Completeness of transfinite evaluation in an extension of the lambda calculus

Abstract

The lambda calculus was conceived by Church, and to some extent by Curry (see [1] and [2]), as a general theory of functions, more precisely as a formalization of two basic functional concepts: application and abstraction. This claim to universality was impossible to maintain, for it soon became apparent that functions in the calculus are required to have properties which are by no means universal. Self-application is certainly a very special feature, but more relevant from our point of view is the existence of fixed points, which is definitely a nonuniversal property of functions. This situation was not ignored, but was considered rather as a form of inconsistency very much related to the paradoxes of set theory (for example, see the discussion of the paradoxical combinator in [2]).More recently, the lambda calculus has been studied by people in computer science, and as a result the algorithmic nature of the system has been more explicitly recognized. Eventually, the right approach was taken by D. Scott (see [4] and [5]) who restricted the objects of the calculus to continuous functionals over complete lattices, which can be combined in structures where application and abstraction are properly interpreted. It soon became clear that complete lattices were not necessary, and more general structures were introduced. On the other hand the requirement of continuity was essential for the construction, and has remained a basic feature. It is one of the purposes of this paper to show that in the general situation continuity can be replaced by monotonicity.The work of Scott was extended by Wadsworth [6], who gave a precise formulation of the operational semantics, and proved the equivalence to Scott's interpretation.