Applicative Control and Computational Complexity

Abstract

We establish a tight correspondence between three major complexity classes and simple syntactic restrictions on applicative programs in the simply type lambda calculus with a recurrence operator. The syntactic retrictions considered are: recurrence arguments cannot be passed as computed values (input-driven terms), abstracted higher-order variables can appear at most once (solitary terms), and abstracted variables cannot be eventually nested (separated terms). We show that the functions over word algebras represented by input-driven terms are precisely the poly-time functions (a result akin to [8] (Chapter 24.2)). When input-driven recurrence is permitted over all finite type, the elementary functions are obtained (a result akin to [1]). When terme are further restricted to solitary ones, even recurrence in all finite type yields only the poly-time functions. Finally, separated terms generate exactly the poly-space functions. The interest in the approach discussed here lies in its simplicity: the complexity charaxcterizations are based on restricted use of standard applicative constructs, rather than a syntactic overlay as in ramified recurrence [3,12,15,7,21]. However, approaches based on ramified recurrence are more powerful than simple syntactic control, as well as more conceptually and methodologically coherent. Thus, the two approaches are complementary and of independent interest.