Term-Generic Logic

  • Andrei Popescu
  • Grigore Roşu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5486)

Abstract

Term-generic logic (TGL) is a first-order logic parameterized with terms defined axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. TGL has a complete Gentzen system generalizing that of first-order logic. A certain fragment of TGL, called Horn2, possesses a much simpler Gentzen system, similar to traditional typing derivation systems of λ-calculi. Horn2 appears to be sufficient for defining a whole plethora of λ-calculi as theories inside the logic. Within intuitionistic TGL, a Horn2 specification of a calculus is likely to be adequate by default. A bit of extra effort shows adequacy w.r.t. classic TGL as well, endowing the calculus with a complete loose semantics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abadi, M., et al.: Explicit substitutions. J. Funct. Program. 1(4), 375–416 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aczel, P.: Frege structures and the notions of proposition, truth and set. In: The Kleene Symposium, pp. 31–59. North Holland, Amsterdam (1980)CrossRefGoogle Scholar
  3. 3.
    Barendregt, H.: Introduction to generalized type systems. J. Funct. Program. 1(2), 125–154 (1991)MathSciNetMATHGoogle Scholar
  4. 4.
    Barendregt, H.P.: The Lambda Calculus. North-Holland, Amsterdam (1984)MATHGoogle Scholar
  5. 5.
    Birkhoff, G.: On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society 31, 433–454 (1935)CrossRefMATHGoogle Scholar
  6. 6.
    Bruijn, N.: λ-calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indag. Math. 34(5), 381–392 (1972)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Diaconescu, R.: Institution-independent Model Theory. Birkhauser, Basel (2008)MATHGoogle Scholar
  8. 8.
    Dowek, G., Hardin, T., Kirchner, C.: Binding logic: Proofs and models. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS, vol. 2514, pp. 130–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proc. 14th LICS Conf., pp. 193–202. IEEE, Los Alamitos (1999)Google Scholar
  10. 10.
    Gallier, J.H.: Logic for computer science. Foundations of automatic theorem proving. Harper & Row (1986)Google Scholar
  11. 11.
    Girard, J.-Y.: Une extension de l’interpretation de Gödel a l’analyse, et son application a l’elimination des coupure dans l’analyse et la theorie des types. In: Fenstad, J. (ed.) 2nd Scandinavian Logic Symposium, pp. 63–92. North Holland, Amsterdam (1971)CrossRefGoogle Scholar
  12. 12.
    Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. Journal of the ACM 39(1), 95–146 (1992)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Goguen, J., Meseguer, J.: Order-sorted algebra I. Theoretical Computer Science 105(2), 217–273 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gunter, C.A.: Semantics of Programming Languages. MIT Press, Cambridge (1992)MATHGoogle Scholar
  15. 15.
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. In: Proc. 2nd LICS Conf., pp. 194–204. IEEE, Los Alamitos (1987)Google Scholar
  16. 16.
    Hofmann, M.: Semantical analysis of higher-order abstract syntax. In: Proc. 14th LICS Conf., pp. 204–213. IEEE, Los Alamitos (1999)Google Scholar
  17. 17.
    McDowell, R.C., Miller, D.A.: Reasoning with higher-order abstract syntax in a logical framework. ACM Trans. Comput. Logic 3(1), 80–136 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Meseguer, J.: General logics. In: Ebbinghaus, H.-D., et al. (eds.) Proceedings, Logic Colloquium 1987, pp. 275–329. North-Holland, Amsterdam (1989)Google Scholar
  19. 19.
    Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Logic 51(1-2), 125–157 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mitchell, J.C.: Foundations for Programming Languages. MIT Press, Cambridge (1996)Google Scholar
  21. 21.
    Pfenning, F., Elliot, C.: Higher-order abstract syntax. In: PLDI 1988, pp. 199–208. ACM Press, New York (1988)Google Scholar
  22. 22.
    Pitts, A.M.: Nominal logic: A first order theory of names and binding. In: Kobayashi, N., Pierce, B.C. (eds.) TACS 2001. LNCS, vol. 2215, pp. 219–242. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  23. 23.
    Popescu, A., Roşu, G.: Term-generic logic. Tech. Rep. Univ. of Illinois at Urbana-Champaign UIUCDCS-R-2009-3027 (2009)Google Scholar
  24. 24.
    Reynolds, J.C.: Towards a theory of type structure. In: Robinet, B. (ed.) Programming Symposium. LNCS, vol. 19, pp. 408–423. Springer, Heidelberg (1974)CrossRefGoogle Scholar
  25. 25.
    Roşu, G.: Extensional theories and rewriting. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1066–1079. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Sannella, D., Tarlecki, A.: Foundations of Algebraic Specifications and Formal Program Development. To appear in Cambridge University Press (Ask authors for current version at tarlecki@mimuw.edu.pl)Google Scholar
  27. 27.
    Sun, Y.: An algebraic generalization of Frege structures - binding algebras. Theoretical Computer Science 211(1-2), 189–232 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Andrei Popescu
    • 1
  • Grigore Roşu
    • 1
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUSA

Personalised recommendations