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Expedited Broda-Damas bracket abstraction

Published online by Cambridge University Press:  12 March 2014

M. W. Bunder
M. W. Bunder*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia, E-mail:martin-bunder@uow.edu.au
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Abstract

A bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schönfinkel [4] algorithm plus two mappings which convert the Schönfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2n − 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1850 - 1857
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Broda, S. and Damas, L., Compact bracket abstraction in combinatory logic, this Journal, vol. 62 (1997), pp, 729–740.Google Scholar
[2]Bunder, M. W., Lambda terms definable as combinators, Theoretical Computer Science, vol. 169 (1996), pp. 3–21.CrossRefGoogle Scholar
[3]Curry, Haskell B. and Feys, Robert, Combinatory logic. Vol. I, North-Holland Publishing Co., Amsterdam, 1958.Google Scholar
[4]Schönfinkel, M., Über die bausteine der mathematische logik, Mathematische Annalen, (1924), pp. 305–316.Google Scholar