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....done for the sequential world [26] In a different direction, Jean Yves Girard [13, 16, 17, 18] has instituted the rapidly growing area of linear logic, a radical modification of traditional logic which appears to have strong connections with theoretical computer science. In several publications [16, 15] Girard has suggested that linear logic should 2 have deeper connections with parallelism and concurrent computation. This suggestion was taken up more formally by Abramsky [1] in an influential series of lectures, and in some unpublished work of Robin Milner [24] The Abramsky view is ....

J-Y. Girard. Linear Logic and Parallelism, manuscript, 1987.

.... seen as a refinement of classical logic sequent calculus where contraction and weakening rules are forbidden [22] Both classical linear logic (CLL) and its intuitionistic fragment, called intuitionistic linear logic (ILL) are studied as models of computation [2] and of semantics of parallelism [20]. 3.1 CLASSICAL LINEAR LOGIC The usual presentation of CLL consists in starting from the CL sequent calculus with suppression of the contraction and weakening rules. Then the equivalence between the additive and multiplicative presentations of the and connectives disappears and then leads to ....

J.Y. Girard. Linear logic and parallelism. In Mathematical Models for the Semantics of Parallelism, LNCS 280, pages 166--182, Roma, Italia, October 1986.

....done for the sequential world [26] In a different direction, Jean Yves Girard [13, 16, 17, 18] has instituted the rapidly growing area of linear logic, a radical modification of traditional logic which appears to have strong connections with theoretical computer science. In several publications [16, 15] Girard has suggested that linear logic should 2 have deeper connections with parallelism and concurrent computation. This suggestion was taken up more formally by Abramsky [1] in an influential series of lectures, and in some unpublished work of Robin Milner [24] The Abramsky view is ....

J-Y. Girard. Linear Logic and Parallelism, manuscript, 1987.

....maximal ideals require Zorn s lemma, a highly non constructive principle. This makes the other approach difficult, the integrated development, where an algorithm is extracted from a constructive existence proof. The notion of integrated and external programming logics was introduced by Girard [Gir86], for a comparison between the integrated and external approach to program development, see [Dyb90] Grobner bases together with an algorithm for computing them, was introduced by Buchberger [Buc65,Buc85] It can be seen as a generalisation of the Euclidian algorithm for computing the greatest ....

J-Y Girard. Linear logic and parallelism. In M. Venturini Zilli, editor, Mathematical Models for the Semantics of Parallelism, number LNCS 280, pages 166--182. Springer-Verlag, September 1986.

....that are related to the one presented here: ffl It is possible to make more of the arithmetic operations on R than we do. Koymans [ 16 ] uses this structure to define a temporal logic based on the reals. ffl This approach can be taken further using the concept of enrichment; Pratt s group in [ 6 ] gives several categories based on the reals with a monoidal structure. These are then used to enrich behavioural categories. Such techniques are comprehensively displayed in Kasangian and Labella s [ 15 ] ffl Another starting point is idea that we can associate a set of predicates with a state ....

....a Petri net view, EATCS Monographs on theoretical computer science, volume 13, Springer Verlag, 1988. 5] S. Brookes and A. Roscoe, An improved failures model for communicating sequential processes, in the Proceedings NSF SERC Seminar on Concurrency, Volume 197, Springer Verlag LNCS, 1985. [6] R. Casley, R. Crew, J. Meseguer, and V. Pratt, Temporal structures, in Category Theory and Computer Science, Volume 389, SpringerVerlag LNCS, 1989. 7] T. Coquand, Categories of embeddings, Theoretical Computer Science, Volume 68 (1989) Pp. 221 237. 8] T. Coquand, C. Gunter, and G. Winskel, ....

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J.-Y. Girard. Linear logic and parallelism. In M. Venturini-Zilli, editor, Mathematical Models for the Semantics of Parallelism, volume 280. Springer-Verlag LNCS, 1987.

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