  Turner's bracket abstraction algorithm is perhaps the most well-known
improvement on simple bracket abstraction algorithms. It is also one of the
most studied bracket abstraction algorithms. The definition of the algorithm in
Turner's original paper is slightly ambiguous and it has been subject to
different interpretations. It has been erroneously claimed in some papers that
certain formulations of Turner's algorithm are equivalent. In this note we
clarify the relationship between various presentations of Turner's algorithm
and we show that some of them are in fact equivalent for translating
lambda-terms in beta-normal form.
