Combining first and higher order rewrite systems with type assignment systems

  • Franco Barbanera
  • Maribel Fernández
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 664)


General computational models obtained by integrating different specific models have recently become a stimulating and promising research subject for the computer science community. In particular, combinations of algebraic term rewriting systems and various λ-calculi, either typed or untyped, have been deeply investigated. In the present paper this subject is addressed from the point of view of type assignment. A powerful type assignment system, the one based on intersection types, is combined with term rewriting systems, first and higher order, and relevant properties of the resulting system, like strong normalization and confluence, are proved.


Reduction Relation Function Symbol Critical Pair Type Assignment Lambda Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. van Bakel, S. Smetsers, and S. Brock. Type assignment in left linear applicative term rewriting systems. In Proc. of CAAP'92. Colloquium on Trees in Algebra and Programming, Rennes, France, 1992, 1992.Google Scholar
  2. 2.
    F. Barbanera. Adding algebraic rewriting to the calculus of constructions: Strong normalization preserved. In Proc. of the 2nd Int. Workshop on Conditional and Typed Rewriting, 1990.Google Scholar
  3. 3.
    F. Barbanera. Combining term rewriting and type assignment systems. International Journal of Foundations of Computer Science, 1:165–184, 1990.Google Scholar
  4. 4.
    H. Barendregt, M. Coppo, and M. Dezani-Ciancaglini. A filter λ-model and the completeness of type assignment. Journal of Symbolic Logic, 48(4):931–940, 1983.Google Scholar
  5. 5.
    H. P. Barendregt. The Lambda Calculus, its Syntax and Semantics. North Holland, Amsterdam, 2nd ed., 1984.Google Scholar
  6. 6.
    Val Breazu-Tannen. Combining algebra and higher-order types. In Proc. 3rd IEEE Symp. Logic in Computer Science, Edinburgh, July 1988.Google Scholar
  7. 7.
    Val Breazu-Tannen and Jean Gallier. Polymorphic rewriting conserves algebraic strong normalization. Theoretical Computer Science, 1990. to appear.Google Scholar
  8. 8.
    F. Cardone and M. Coppo. Two extensions of Curry's type inference system. In P. Odifreddi, editor, Logic and Computer Sciennce. Academic Press, 1990.Google Scholar
  9. 9.
    Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 243–309. North-Holland, 1990.Google Scholar
  10. 10.
    Daniel J. Dougherty. Adding algebraic rewriting to the untyped lambda calculus. In Proc. 4th Rewriting Techniques and Applications, Como, LNCS 488, 1991.Google Scholar
  11. 11.
    J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.Google Scholar
  12. 12.
    M. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF. Lecture Notes in Computer Science, 78, 1979.Google Scholar
  13. 13.
    R. Hindley. Types with intersection, an introduction. Formal aspects of Computing, 1990.Google Scholar
  14. 14.
    R. Hindley and J. Seldin. Introduction to Combinators and λ-calculus. Cambridge University Press, 1986.Google Scholar
  15. 15.
    Jean-Pierre Jouannaud and Mitsuhiro Okada. Executable higher-order algebraic specification languages. In Proc. 6th IEEE Symp. Logic in Computer Science, Amsterdam, pages 350–361, 1991.Google Scholar
  16. 16.
    J. W. Klop. Term rewriting systems: a tutorial. EATCS Bulletin, 32:143–182, June 1987.Google Scholar
  17. 17.
    D. Leivant. Typing and computational properties of lambda expressions. Theoretical Computer Science, 44:51–68, 1986.Google Scholar
  18. 18.
    M. H. A. Newman. On theories with a combinatorial definition of ‘equivalence'. Ann. Math., 43(2):223–243,1942.Google Scholar
  19. 19.
    T. Nipkow. Higher order critical pairs. In Proc. IEEE Symp. on Logic in Comp. Science, Amsterdam, 1991.Google Scholar
  20. 20.
    Mitsuhiro Okada. Strong normalizability for the combined system of the types lambda calculus and an arbitrary convergent term rewrite system. In Proc. ISSAC 89, Portland, Oregon, 1989.Google Scholar
  21. 21.
    G. Pottinger. A type assignment for the strongly normalizable λ-terms. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 561–578. Academic Press, 1980.Google Scholar
  22. 22.
    Y. Toyama. Counterexamples to termination for the direct sum of term rewriting systems. Information Processing Letters, 25:141–143, April 1987.Google Scholar
  23. 23.
    D. A. Turner. Miranda: A non-strict functional language with polymorphic types. In Functional Programming Languages and Computer Architecture, Nancy, LNCS 201. Springer-Verlag, September 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Franco Barbanera
    • 1
  • Maribel Fernández
    • 2
  1. 1.Dipartimento di InformaticaTorinoItaly
  2. 2.CNRS/Université de Paris-SudOrsay CedexFrance

Personalised recommendations