Abstract. We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead compute and recompute small parts of the intermediate result on demand. In this paper, we study how the implementation of such techniques can be supported by functional programming languages. Our approach is based on modelling computation by interaction using the Int construction of Joyal, Street & Verity. We derive functional programming constructs from the structure obtained by applying the Int construction to a term model of a given functional language. The thus derived functional language is formulated by means of a type system inspired Baillot & Terui’s Dual Light Affine Logic. We assess its expressiveness by showing that it captures LOGSPACE. 1

We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new two-sorted algebra which is very similar to Cook and Bellantoni’s famous two-sorted algebra B for polynomial time [4]. The second theory describes logarithmic space by justifying concatenation- and sharply bounded recursion. All theories contain the predicates W representing words, and V representing temporary inaccessible words. They are inspired by Cantini’s theories [6] formalising B. 1.