Another algorithm for bracket abstraction

Abstract

This short article presents an algorithm for bracket abstraction [1] which avoids a combinatorial explosion in the size of the resulting expression when applied repeatedly for abstraction in a series of variables. It differs from a previous solution [2] in introducing only a finite number of additional combinators and in not requiring that all the variables to be abstracted be treated together in a single operation.

Translating Higher-Order Clauses to First-Order Clauses

“…In an attempt to improve the success rate, we decided to experiment with two alternatives to the venerable Curry combinators: Turner's extended combinator set [24] and λ-lifting. We were surprised to discover that they yielded no convincing improvement.…”

“…The identity combinator, I, can be defined by SKK. As Turner relates [24], Curry improved upon this system by introducing two new combinators, B and C, to handle special cases of S. The full set can be defined as follows.…”

“…Turner [24] introduced three new combinators, S , B and C , in order to address the quadratic behaviour shown above. Peyton Jones [17] claims that Turner's system can be further improved if B is replaced by B * , yielding the following definitions of the new combinators.…”

“…We have also addressed the question of how to remove λ-abstractions from the HOL problems. We have implemented both λ-lifting, which replaces λ-abstractions by newly-defined functions [6], and two different combinator translations [24]. We have carried out extensive experiments on all of these translations, using the provers E [20], SPASS [26] and Vampire 8.1 [19].…”

What is an Efficient Implementation of the lambda-calculus?

“…The combinators S,K,I,B,C were first defined by Schönfinkel [Sch24] under the names S,C,I,Z,T. Turner introduced the combinators S , B , C [Tur79a]. Functional abstraction (the translation process) is described in [Sch24,CuFe58,Tur79a].…”