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Memoryful geometry of Interaction II: recursion and adequacy

Published:11 January 2016Publication History
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Abstract

A general framework of Memoryful Geometry of Interaction (mGoI) is introduced recently by the authors. It provides a sound translation of lambda-terms (on the high-level) to their realizations by stream transducers (on the low-level), where the internal states of the latter (called memories) are exploited for accommodating algebraic effects of Plotkin and Power. The translation is compositional, hence ``denotational,'' where transducers are inductively composed using an adaptation of Barbosa's coalgebraic component calculus. In the current paper we extend the mGoI framework and provide a systematic treatment of recursion---an essential feature of programming languages that was however missing in our previous work. Specifically, we introduce two new fixed-point operators in the coalgebraic component calculus. The two follow the previous work on recursion in GoI and are called Girard style and Mackie style: the former obviously exhibits some nice domain-theoretic properties, while the latter allows simpler construction. Their equivalence is established on the categorical (or, traced monoidal) level of abstraction, and is therefore generic with respect to the choice of algebraic effects. Our main result is an adequacy theorem of our mGoI translation, against Plotkin and Power's operational semantics for algebraic effects.

References

  1. S. Abramsky, R. Jagadeesan, and P. Malacaria. Full abstraction for PCF. Information and Computation, 163(2):409–470, 2000. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. S. Abramsky, E. Haghverdi, and P. J. Scott. Geometry of Interaction and linear combinatory algebras. Math. Struct. in Comp. Sci., 12(5): 625–665, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. U. Dal Lago and U. Schöpp. Type inference for sublinear space functional programming. In Programming Languages and Systems, volume 6461 of Lecture Notes in Computer Science, pages 376–391. Springer Berlin Heidelberg, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. V. Danos and L. Regnier. Local and asynchronous beta-reduction (an analysis of girard’s execution formula). In Logic in Computer Science, 1993. LICS ’93., Proceedings of Eighth Annual IEEE Symposium on, pages 296–306, 1993.Google ScholarGoogle ScholarCross RefCross Ref
  5. J. Egger, R. E. Møgelberg, and A. Simpson. Enriching an effect calculus with linear types. In E. Grädel and R. Kahle, editors, Computer Science Logic, volume 5771 of Lecture Notes in Computer Science, pages 240–254. Springer Berlin Heidelberg, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. M. Fernández and I. Mackie. Call-by-value λ-graph rewriting without rewriting. In A. Corradini, H. Ehrig, H.-J. Kreowski, and G. Rozenberg, editors, Graph Transformation, volume 2505 of Lecture Notes in Computer Science, pages 75–89. Springer Berlin Heidelberg, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. D. Ghica. Geometry of Synthesis: a structured approach to VLSI design. In POPL 2007, pages 363–375. ACM, 2007. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. D. R. Ghica and A. I. Smith. Geometry of synthesis II: from games to delay-insensitive circuits. Electr. Notes Theor. Comput. Sci., 265: 301–324, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. D. R. Ghica and A. I. Smith. Geometry of synthesis III: resource management through type inference. In T. Ball and M. Sagiv, editors, Proceedings of the 38th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2011, Austin, TX, USA, January 26- 28, 2011, pages 345–356. ACM, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. D. R. Ghica, A. I. Smith, and S. Singh. Geometry of Synthesis IV: compiling affine recursion into static hardware. In ICFP, pages 221– 233, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. J.-Y. Girard. Geometry of interaction 1: Interpretation of system F. In S. V. R. Ferro, C. Bonotto and A. Zanardo, editors, Logic Colloquium ’88 Proceedings of the Colloquium held in Padova, volume 127 of Studies in Logic and the Foundations of Mathematics, pages 221–260. Elsevier, 1989.Google ScholarGoogle Scholar
  12. J. Y. Girard. Geometry of interaction 2: Deadlock-free algorithms. In Proceedings of the International Conference on Computer Logic, COLOG-88, pages 76–93, New York, NY, USA, 1990. Springer-Verlag New York, Inc. ISBN 0-387-52335-9. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. E. Haghverdi. Towards a geometry of recursion. Theor. Comput. Sci., 412(20):2015–2028, Apr. 2011. ISSN 0304-3975.. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. M. Hasegawa. On traced monoidal closed categories. Math. Struct. in Comp. Sci., 19(2):217–244, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. N. Hoshino. A modified GoI interpretation for a linear functional programming language and its adequacy. In FoSSaCS 2011, volume 6604 of Lect. Notes Comp. Sci., pages 320–334. Springer, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. N. Hoshino, K. Muroya, and I. Hasuo. Memoryful Geometry of Interaction: from coalgebraic components to algebraic effects. In CSL-LICS 2014, page 52. ACM, 2014. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. J. Hyland and C.-H. Ong. On full abstraction for PCF: I, II, and III. Information and Computation, 163(2):285–408, 2000. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. A. Joyal, R. Street, and D. Verity. Traced monoidal categories. Math. Proc. Cambridge Phil. Soc., 119(3):447–468, 1996.Google ScholarGoogle ScholarCross RefCross Ref
  19. U. D. Lago, C. Faggian, B. Valiron, and A. Yoshimizu. Parallelism and synchronization in an infinitary context. In LICS, 2015.Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. O. Laurent. A token machine for full Geometry of Interaction. In TLCA, pages 283–297, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. S. Mac Lane. Categories for the working mathematician. Springer, 1998.Google ScholarGoogle Scholar
  22. I. Mackie. The Geometry of Interaction machine. In POPL 1995, pages 198–208. ACM, 1995. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. E. Moggi. Computational lambda-calculus and monads. Tech. Report, pages 1–23, 1988.Google ScholarGoogle Scholar
  24. J. S. Pinto. Implantation Parallèle avec la Logique Linéaire (Applications des Réseaux d’Interaction et de la Géométrie de l’Interaction). PhD thesis, École Polytechnique, 2001. Main text in English.Google ScholarGoogle Scholar
  25. G. Plotkin and J. Power. Adequacy for algebraic effects. In FoSSaCS 2001, volume 2030 of Lect. Notes Comp. Sci., pages 1–24. Springer, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. G. Plotkin and J. Power. Algebraic operations and generic effects. Appl. Categorical Struct., 11(1):69–94, 2003.Google ScholarGoogle Scholar
  27. J. Power and E. Robinson. Premonoidal categories and notions of computation. Math. Struct. in Comp. Sci., 7(5):453–468, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. U. Schöpp. Computation-by-interaction with effects. In H. Yang, editor, Programming Languages and Systems, volume 7078 of Lecture Notes in Computer Science, pages 305–321. Springer Berlin Heidelberg, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. U. Schöpp. Call-by-value in a basic logic for interaction. In J. Garrigue, editor, Programming Languages and Systems, volume 8858 of Lecture Notes in Computer Science, pages 428–448. Springer International Publishing, 2014.Google ScholarGoogle Scholar
  30. U. Schöpp. From call-by-value to interaction by typed closure conversion. In APLAS, 2015. To appear.Google ScholarGoogle ScholarCross RefCross Ref

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        • Published in

          cover image ACM SIGPLAN Notices
          ACM SIGPLAN Notices  Volume 51, Issue 1
          POPL '16
          January 2016
          815 pages
          ISSN:0362-1340
          EISSN:1558-1160
          DOI:10.1145/2914770
          • Editor:
          • Andy Gill
          Issue’s Table of Contents
          • cover image ACM Conferences
            POPL '16: Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
            January 2016
            815 pages
            ISBN:9781450335492
            DOI:10.1145/2837614

          Copyright © 2016 ACM

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          New York, NY, United States

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          • Published: 11 January 2016

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