Light affine lambda calculus and polynomial time strong normalization

Kazushige Terui
2007 Archive for Mathematical Logic  
Light Linear Logic (LLL) and its variant, Intuitionistic Light Affine Logic (ILAL), are logics of polynomial time computation. It is known that every polynomial time function is representable by a proof of these logics via the proofs-as-programs correspondence, and conversely that there is a specific reduction strategy which normalizes a given proof in polynomial time. While the latter may well be considered as the polytime "weak" normalization theorem, it has not been known whether a "strong"
more » ... orm of polytime normalization theorem holds or not. In this paper, we introduce an untyped term calculus, called Light Affine Lambda Calculus (λla), based on the essential ideas of Light Logics. It amounts to a bi-modal λ-calculus with certain constraints, endowed with a very simple notion of reduction. We then prove the polytime strong normalization theorem for this calculus: any reduction strategy normalizes a given λla term in a polynomial number of reduction steps, and indeed in polynomial time. Since proofs of ILAL are structurally representable by terms of λla, we may conclude that the same holds for ILAL.
doi:10.1007/s00153-007-0042-6 fatcat:krmcs76flfe6thyjhkddc57toa