Pure functions in APL and J

Any expression in combinatory logic made up of combinators and variables can be abstracted into a pure combinator expression applied to a sequence of variables. Because there are great similarities between combinators and certain APL operators, a similar result obtains in many APL dialects. However, rewriting arbitrary APL expressions as pure functions requires new operators, not provided as primitives by any dialect. This paper defines functional completeness, gives a construction for achieving it, proves a conjecture of Kenneth Iverson that J is functionally complete, and shows how closely the major APL dialects have approached these conditions.