Lambda lifting

Abstract: The description for this book, The Calculi of Lambda Conversion. (AM-6), will be forthcoming.

Abstract: Lambda lifting is a technique for transforming a functional program with local function definitions, possibly with free variables in the function definitions, into a program consisting only of global function (combinator) definitions which will be used as rewrite rules. Different ways of doing lambda lifting are presented, as well as reasons for rejecting or selecting the method used in our Lazy ML compiler. A functional program implementing the chosen algorithm is given.

Abstract: An elementary, purely algebraic definition of model for the untyped lambda calculus is given. This definition is shown to be equivalent to the natural semantic definition based on environments. These definitions of model are consistent with, and yield a completeness theorem for, the standard axioms for lambda convertibility. A simple construction of models for lambda calculus is reviewed. The algebraic formulation clarifies the relation between combinators and lambda terms.

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.