Space-Efficient Computation by Interaction

A Type System for Logarithmic Space
  • Ulrich Schöpp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)


We introduce a typed functional programming language for logarithmic space. Its type system is an annotated subsystem of Hofmann’s polytime LFPL. To guide the design of the programming language and to enable the proof of logspace-soundness, we introduce a realisability model over a variant of the Geometry of Interaction. This realisability model, which takes inspiration from Møller-Neergaard and Mairson’s work on BC \(^{\rm --}_{\epsilon}\), provides a general framework for modelling space-restricted computation.


Turing Machine Memory Location Simple Object Linear Logic Memory Block 
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  1. 1.
    Abramsky, S., Ghica, D.R., Murawski, A.S., Ong, C.-H.L.: Applying game semantics to compositional software modeling and verification. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 421–435. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Abramsky, S., Jagadeesan, R.: Games and full completeness for multiplicative linear logic (extended abstract). In: Shyamasundar, R.K. (ed.) FSTTCS 1992. LNCS, vol. 652, pp. 291–301. Springer, Heidelberg (1992)Google Scholar
  4. 4.
    Bellantoni, S.J.: Predicative Recursion and Computational Complexity. PhD thesis, University of Toronto (1992)Google Scholar
  5. 5.
    Dal Lago, U., Hofmann, M.: Quantitative models and implicit complexity. In: Ramanujam, R., Sen, S. (eds.) FSTTCS 2005. LNCS, vol. 3821, pp. 189–200. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Danos, V., Herbelin, H., Regnier, L.: Game semantics & abstract machines. In: LICS 1996, pp. 394–405 (1996)Google Scholar
  7. 7.
    Ghica, D.R., McCusker, G.: Reasoning about idealized algol using regular languages. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 103–115. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Hahgverdi, E.: A Categorical Approach to Linear Logic, Geometry of Proofs and Full Completeness. PhD thesis, University of Ottawa (2000)Google Scholar
  9. 9.
    Hofmann, M.: Linear types and non-size-increasing polynomial time computation. Information and Computation 183, 57–85 (1999)CrossRefGoogle Scholar
  10. 10.
    Hofmann, M.: Semantics of linear/modal lambda calculus. Journal of Functional Programming 9(3), 247–277 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hofmann, M.: Programming languages capturing complexity classes. SIGACT News 31(1), 31–42 (2000)CrossRefGoogle Scholar
  12. 12.
    Hyland, J.M.E., Schalk, A.: Glueing and orthogonality for models of linear logic. Theoretical Computer Science 294(1–2), 183–231 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jones, N.D.: logspace and ptime characterized by programming languages. Theoretical Computer Science 228(1–2), 151–174 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kristiansen, L.: Neat function algebraic characterizations of LOGSPACE andLINESPACE. Computational Complexity 14, 72–88 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Longley, J.R.: Realizability Toposes and Language Semantics. PhD thesis, University of Edinburgh (1994)Google Scholar
  16. 16.
    Mackie, I.: The geometry of interaction machine. In: POPL 1995 (1995)Google Scholar
  17. 17.
    Møller-Neegaard, P.: Complexity Aspects of Programming Language Design. PhD thesis, Brandeis University (2004)Google Scholar
  18. 18.
    Neergaard, P.M.: A functional language for logarithmic space. In: Chin, W.-N. (ed.) APLAS 2004. LNCS, vol. 3302, pp. 311–326. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Sannella, D., Hofmann, M., Aspinall, D., Gilmore, S., Stark, I., Beringer, L., Loidl, H.-W., MacKenzie, K., Momigliano, A., Shkaravska, O.: Mobile Resource Guarantees. Trends in Functional Programing 6 (2005) IntellectGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ulrich Schöpp
    • 1
  1. 1.Ludwig-Maximilians-Universität MünchenMünchenGermany

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