On properties of B-terms

01/30/2019

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by   Mirai Ikebuchi, et al.

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B-terms are built from the B combinator alone defined by B≡λ f.λ g.λ x. f (g x), which is well known as a function composition operator. This paper investigates an interesting property of B-terms, that is, whether repetitive right applications of a B-term cycles or not. We discuss conditions for B-terms to have and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of B-terms which have the property and show that there are infinitely many B-terms which do not have the property. Also, we introduce a canonical representation of B-terms that is useful to detect cycles, or equivalently, to prove the property, with an efficient algorithm.