On repetitive right application of B-terms

Abstract

B-terms are built from the B combinator alone defined by B f g 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 circulates or not. We discuss conditions for B-terms to 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 does 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.