Weak Affine Light Typing: Polytime intensional expressivity, soundness and completeness

TL;DR

WALT is a type system for lambda calculus that captures polynomial-time computability, generalizes stratification in Light Systems, and embeds the QlSRN fragment, revealing stratification of arguments in a proof-theoretic manner.

Contribution

It introduces WALT, a new light affine typing system that is both sound and complete for polynomial time, generalizing stratification and embedding QlSRN.

Findings

WALT is poly-time sound and complete.

It generalizes stratification of deductions in Light Systems.

It embeds QlSRN, highlighting argument stratification.

Abstract

Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset of lambda-terms in System F. WALT is poly-time sound: if a lambda-term M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. In particular, the evaluation can proceed under any strategy of a rewriting relation, obtained as a mix of both call-by-name/call-by-value beta-reductions. WALT is poly-time complete since it can represent any poly-time Turing machine. WALT weakens, namely generalizes, the notion of stratification of deductions common to some Light Systems -- we call as such those logical systems, derived from Linear logic, to characterize FP, the set of Polynomial functions -- . A weaker stratification allows to define a compositional embedding of the Quasi-linear fragment QlSRN of Safe recursion on notation (SRN) into WALT. QlSRN is SRN, which is a recursive-theoretical system characterizing FP, where only the composition scheme is restricted to linear safe variables. So, the expressivity of WALT is stronger, as compared to the known Light Systems. In particular, using the types, the embedding puts in evidence the stratification of normal and safe arguments hidden in QlSRN: the less an argument is impredicative, the deeper, in a formal, proof-theoretical sense, gets its representation in WALT.