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Strong reduction and normal form in combinatory logic1

Published online by Cambridge University Press:  12 March 2014

Bruce Lercher
Bruce Lercher*
Affiliation:
State University of New York at Binghamton (Harpur College)
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Extract

The notion of strong reduction is introduced in Curry and Feys' book Combinatory logic [1] as an analogue, in the theory of combinatore, to reduction (more exactly, βη-reduction) in the theory of λ-conversion. The existence of an analogue and its possible importance are suggested by an equivalence between the theory of combinatore and λ-conversion, and the Church-Rosser theorem in λ-conversion. This theorem implies that if a formula X is convertible to a formula X* which cannot be further reduced—is irreducible, or in normal form—then X is convertible to X* by a reduction alone. Moreover, the reduction may be performed in a certain prescribed order.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 32 , Issue 2 , August 1967 , pp. 213 - 223
Copyright
Copyright © Association for Symbolic Logic 1967

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Footnotes

1

Presented, in a somewhat different version, to the Association for Symbolic Logic on December 27, 1963. This work was supported in part by The Research Foundation of State University of New York on fellowship FRF 64–40–30.

References

[1]Curry, H. B. and Feys, R., Combinatory logic, Vol. I, North-Holland, Amsterdam, 1958.Google Scholar
[2]Curry, H. B., Combinatory logic, Vol. II, in preparation.Google Scholar
[3]Hindley, Roger, Axioms for strong reduction in combinatory logic, this Journal, Vol. 32 (1967), pp. 224–236.Google Scholar
[4]Lercher, Bruce, Strong reduction and recursion in combinatory logic, Ph.D. Thesis, The Pennsylvania State University, University Park, Pa., 1963.Google Scholar
[5]Sanchis, L. E., Normal combinations and the theory of types, Ph.D. Thesis, The Pennsylvania State University, University Park, Pa., 1963.Google Scholar
[6]Sanchis, L. E.. Types in combinatory logic, Notre Dame Journal of Formal Logic, Vol. 5 (1964), pp. 161–180.CrossRefGoogle Scholar