, 1987, pages 1–102.
Jean-Yves Girard, Linear Logic and Parallelism, in: Proc. School on Semantics of Parallelism, IAC, CNR, Roma, 1986.
Jean-Yves Girard, Towards a Geometry of Interaction, in: J. W. Gray and A. Scedrov (eds.), Proc. A. M. S. Conference on Categories in Computer Science and Logic, Boulder, 1987.
Carl Gunter and Vijay Gehlot, A Proof-theoretic Operational Semantics for True Concurrency, Preliminary Report, 1989.
W. A. Howard, The Formulae-as-Types Notion of Construction, in: J. P. Seldin and J. R. Hindley (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London, 1980, pages 479–490.
Yves Lafont, The Linear Abstract Machine, Theoretical Computer Science, 59, 1988, pages 157–180.
Yves Lafont, Introduction to Linear Logic, Lecture notes for the Summer School on Constructive Logic and Category Theory, Isle of Thorns, August 1988.
Yves Lafont, From Linear Algebra to Linear Logic, Preliminary Draft, November 1988.
Joachim Lambek, Deductive Systems and Categories I, Mathematical Systems Theory, 2, 1968, pages 287–318.
Joachim Lambek, Deductive Systems and Categories II, in: Category Theory, Homology Theory and their Applications I, Volume 86 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1969, pages 76–122.
Joachim Lambek, Deductive Systems and Categories III, in: F. W. Lawvere (ed.), Toposes, Algebraic Geometry and Logic, Volume 274 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1972, pages 57–82.
F. W. Lawvere, Adjointness in Foundations, Dialectica, 23, 1969, pages 281–296.
Saunders MacLane, Natural Associativity and Commutativity, Rice University Studies, 49, 1963, pages 28–46.
Saunders MacLane, Categories for the Working Mathematician, Volume 5 of Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1971.
Saunders MacLane (ed.), Coherence in Categories, Volume 281 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1972.
Saunders MacLane, Why Commutative Diagrams Coincide with Equivalent Proofs, in: S. A. Amitsur, D. J. Saltman and G. B. Seligman (eds.), Algebraists' Homage: Papers in Ring Theory and Related Topics, Volume 13 of Contemporary Mathematics, American Mathematical Society, Providence, 1982, pages 387–401.
José Meseguer and Ugo Montanari, Petri Nets Are Monoids: A New Algebraic Foundation for Net Theory, in: Proc. Logic in Computer Science, Edinburgh, 1988, pages 155–164.
José Meseguer and Ugo Montanari, Petri Nets Are Monoids, Technical Report SRI-CSL-88-3, C.S.Lab., SRI International, January 1988, submitted for publication.
Robin Milner, Interpreting One Concurrent Calculus in Another, in: Proc. Int. Conf. on Fifth Generation Computer Systems, Tokyo, 1988, pages 321–326.
Valeria C. V. de Paiva, The Dialectica Categories, Ph. D. thesis, University of Cambridge, 1988.
Dag Prawitz, Natural Deduction: A Proof-Theoretical Study, Almqvist and Wiksell, Stockholm, 1965.
Wolfgang Reisig, Petri Nets: An Introduction, Springer-Verlag, Berlin, 1985.
R. A. G. Seely, Linear Logic, *-Autonomous Categories and Cofree Coalgebras, in: J. W. Gray and A. Scedrov (eds.), Proc. A. M. S. Conference on Categories in Computer Science and Logic, Boulder, 1987.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
MartÃ-Oliet, N., Meseguer, J. (1989). From petri nets to linear logic. In: Pitt, D.H., Rydeheard, D.E., Dybjer, P., Pitts, A.M., Poigné, A. (eds) Category Theory and Computer Science. Lecture Notes in Computer Science, vol 389. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0018359
Download citation
DOI: https://doi.org/10.1007/BFb0018359
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51662-0
Online ISBN: 978-3-540-46740-3
eBook Packages: Springer Book Archive