Expedited Broda-Damas Bracket Abstraction
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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 Schonfinkel [4] algorithm plus two mappings which convert the Schonfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2 n − 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.
[1] Sabine Broda,et al. Compact bracket abstraction in combinatory logic , 1997, Journal of Symbolic Logic.
[2] William C. Frederick,et al. A Combinatory Logic , 1995 .
[3] Martin W. Bunder,et al. Lambda Terms Definable as Combinators , 1996, Theor. Comput. Sci..
[4] M. Schönfinkel. Über die Bausteine der mathematischen Logik , 1924 .