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Light Affine Set Theory: A Naive Set Theory of Polynomial Time

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Abstract

In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL.

In this paper, we consider a naive set theory based on Intuitionistic Light Affine Logic (ILAL), a simplification of LLL introduced by [1], and call it Light Affine Set Theory (LAST). The simplicity of LAST allows us to rigorously verify its polytime character. In particular, we prove that a function over {0, 1}* is computable in polynomial time if and only if it is provably total in LAST.

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References

  1. Asperti, A., ‘Light affine logic’, Proceedings of the Thirteenth Annual IEEE Symposium on Logic in Computer Science, 1998, pp. 300-308.

  2. Asperti, A., and L. Roversi, ‘Intuitionistic light affine logic (proof-nets, normalization complexity, expressive power, programming notation)’, ACM Transactions on Computational Logic 3(1): 137-175, 2002.

    Google Scholar 

  3. Baillot, P., ‘Stratified coherent spaces: a denotational semantics for light linear logic’, Theoretical Computer Science, to appear.

  4. Barendregt, H. P., The Lambda Calculus: Its Syntax and Semantics, Elsevier North-Holland, 1981.

    Google Scholar 

  5. Cantini, A., ‘The undecidability of Grishin's set theory’, Studia Logica 74: 345-368, 2003.

    Google Scholar 

  6. Danos, V., and J.-B. Joinet, ‘Linear logic & elementary time’, Information and Computation 183(1): 123-137, 2003.

    Google Scholar 

  7. Girard, J.-Y., ‘Light linear logic’, Information and Computation 14(3): 175-204, 1998.

    Google Scholar 

  8. Girard, J.-Y., ‘Linear logic’, Theoretical Computer Science 50: 1-102, 1987.

    Google Scholar 

  9. Grishin, V. N., ‘A nonstandard logic and its application to set theory’, In Studies in Formalized Languages and Nonclassical Logics (Russian), Izdat, Nauka, Moskow, 1974, pp. 135-171.

  10. Grishin, V. N., ‘Predicate and set theoretic calculi based on logic without contraction rules’ (Russian), Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 45(1): 47-68, 1981. English translation in Math. USSR Izv. 18(1): 41–59, 1982.

    Google Scholar 

  11. Hopcroft, J., and J. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Mass, 1979.

    Google Scholar 

  12. Kanovitch, M, M. Okada, and A. Scedrov, ‘Phase semantics for light linear logic’, Theoretical Computer Science 244(3): 525-549, 2003.

    Google Scholar 

  13. Komori, Y., ‘Illative combinatory logic based on BCK-logic’, Mathematica Japonica 34(4): 585-596, 1989.

    Google Scholar 

  14. Lincoln, P., A. Scedrov, and N. Shankar, ‘Decision problems for second order linear logic’, Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science, 1995, pp. 476-485.

  15. Murawski, A. S., and C.-H. L. Ong, ‘Discreet games, light affine logic and PTIME computation’, Proceedings of Computer Science Logic 2000, Springer-Verlag, LNCS 1862, 2000, pp. 427-441.

    Google Scholar 

  16. Neergaard, P., and H. Mairson, ‘LAL is square: Representation and expressiveness in light affine logic’, presented at the Fourth International Workshop on Implicit Computational Complexity, 2002.

  17. Peterson, U., ‘Logic without contraction as based on inclusion and unrestricted abstraction’, Studia Logica 64(3): 365-403, 2000.

    Google Scholar 

  18. Schwichtenberg, H., and A. S. Troelstra, Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1996.

  19. Shirahata, M., ‘A linear conservative extension of Zermelo-Fraenkel set theory’, Studia Logica 56: 361-392, 1996.

    Google Scholar 

  20. Shirahata, M., ‘Fixpoint theorem in linear set theory’, unpublished manuscript, available at http://www.fbc.keio.ac.jp/~sirahata/Research, 1999.

  21. Terui, K., ‘Light affine lambda calculus and polytime strong normalization’, Proceedings of the Sixteenth Annual IEEE Symposium on Logic in Computer Science, 2001, pp. 209-220. The full version is available at http://research.nii.ac.jp/~terui.

  22. Terui, K., Light Logic and Polynomial Time Computation, PhD thesis, Keio University, 2002. Available at http://research.nii.ac.jp/~terui.

  23. White, R., ‘A demonstrably consistent type-free extension of the logic BCK’, Mathematica Japonica 32(1): 149-169, 1987.

    Google Scholar 

  24. White, R., ‘A consistent theory of attributes in a logic without contraction’, Studia Logica 52: 113-142, 1993.

    Google Scholar 

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Terui, K. Light Affine Set Theory: A Naive Set Theory of Polynomial Time. Studia Logica 77, 9–40 (2004). https://doi.org/10.1023/B:STUD.0000034183.33333.6f

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  • DOI: https://doi.org/10.1023/B:STUD.0000034183.33333.6f

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