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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Standard and normal reductions


Author: R. Hindley
Journal: Trans. Amer. Math. Soc. 241 (1978), 253-271
MSC: Primary 03B40
DOI: https://doi.org/10.1090/S0002-9947-1978-0492300-7
MathSciNet review: 492300
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Abstract: Curry and Feys' original standardization proof for $ \lambda \beta $-reduction is analyzed and generalized to $ \lambda \beta \eta $-reductions with extra operators.

There seem to be two slightly different definitions of 'standard reduction' in current use, without any awareness that they are different; it is proved that although these definitions turn out to be equivalent for $ \lambda \beta $-reduction, they become different for $ \lambda \beta \eta $ and for reductions involving extra operators, for example the recursion operator.

Normal reductions are also studied, and it is shown that the basic normal-reduction theorem stays true when fairly simple operators like Church's $ \delta $ and Curry's iterator Z are added, but fails for more complicated ones like the recursion operator R.

Finally, a table is given summarizing the results, and showing how far the main theorems on $ \lambda \beta $-reductions extend to reductions with various extra operators.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0492300-7
Keywords: Reduction, lambda-conversion, combinator, standard reduction, normal reduction, Gross reduction, delta-conversion, recursion combinator
Article copyright: © Copyright 1978 American Mathematical Society