λ-Calculus and Computer Science Theory pp 62-82 | Cite as

# Infinite normal forms for the λ-calculus

Conference paper

First Online:

## Abstract

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, ℂ_{fin}. The D_{∞}-value of X ε Λ is characterized by C(X) ε ℂ_{fin}. Extending the syntactical structure of ℂ_{fin} into ℂ_{inf}, 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 ℂ_{inf} Since ℂ_{inf} is deduced from Λ independent of D_{∞}, ℂ_{inf} can be said to give a natural lattice structure of Λ.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Barendregt, H.P., Some extensional term models for combinatory logics and λ-calculi, Thesis, Utrecht (1971).Google Scholar
- [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]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]Nakajima, R., Ph.D. Thesis, University of California at Berkeley (to appear).Google Scholar
- [5]Park, D., The Y-combinator in Scott's λ-calculus models, Symposium on Theory of Programming, University of Warwick (1970).Google Scholar
- [6]Plotkin, C.D., The λ-calculus is ω-incomplete, SAI-RM-2, School of Artificial Intelligence, University of Edinburgh (1973).Google Scholar
- [7]Reynolds, J., Lattice theoretic approach to theory of computation, Unpublished lecture notes, Syracuse University (1971).Google Scholar
- [8]Scott, D., Outline of a mathematical theory of computation, Oxford Monograph PRG-2, Oxford University (1970).Google Scholar
- [9]Scott, D., Continuous lattices, Oxford Monograph PRG-7, Oxford University (1972).Google Scholar
- [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]Scott, D., The lattice of flow diagrams, Semantics of Algorithmic Languages, Springer Lecture Notes in Mathematics, Vol. 188 (1971), 311–366.Google Scholar
- [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]Wadsworth, C.P., Approximate reductions and λ-calculus models, SIAM Journal of Computing (to appear).Google Scholar
- [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