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Unification and Logarithmic Space

  • Conference paper
Rewriting and Typed Lambda Calculi (RTA 2014, TLCA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8560))

Abstract

We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof theory and more specifically linear logic and Geometry of Interaction.

We show how unification can be used to build a model of computation by means of specific subalgebras associated to finite permutation groups.

We then prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. We also show that the construction can naturally encode pointer machines, an intuitive way of understanding logarithmic space computing.

This work was partly supported by the ANR-10-BLAN-0213 Logoi and the ANR-11-BS02-0010 Récré.

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References

  1. Girard, J.Y.: Linear logic. Theoret. Comput. Sci. 50(1), 1–101 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Girard, J.Y., Scedrov, A., Scott, P.J.: Bounded linear logic: a modular approach to polynomial-time computability. Theoret. Comput. Sci. 97(1), 1–66 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Girard, J.Y.: Light linear logic. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 145–176. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  4. Schöpp, U.: Stratified bounded affine logic for logarithmic space. In: LICS, pp. 411–420. IEEE Computer Society (2007)

    Google Scholar 

  5. Dal Lago, U., Hofmann, M.: Bounded linear logic, revisited. LMCS 6(4) (2010)

    Google Scholar 

  6. Gaboardi, M., Marion, J.Y., Ronchi Della Rocca, S.: An implicit characterization of PSPACE. ACM Trans. Comput. 13(2), 18:1–18:36 (2012)

    Google Scholar 

  7. Baillot, P., Mazza, D.: Linear logic by levels and bounded time complexity. Theoret. Comput. Sci. 411(2), 470–503 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Girard, J.Y.: Towards a geometry of interaction. In: Gray, J.W., Ščedrov, A. (eds.) Proceedings of the AMS Conference on Categories, Logic and Computer Science. Categories in Computer Science and Logic, vol. 92, pp. 69–108. AMS (1989)

    Google Scholar 

  9. Asperti, A., Danos, V., Laneve, C., Regnier, L.: Paths in the lambda-calculus. In: LICS, pp. 426–436. IEEE Computer Society (1994)

    Google Scholar 

  10. Laurent, O.: A token machine for full geometry of interaction (extended abstract). In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 283–297. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  11. Girard, J.Y.: Geometry of interaction 1: Interpretation of system F. Studies in Logic and the Foundations of Mathematics 127, 221–260 (1989)

    Article  MathSciNet  Google Scholar 

  12. Baillot, P., Pedicini, M.: Elementary complexity and geometry of interaction. Fund. Inform. 45(1-2), 1–31 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Girard, J.Y.: Normativity in logic. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds.) Epistemology Versus Ontology. Logic, Epistemology, and the Unity of Science, vol. 27, pp. 243–263. Springer (2012)

    Google Scholar 

  14. Aubert, C., Seiller, T.: Characterizing co-NL by a group action. Arxiv preprint abs/1209.3422 (2012)

    Google Scholar 

  15. Aubert, C., Seiller, T.: Logarithmic space and permutations. Arxiv preprint abs/1301.3189 (2013)

    Google Scholar 

  16. Girard, J.Y.: Geometry of interaction III: accommodating the additives. In: Girard, J.Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic. London Mathematical Society Lecture Note Series, vol. 222, pp. 329–389. CUP (1995)

    Google Scholar 

  17. Girard, J.Y.: Three lightings of logic. In: Ronchi Della Rocca, S. (ed.) CSL. LIPIcs, vol. 23, pp. 11–23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

    Google Scholar 

  18. Knight, K.: Unification: A multidisciplinary survey. ACM Comput. Surv. 21(1), 93–124 (1989)

    Article  MATH  Google Scholar 

  19. Dwork, C., Kanellakis, P.C., Mitchell, J.C.: On the sequential nature of unification. J. Log. Program. 1(1), 35–50 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  20. Dwork, C., Kanellakis, P.C., Stockmeyer, L.J.: Parallel algorithms for term matching. SIAM J. Comput. 17(4), 711–731 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Aubert, C.: Linear Logic and Sub-polynomial Classes of Complexity. PhD thesis, Université Paris 13–Sorbonne Paris Cité (2013)

    Google Scholar 

  22. Hartmanis, J.: On non-determinancy in simple computing devices. Acta Inform. 1(4), 336–344 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lange, K.J., McKenzie, P., Tapp, A.: Reversible space equals deterministic space. J. Comput. System Sci. 60(2), 354–367 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CNRS, I2M, UMR 7373, Aix-Marseille Université, 13453, Marseille, France

    Clément Aubert & Marc Bagnol

Authors
  1. Clément Aubert
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  2. Marc Bagnol
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Editors and Affiliations

  1. Inria, 23 avenue d’Italie, CS 81321, 75214, Paris Cedex 13, France

    Gilles Dowek

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Aubert, C., Bagnol, M. (2014). Unification and Logarithmic Space. In: Dowek, G. (eds) Rewriting and Typed Lambda Calculi. RTA TLCA 2014 2014. Lecture Notes in Computer Science, vol 8560. Springer, Cham. https://doi.org/10.1007/978-3-319-08918-8_6

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  • DOI: https://doi.org/10.1007/978-3-319-08918-8_6

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08917-1

  • Online ISBN: 978-3-319-08918-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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