Lambda calculus schemata

Abstract

A lambda calculus schema is an expression of the lambda calculus augmented by uninterpreted constant and function symbols and thus is an abstraction of programming languages such as LISP which permit functions to be passed to or returned from other functions. We then consider two natural implementation strategies: the retention strategy in which all variable bindings are retained until no longer needed (implying the use of some sort of garbage collected store) and the deletion strategy, modelled after the usual stack implementation of ALGOL-60, in which variable bindings are destroyed when control leaves the procedure (or block) in which they were created. Berry shows that the deletion strategy implementation is not “correct” for a wide class of languages in the sense that it is not equivalent to natural extensions of the copy rule of ALGOL to such languages, whereas the retention strategy is correct in that sense. We show, however, that no real power is lost in restricting oneself to a deletion strategy implementation, for any program can be translated into an equivalent one which will work correctly under such an implementation. The proof makes no use of the particular primitive functions and data of the language and hence is true of the corresponding schemata under all interpretations.