Infinite normal forms for the λ-calculus

  • Reiji Nakajima
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 37)


The notion of C-function is introduced to λ-calculus with η-convertibility as a generalization of normal forms. C is a function from the λ-expressions, Λ, onto a partially ordered set, ℂfin. The D-value of X ε Λ is characterized by C(X) ε ℂfin. Extending the syntactical structure of ℂfin into ℂinf, we generalize Λ to Λ, the infinite λ-expressions. The lattice topology of Λ and Λ induced by D is equivalent to the lattice topology of ℂinf Since ℂinf is deduced from Λ independent of D, ℂinf can be said to give a natural lattice structure of Λ.


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  1. [1]
    Barendregt, H.P., Some extensional term models for combinatory logics and λ-calculi, Thesis, Utrecht (1971).Google Scholar
  2. [2]
    Böhm, C., Alcune proprieta della forme β-η-normali del λ-κ-calcolo, Publicazioni dell'Istituto per le Applicazioni Del Calcolo, No. 696, Rome (1968).Google Scholar
  3. [3]
    Morris, J. and Nakajima, R., Mechanical characterization of the partial order in lattice model, D, of the λ-calculus, Technical Report No. 18, Department of Computer Science, University of California at Berkeley (1973).Google Scholar
  4. [4]
    Nakajima, R., Ph.D. Thesis, University of California at Berkeley (to appear).Google Scholar
  5. [5]
    Park, D., The Y-combinator in Scott's λ-calculus models, Symposium on Theory of Programming, University of Warwick (1970).Google Scholar
  6. [6]
    Plotkin, C.D., The λ-calculus is ω-incomplete, SAI-RM-2, School of Artificial Intelligence, University of Edinburgh (1973).Google Scholar
  7. [7]
    Reynolds, J., Lattice theoretic approach to theory of computation, Unpublished lecture notes, Syracuse University (1971).Google Scholar
  8. [8]
    Scott, D., Outline of a mathematical theory of computation, Oxford Monograph PRG-2, Oxford University (1970).Google Scholar
  9. [9]
    Scott, D., Continuous lattices, Oxford Monograph PRG-7, Oxford University (1972).Google Scholar
  10. [10]
    Scott, D., Lattice theory, data types and semantics, Formal Semantics of Programming Languages, Courant Computer Science Symposium 2 (1970), 65–106.Google Scholar
  11. [11]
    Scott, D., The lattice of flow diagrams, Semantics of Algorithmic Languages, Springer Lecture Notes in Mathematics, Vol. 188 (1971), 311–366.Google Scholar
  12. [12]
    Wadsworth, C.P., The relation between λ-expressions and their denotations in Scott's models for the λ-calculus, SIAM Journal of Computing (to appear).Google Scholar
  13. [13]
    Wadsworth, C.P., Approximate reductions and λ-calculus models, SIAM Journal of Computing (to appear).Google Scholar
  14. [14]
    Wadsworth, C.P., A general form of a theorem of Böhm and its application to Scott's model for the λ-calculus (to appear).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • Reiji Nakajima
    • 1
  1. 1.Computer Science DivisionUniversity of CaliforniaBerkeley

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