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A taste of linear logic

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Mathematical Foundations of Computer Science 1993 (MFCS 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 711))

Abstract

This tutorial paper provides an introduction to intuitionistic logic and linear logic, and shows how they correspond to type systems for functional languages via the notion of ‘Propositions as Types”. The presentation of linear logic is simplified by basing it on the Logic of Unity. An application to the array update problem is briefly discussed.

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References

  1. S. Abramsky, Computational interpretations of linear logic. Presented at Workshop on Mathematical Foundations of Programming Language Semantics, 1990. To appear in Theoretical Computer Science.

    Google Scholar 

  2. N. Benton, G. Bierman, V. de Paiva, and M. Hyland, Type assignment for intuitionistic linear logic. Draft paper, August 1992.

    Google Scholar 

  3. J. Chirimar, C. A. Gunter, and J. G. Riecke. Linear ML. In Symposium on Lisp and Functional Programming, ACM Press, San Francisco, June 1992.

    Google Scholar 

  4. H. B. Curry and R. Feys, Combinatory Logic, North Holland, 1958.

    Google Scholar 

  5. J.-Y. Girard, Interprétation functionelle et élimination des coupures dans l'arithmétique d'ordre supérieure. Ph.D. thesis, Université Paris VII, 1972.

    Google Scholar 

  6. J.-Y. Girard, Linear logic. Theoretical Computer Science, 50:1–102, 1987.

    Google Scholar 

  7. J.-Y. Girard, On the unity of logic. Manuscript, 1991.

    Google Scholar 

  8. J.-Y. Girard, Y. Lafont, and P. Taylor, Proofs and types, Cambridge University Press, 1989.

    Google Scholar 

  9. R. Hindley, The principal type scheme of an object in combinatory logic. Trans. Am. Math. Soc., 146:29–60, December 1969.

    Google Scholar 

  10. S. Holmström, A linear functional language. Draft paper, Chalmers University of Technology, 1988.

    Google Scholar 

  11. W. A. Howard, The formulae-as-types notion of contstruction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism, Academic Press, 1980. (The original version was circulated privately in 1969.)

    Google Scholar 

  12. Y. Lafont, The linear abstract machine. Theoretical Computer Science, 59:157–180, 1988.

    Google Scholar 

  13. P. Lincoln and J. Mitchell, Operational aspects of linear lambda calculus. In 7'th Symposium on Logic in Computer Science, IEEE Press, Santa Cruz, California, June 1992.

    Google Scholar 

  14. R. Milner, A theory of type polymorphism in programming. J. Comput. Syst. Sci., 17:348–375, 1978.

    Google Scholar 

  15. V. Pratt, Event spaces and their linear logic. In AMAST '91: Algebraic Methodology And Software Technology, Iowa City, Springer Verlag LNCS, 1992.

    Google Scholar 

  16. U. S. Reddy, A typed foundation for directional logic programming. In E. Lamma and P. Mello, editors, Extensions of logic programming, Lecture Notes in Artificial Intelligence 660, Springer-Verlag, 1993.

    Google Scholar 

  17. J. C. Reynolds, Towards a theory of type structure. In B. Robinet, editor, Proc. Colloque sur la Programmation, LNCS 19, Springer-Verlag.

    Google Scholar 

  18. J. C. Reynolds, Three approaches to type structure. In Mathematical Foundations of Software Development, LNCS 185, Springer-Verlag, 1985.

    Google Scholar 

  19. R. A. G. Seely, Linear logic, *-autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

    Google Scholar 

  20. A. S. Troelstra, Lectures on Linear Logic. CSLI Lecture Notes, 1992.

    Google Scholar 

  21. P. Wadler, Linear types can change the world! In IFIP TC 2 Working Conference on Programming Concepts and Methods, Sea of Galilee, Israel, April 1990. Published as M. Broy and C. Jones, editors, Programming Concepts and Methods, North Holland, 1990.

    Google Scholar 

  22. P. Wadler, Is there a use for linear logic? In Conference on Partial Evaluation and Semantics-Based Program Manipulation (PEPM), ACM Press, New Haven, Connecticut, June 1991.

    Google Scholar 

  23. P. Wadler, There's no substitute for linear logic. Presented at Workshop on Mathematical Foundations of Programming Language Semantics, Oxford, April 1992.

    Google Scholar 

  24. P. Wadler, A syntax for linear logic. Presented at Conference on Mathematical Foundations of Programming Language Semantics, New Orleans, April 1993.

    Google Scholar 

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Andrzej M. Borzyszkowski Stefan Sokołowski

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© 1993 Springer-Verlag Berlin Heidelberg

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Wadler, P. (1993). A taste of linear logic. In: Borzyszkowski, A.M., Sokołowski, S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57182-5_12

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  • DOI: https://doi.org/10.1007/3-540-57182-5_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57182-7

  • Online ISBN: 978-3-540-47927-7

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