The Space of Interaction

Abstract

The space complexity of functional programs is not well understood. In particular, traditional implementation techniques are tailored to time efficiency, and space efficiency induces time inefficiencies, as it prefers re-computing to saving. Girard's geometry of interaction underlies an alternative approach based on the interaction abstract machine (IAM), claimed as space efficient in the literature. It has also been conjectured to provide a reasonable notion of space for the λ-calculus, but such an important result seems to be elusive.In this paper we introduce a new intersection type system precisely measuring the space consumption of the IAM on the typed term. Intersection types have been repeatedly used to measure time, which they achieve by dropping idempotency, turning intersections into multisets. Here we show that the space consumption of the IAM is connected to a further structural modification, turning multisets into trees. Tree intersection types lead to a finer understanding of some space complexity results from the literature. They also shed new light on the conjecture about reasonable space: we show that the usual way of encoding Turing machines into the λ-calculus cannot be used to prove that the space of the IAM is a reasonable cost model.

Multi Types and Reasonable Space (Long Version)

“…in the execution of the encoding of Turing machines into the -calculus, the exponential gap does not show up. 8 We have already proved the soundness of the type system w.r.t. termination, which does not depend on the weighting system.…”

“…In a recent work [8], Accattoli, Dal Lago, and Vanoni pushed the correspondence into a new direction, by extracting the first space bounds from multi type derivations. Additionally, they use these bounds to disprove a conjecture about reasonable space for the -calculus.…”

The Theory of Call-by-Value Solvability (long version)

“…In particular, for head reduction, which in CbN is the reduction characterizing solvability. De Carvalho's seminal work has been extended to many notions of reduction and formalisms.A first wave was inspired directly from his original work [21,34,35,47,61], and a second wave [6,7,10,11,16,23,31,[55][56][57] started after the revisitation of de Carvalho's technique by Accattoli et al [8].…”

Reasonable Space for the $λ$-Calculus, Logarithmically

“…The belief in the conjecture was re-inforced by further uses of IAM-like mechanisms for space parsimony related to circuits, by Ghica [22], and for characterizing L, by Mazza [30]. However, in 2021, Accattoli et al have essentially disproved the conjecture: the space used by the IAM to evaluate the reference encoding of TMs is unreasonable [9] (as well as time inefficient [8]). While one might look for different encodings, the unreasonable behavior of the IAM concerns the modeling of recursion via fix-point combinators, which is hardly avoidable by any encoding.…”