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From Petri nets to linear logic†

Published online by Cambridge University Press:  04 March 2009

Narciso Martí-Oliet  and
José Meseguer
Narciso Martí-Oliet
Affiliation:
SRI International, Menlo Park, CA 94025, USA and Center for the Study of Language and Information, Stanford University, Stanford, CA 94305, USA
José Meseguer
Affiliation:
SRI International, Menlo Park, CA 94025, USA and Center for the Study of Language and Information, Stanford University, Stanford, CA 94305, USA
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Abstract

Linear logic has recently been 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 the objects of which are states, and the morphisms of which 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.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 1 , Issue 1 , March 1991 , pp. 69 - 101
Copyright
Copyright © Cambridge University Press 1991

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References

Abramsky, S. and Vickers, S. (1988) Linear process logic. Notes by Steve Vickers.Google Scholar
Asperti, A. (1987) A logic for concurrency. Unpublished manuscript.Google Scholar
Asperti, A., Ferrari, G. L. and Gorrieri, R. (1990) Implicative formulae in the ‘proofs as computations’ analogy. In: Proc. 17th Annual ACM Symp. on Principles of Programming Languages, San Francisco, California, pp. 59–71.Google Scholar
Barr, M. (1979) *-Autonomous Categories, Springer Lecture Notes in Mathematics 752. Springer-Verlag.CrossRefGoogle Scholar
Degano, P., Meseguer, J. and Montanari, U. (1989) Axiomatizing net computations and processes. In: Proc. 4th Annual Symp. on Logic in Computer Science, Asilomar, California, pp. 175–85.Google Scholar
Engberg, U. and Winskel, G. (1990) Petri nets as models of linear logic. In: Arnold, A., ed., CAAP’90, 15th. Colloquium on Trees in Algebra and Programming, Copenhagen, Denmark, May 1990, Springer Lecture Notes in Computer Science 431. Springer-Verlag, pp. 147–61.Google Scholar
Genrich, H. J. and Lautenbach, K. (1981) System modelling with high-level Petri nets. Theoret. Comput. Sci. 13 109–36.CrossRefGoogle Scholar
Girard, J.-Y. (1987a) Linear logic. Theoret. Comput. Sci. 50 1–102.CrossRefGoogle Scholar
Girard, J.-Y. (1987b) Linear logic and parallelism. In: Zilli, Marisa Venturini, ed., Mathematical Models for the Semantics of Parallelism, Advanced School, Rome, Italy, September 1986, Springer Lecture Notes in Computer Science 280. Springer-Verlag, pp. 166–82.Google Scholar
Girard, J.-Y. (1989) Towards a geometry of interaction. In: Gray, J. W. and Scedrov, A., eds, Categories in Computer Science and Logic, Boulder, June 1987, Vol. 92 of Contemporary Mathematics, American Mathematical Society, pp. 69–108.Google Scholar
Gunter, C. and Gehlot, V. (1989) Nets as tensor theories. Technical Report MS-CIS-89–68 Logic & Computation 17, Department of Computer and Information Science, University of Pennsylvania.Google Scholar
Howard, W. A. (1980) The formulae-as-types notion of construction. In: Seldin, J. P. and Hindley, J. R., eds, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, pp. 479–90.Google Scholar
Joyal, A and Street, R. (1988) Braided Tensor Categories, Macquarie Mathematics Reports.Google Scholar
Joyal, A (1989) The Geometry of Tensor Calculus I, Macquarie Mathematics Reports.Google Scholar
Lafont, Y. (1988a) The linear abstract machine. Theoret. Comput. Sci. 59 157–80.CrossRefGoogle Scholar
Lafont, Y. (1988b) Introduction to linear logic. Lecture Notes for the Summer School on Constructive Logic and Category Theory, Isle of Thorns.Google Scholar
Lafont, Y. (1988c) From linear algebra to linear logic. Manuscript.Google Scholar
Lambek, J. (1968) Deductive systems and categories I. Math. Sys. Theory 2 287–318.CrossRefGoogle Scholar
Lambek, J. (1969) Deductive systems and categories II. In: Category Theory, Homology Theory and their Applications I, Springer Lecture Notes in Mathematics 86. Springer-Verlag, pp. 76–122.CrossRefGoogle Scholar
Lamberk, J. (1972) Deductive systems and categories III. In: Lawvere, F. W., ed., Toposes, Algebraic Geometry and Logic, Springer Lecture Notes in Mathematics 274. Springer-Verlag, pp. 57–82.Google Scholar
Lawvere, F. W. (1969) Adjointness in foundations. Dialectica 23 281–96.CrossRefGoogle Scholar
MacLane, S. (1963) Natural associativity and commutativity. Rice University Studies 49 28–46.Google Scholar
MacLane, S. (1971) Categories for the Working Mathematician, Vol. 5 of Graduate Texts in Mathematics, Springer-Verlag.CrossRefGoogle Scholar
MacLane, S. (1972) ed., Coherence in Categories, Springer Lecture Notes in Mathematics 281. Springer-Verlag.Google Scholar
MacLane, S. (1982) Why commutative diagrams coincide with equivalent proofs. In: Amitsur, S. A., Saltman, D. J. and Seligman, G. B., eds, Algebraists’ Homage: Papers in Ring Theory and Related Topics, Vol. 13 of Contemporary Mathematics, American Mathematical Society, pp. 387–401.Google Scholar
Martí-Oliet, N. and Meseguer, J. (1989) From Petri nets to linear logic. In: Pitt, D. H.et al., eds, Category Theory and Computer Science, Manchester, UK, September 1989, Springer Lecture Notes in Computer Science 389. Springer-Verlag, pp. 313–40.Google Scholar
Martí-Oilet, N. (1990a) Duality in closed and linear categories. Technical Report SRI-CSL-90–01, Computer Science Laboratory, SRI International.Google Scholar
Martí-Oilet, N. (1990b) An algebraic axiomatization of linear logic models. In: Reed, G. M., Roscoe, A. W. and Wachter, R., eds, Topology in Computer Science, Oxford University Press.Google Scholar
Meseguer, J. and Montanari, U. (1988a) Petri nets arc monoids: A new algebraic foundation for net theory. In: Proc. 3rd. Ann. Symp. on Logic in Computer Science, Edinburgh, Scotland, pp. 155–64.Google Scholar
Meseguer, J. (1990) Petri nets are monoids. Informat. Computat. 88 105–55.CrossRefGoogle Scholar
Milner, R. (1988) Interpreting one concurrent calculus in another. In: Proc. Int. Conf on Fifth Generation Computer Systems, Tokyo, Japan, pp. 321–6.Google Scholar
Mints, G. (1989) Some information on linear logic, manuscript.Google Scholar
de Paiva, V. C. V. (1988) The dialectica categories, PhD thesis, University of Cambridge.CrossRefGoogle Scholar
Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study, Almqvist and Wiksell.Google Scholar
Reisig, W. (1985) Petri Nets: An Introduction, Vol. 4 of EATCS Monographs on Theoretical Computer Science, Springer-Verlag.Google Scholar
Seely, R. A. G. (1989) Linear logic, *-autonomous categories and cofree coalgebras. In: Gray, J. W. and Scedrov, A., eds, Categories in Computer Science and Logic, Boulder, June 1987, Vol. 92 of Contemporary Mathematics, American Mathematical Society, pp. 371–82.CrossRefGoogle Scholar
Yetter, D. N. (1990) Quantales and (non-commutative) linear logic. J. Symb. Logic 55 41–64.CrossRefGoogle Scholar