Linear Realizability

  • Naohiko Hoshino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4646)


We define a notion of relational linear combinatory algebra (rLCA) which is a generalization of a linear combinatory algebra defined by Abramsky, Haghverdi and Scott. We also define a category of assemblies as well as a category of modest sets which are realized by rLCA. This is a linear style of realizability in a way that duplicating and discarding of realizers is allowed in a controlled way. Both categories form linear-non-linear models and their coKleisli categories have a natural number object. We construct some examples of rLCA’s which have some relations to well known PCA’s.


Natural Transformation Full Subcategory Monoidal Category Left Adjoint Monoidal Functor 


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  1. Abramsky, S., Haghverdi, E., Scott, P.J.: Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12(5), 625–665 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. Abramsky, S., Lenisa, M.: Linear realizability and full completeness for typed lambda-calculi. Ann. Pure Appl. Logic 134(2-3), 122–168 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. Benton, P.N.: A mixed linear and non-linear logic: Proofs, terms and models (extended abstract). In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 121–135. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. Bierman, G.M.: What is a categorical model of intuitionistic linear logic? In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 78–93. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  5. Hyland, J.M.E., Ong, C.-H.L.: Modified realizability toposes and strong normalization proofs. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 179–194. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  6. Hoshino, N.: Linear realizability. Master’s thesis, Kyoto University (2007)Google Scholar
  7. Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North Holland, Amsterdam (1999)MATHGoogle Scholar
  8. Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Cambridge Phil. Soc. 119(3) (1996)Google Scholar
  9. Kleene, S.C., Vesley, R.E.: The Foundations of Intuitionistic Mathematics, especially in relation to recursive functions. North-Holland, Amsterdam (1965)Google Scholar
  10. Lafont, Y.: Soft linear logic and polynomial time. Theor. Comput. Sci. 318(1-2), 163–180 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. Longley, J.: Realizability Toposes and Language Semantics. PhD thesis, Edinburgh University (1995)Google Scholar
  12. Simpson, A.K.: Reduction in a linear lambda-calculus with applications to operational semantics. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 219–234. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Naohiko Hoshino
    • 1
  1. 1.RIMS, Kyoto University 

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