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Compact bracket abstraction in combinatory logic

Published online by Cambridge University Press:  12 March 2014

Sabine Broda
LIACC, Universidade do Porto, R. do Campo Alegre 823, 4150 Porto, PortugalPhone: 351 2 6001672, Fax: 351 2 6003654, E-mail:
Luís Damas
LIACC, Universidade do Porto, R. do Campo Alegre 823, 4150 Porto, Portugal Phone: 351 2 6001672 Fax: 351 2 6003654, E-mail:


Translations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schönfinkel [9]. The algorithm introduces at most one combinator for each abstraction in the initial Lambda term. This avoids explosive term growth during successive abstractions and makes the system suitable for practical applications. We prove the correctness of the algorithm and establish some relations between the combinatory system and the Lambda calculus.

Research Article
Copyright © Association for Symbolic Logic 1997

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