Light affine lambda calculus and polytime strong normalization

Light linear logic (LLL) and its variant, intuitionistic light affine logic (ILAL), are logics of polytime computation. All polynomial-time functions are representable by proofs of these logics (via the proofs-as-programs correspondence), and, conversely, that there is a specific reduction (cut-elimination) strategy which normalizes a given proof in polynomial time (the latter may well be called the polytime “weak” normalization theorem). In this paper, we introduce an untyped term calculus, called the light affine lambda calculus (λ_LA), generalizing the essential ideas of light logics into an untyped framework. It is a simple modification of the λ-calculus, and has ILAL as a type assignment system. Then, in this generalized setting, we prove the polytime “strong” normalization theorem: any reduction strategy normalizes a given λ_LA term (of fixed depth) in a polynomial number of reduction steps, and indeed in polynomial time