Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic

  • Damien Pous
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6247)

Abstract

We prove ”untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic, and give rise to a substantial optimisation of standard proof search algorithms.

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References

  1. 1.
    Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2(3), 297–347 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bellin, G., Fleury, A.: Planar and braided proof-nets for MLL with mix. Archive for Mathematical Logic 37, 309–325 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bellin, G., Scott, P.: On the π-calculus and linear logic. TCS 135, 11–65 (1994)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Braibant, T., Pous, D.: Coq library: ATBR, algebraic tools for binary relations (May 2009), http://sardes.inrialpes.fr/~braibant/atbr/
  5. 5.
    Braibant, T., Pous, D.: An efficient coq tactic for deciding Kleene algebras. In: Kaufmann, M., Paulson, L.C. (eds.) Interactive Theorem Proving. LNCS, vol. 6172, pp. 163–178. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Burmeister, P.: Partial Algebra. In: Algebras and Orders. Kluwer Pub., Dordrecht (1993)Google Scholar
  7. 7.
    Diaconescu, R.: An encoding of partial algebras as total algebras. Information Processing Letters 109(23-24), 1245–1251 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dojer, N.: Applying term rewriting to partial algebra theory. Fundamenta Informaticae 63(4), 375–384 (2004)MATHMathSciNetGoogle Scholar
  9. 9.
    Doornbos, H., Backhouse, R., van der Woude, J.: A calculational approach to mathematical induction. TCS 179(1-2), 103–135 (1997)MATHCrossRefGoogle Scholar
  10. 10.
    Freyd, P., Scedrov, A.: Categories, Allegories. North-Holland, Amsterdam (1990)MATHGoogle Scholar
  11. 11.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at substructural logics. Stud. in Log. and Found. of Math. 151, 532 (2007)MathSciNetGoogle Scholar
  12. 12.
    Girard, J.-Y.: Linear logic. TCS 50, 1–102 (1987)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Grégoire, B., Mahboubi, A.: Proving equalities in a commutative ring done right in Coq. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 98–113. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Harrison, J.: A HOL decision procedure for elementary real algebra. In: Joyce, J.J., Seger, C.-J.H. (eds.) HUG 1993. LNCS, vol. 780, pp. 426–435. Springer, Heidelberg (1994)Google Scholar
  15. 15.
    Jipsen, P.: Semirings to residuated Kleene lattices. Stud. Log. 76(2), 291–303 (2004)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jipsen, P., Tsinakis, C.: A survey of residuated lattices. Ord. Alg. Struct. (2002)Google Scholar
  17. 17.
    Kanovich, M.: The complexity of neutrals in linear logic. In: Proc. LICS, pp. 486–495. IEEE, Los Alamitos (1995)Google Scholar
  18. 18.
    Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Automata Studies, pp. 3–41. Princeton University Press, Princeton (1956)Google Scholar
  19. 19.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kozen, D.: Kleene algebra with tests. Trans. PLS 19(3), 427–443 (1997)Google Scholar
  21. 21.
    Kozen, D.: Typed Kleene algebra. Technical Report 98-1669, Cornell Univ. (1998)Google Scholar
  22. 22.
    Krob, D.: Complete systems of B-rational identities. TCS 89(2), 207–343 (1991)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mossakowski, T.: Relating CASL with other specification languages: the institution level. TCS 286(2), 367–475 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Norrish, M.: Complete integer decision procedures as derived rules in HOL. In: Basin, D., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 71–86. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  26. 26.
    Okada, M., Terui, K.: The finite model property for various fragments of intuitionistic linear logic. Journal of Symbolic Logic 64(2), 790–802 (1999)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ono, H., Komori, Y.: Logics without the contraction rule. Journal of Symbolic Logic 50(1), 169–201 (1985)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Pentus, M.: Lambek calculus is NP-complete. TCS 357(1-3), 186–201 (2006)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Pous, D.: Web appendix of this paper, http://sardes.inrialpes.fr/~pous/utas
  30. 30.
    Pratt, V.R.: Action logic and pure induction. In: van Eijck, J. (ed.) JELIA 1990. LNCS, vol. 478, pp. 97–120. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  31. 31.
    Regnier, L.: Lambda-calcul et réseaux. Thèse de doctorat, Univ. Paris VII (1992)Google Scholar
  32. 32.
    Wille, A.: A Gentzen system for involutive residuated lattices. Algebra Universalis 54, 449–463 (2005)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Yetter, D.: Quantales and (noncommutative) linear logic. Journal of Symbolic Logic 55(1), 41–64 (1990)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Damien Pous
    • 1
  1. 1.CNRS (LIG, UMR 5217) 

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