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An abstraction algorithm for combinatory logic

Published online by Cambridge University Press:  12 March 2014

S. Kamal Abdali
S. Kamal Abdali*
Affiliation:
Rensselaer Polytechnic Institute, Troy, New York 12181
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Extract

This note presents a practical algorithm for carrying out abstraction on combinatory terms. The well-known abstraction algorithms [1, pp. 188ff.] defining abstracts in terms of the combinator sets { S, K }, { B, C, K, W }, etc. operate on one variable at a time, and result in rather long abstracts when several variables are involved. These algorithms are not practical for the applications of the combinatory logic to the theory of computing which make much use of multi-variable abstraction (e.g., [2], [3]). The present algorithm performs the abstraction with respect to all specified variables in a single step, and yields abstracts in a concise form (with sizes proportional to those of given combinatory terms).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 1 , March 1976 , pp. 222 - 224
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

The material in this paper is derived from the author's Ph.D. dissertation [3], written under the supervision of Professor George W. Petznick. The author is also thankful to Professor Haskell B. Curry for several suggestions, including the present definition of (in formulae (3)). The preparation of this paper was supported by NSF grant GJ 25393.

References

REFERENCES

[1] Curry, H. B. and Feys, R., Combinatory logic, vol. 1, North-Holland, Amsterdam, 1958.Google Scholar
[2] Orgass, R. J. and Fitch, F. B., A theory of programming languages, Studium Generale, vol. 22 (1969), pp. 113–136.Google ScholarPubMed
[3] Abdali, S. K., A combinatory logic model of programming languages, Ph.D. Dissertation, University of Wisconsin, 1974.CrossRefGoogle Scholar
[4] Curry, H. B., Apparent variables from the standpoint of combinatory logic, Annals of Mathematics (2), vol. 34 (1933), pp. 381–404.CrossRefGoogle Scholar