Computation by interaction for space-bounded functional programming

Abstract

We consider the problem of supporting sublinear space programming in a functional programming language. 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 core functional language intml is formulated by means of a type system inspired by Baillot & Terui's Dual Light Affine Logic. It can be understood as a programming language simplification of Stratified Bounded Affine Logic. We show that it captures the classes flogspace and nflogspace of the functions computable in deterministic logarithmic space and in non-deterministic logarithmic space, respectively. We illustrate the expressiveness of intml by showing how typical graph algorithms, such a test for acyclicity in undirected graphs, can be represented in it.

The Machinery of Interaction

“…The key point is that the token does not store information about every single β-redex, thus disentangling space-consumption from time-consumption. In other words, GoI machines are good candidates for space-efficient implementation schemes, as first shown by Schöpp and coauthors [23,51]. The price to pay is that the machine wastes a lot of time to retrieve β-redexes, so that time is sacrificed for space.…”

“…The GoI has been studied in relationship with implementations of functional languages, by Gonthier, Abadi and Levy, who studied Lévy's optimal evaluation [34], and by Mackie with his GoI machine for PCF [41], Gödel System T [42], and-with Fernandez-for call-by-value [30]. The space-efficiency studied by Dal Lago and Schöpp [23] is exploited by Mazza in [44] and, together with Terui, in [45]. Dal Lago and coauthors have also introduced variants of the IAM acting on proof nets for a number of extensions of the λcalculus [19][20][21]24].…”

The Abstract Machinery of Interaction (Long Version)

“…The key point is that the token does not store information about every single β-redex, thus disentangling space-consumption from timeconsumption. In other words, GoI machines are good candidates for space-efficient implementation schemes, as first shown by Dal Lago and Schöpp [DLS16]. The price to pay is that the machine wastes a lot of time to retrieve β-redexes, so that time is sacrificed for space.…”

“…The GoI has also been studied in relationship with implementation of functional languages, by Gonthier, Abadi and Levy as a proof methodology in the study of optimal implementations [GAL92], and by Mackie with his GoI machine for PCF [Mac95] and Gödel System T [Mac17]. Recently, the space-efficiency studied by Dal Lago and Schöpp [DLS16] has been exploited by Mazza in [Maz15] and, together with Terui, in [MT15]. Dal Lago and coauthors have also introduced variants of the IAM acting on proof nets for a number of extensions of the λ-calculus…”

The Dynamic Geometry of Interaction Machine: A Call-by-need Graph Rewriter

“…The interactionbased approach is also convenient for the complexity analysis of programs, e.g. Dal Lago and Schöpp's IntML type system of logarithmic-space evaluation [7], and Dal Lago et al's linear dependent type system of polynomialtime evaluation [5,6].…”