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SN combinators and partial combinatory algebras

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Rewriting Techniques and Applications (RTA 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1379))

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Abstract

We introduce an intersection typing system for combinatory logic. We prove the soundness and completeness for the class of partial combinatory algebras. We derive that a term of combinatory logic is typeable iff it is SN. Let F be the class of non-empty filters which consist of types. Then F is an extensional non-total partial combinatory algebra. Furthermore, it is a fully abstract model with respect to the set of sn c terms of combinatory logic. By F, we can solve Bethke-Klop's question; “find a suitable representation of the finally collapsed partial combinatory algebra of P”. Here, P is a partial combinatory algebra, and is the set of closed sn terms of combinatory logic modulo the inherent equality. Our solution is the following: the finally collapsed partial combinatory algebra of P is representable in F. To be more precise, it is isomorphically embeddable into F.

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References

  1. Y. Akama. A λ-to-CL translation for strong normalization. In Ph. de Groote and J.R. Hindley, eds., TLCA, Proceedings, vol. 1210 of LNCS, pp. 1–10. 1997.

    Google Scholar 

  2. H.P. Barendregt. The Lambda Calculus, Its Syntax and Semantics. North-Holland, second edition, 1984.

    Google Scholar 

  3. M. Beeson. Foundations of Constructive Mathematics. 1984.

    Google Scholar 

  4. A. Berarducci and B. Intrigila. Some new results on easy lambda-terms. TCS, 121:71–88, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Bethke. On the existence of extensional partial combinatory algebras. JSL, 52(3):819–833, 1987.

    MathSciNet  MATH  Google Scholar 

  6. I. Bethke and J. W. Klop. Collapsing partial combinatory algebras. In G. Dowek et al., eds., HOA, Proceedings, vol. 1074 of LNCS, pp. 57–73. 1996.

    Google Scholar 

  7. N. Cagman and J.R. Hindley. Combinatory weak reduction in lambda calculus. Technical report, University of Wales Swansea, Swansea, U.K., 1997.

    Google Scholar 

  8. M. Dezani-Ciancaglini and J.R. Hindley. Intersection types for combinatory logic. TCS, 100(2):303–324, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Egidi, F. Honsell, and S. Ronchi Della Rocca. Operational, denotational and logical descriptions: A case study. Fundamenta Informaticae, 16:149–169, 1992.

    MathSciNet  MATH  Google Scholar 

  10. B. Gramlich. Termination and Confluence Properties of Structured Rewrite Systems. PhD thesis, Universität Kaiserslautern, 1996.

    Google Scholar 

  11. F. Honsell and S. Ronchi della Rocca. Models for theories of functions strictly depending on all their arguments. Internal report, Department of Computer Science, Turin, Italy, 1984.

    Google Scholar 

  12. J. Martin E. Hyland. A syntactic characterization of the equality in some models of the lambda calculus. J. London Math. Soc. (2), 12:361–370, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  13. J.M.E. Hyland and C.-H. L. Ong. Modified realizability toposes and strong normalization proofs (extended abstract). In M. Bezem and J.F. Groote, eds., TLCA, Proceedings, vol. 664 of LNCS, pp. 179–194. 1993.

    Google Scholar 

  14. G.Jacopini. A condition for identifying two elements of whatever model of combinatory logic. In C.Böhm, editor, λ-calculus and computer science theory, vol. 37 of LNCS, pp. 213–219. 1975.

    Google Scholar 

  15. G. Jacopini and M. Venturini Zilli. Easy terms in the lambda-calculus. Fundamenta Informaticae, 8(2):225–233, 1985.

    MathSciNet  MATH  Google Scholar 

  16. J.W. Klop. Term rewriting systems. In S. Abramsky et al., editors, Handbook of Logic in Computer Science, volume 2, pages 2–117. Oxford UP, 1992.

    Google Scholar 

  17. T. Kurata. Subtype relations yielding filter lambda-models. In T. Ida, A. Ohori, and M. Takeichi, eds., Second Fuji International Workshop on Functional and Logic Programming, Proceedings, pp. 210–229. World Scientific, 1997.

    Google Scholar 

  18. R. Pino Perez. An extensional partial combinatory algebra based on λ-terms. In A. Tarlecki, editor, MFCS, Proceedings, vol. 520 of LNCS, pp. 387–396. 1991.

    Google Scholar 

  19. G. Plotkin. Call-by-name, call-by-value, and the λ-calculus. TCS, 1:125–159, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. van Bakel. Complete restrictions of the intersection type discipline. TCS, 102:135–163, 1992.

    Article  MATH  Google Scholar 

  21. C. P. Wadsworth. The relation between computational and denotational properties for scott's D∞-models of the lambda calaulus. SIAM J. Computing, 5:488–521, 1976.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Information Science, Tokyo University, Bunkyou-ku, 113, Tokyo

    Yohji Akama

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  1. Yohji Akama
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Tobias Nipkow

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© 1998 Springer-Verlag

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Akama, Y. (1998). SN combinators and partial combinatory algebras. In: Nipkow, T. (eds) Rewriting Techniques and Applications. RTA 1998. Lecture Notes in Computer Science, vol 1379. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0052378

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  • DOI: https://doi.org/10.1007/BFb0052378

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64301-2

  • Online ISBN: 978-3-540-69721-3

  • eBook Packages: Springer Book Archive

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