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

arXiv:0712.4222v2 [cs.LO] 31 Mar 2008 Weak Affine Light Typing: Polytime intensional expressivity, soundness and completeness Luca Roversi ∗† November 1, 2018 Abstract Ridefinite Quasi-linear come Composition-linear Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset of λ-terms in System F. WALT is poly-time sound: if a λ-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 β-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. Contents 1 Introduction 2 2 Weak Affine Light Typing (WALT) 3 3 Dynamic properties 9 4 Polytime soundness 12 4.1 Weak polytime soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1.1 Proving Proposition 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.2 Proving Proposition 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Strong polytime soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Quasi-linear safe recursion on notation (QlSRN) 17 6 Programming combinators in WALT 17 6.1 Basic data-types in WALT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Core combinators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 Iterators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.4 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ∗Dipartimento di Informatica, Universit`a di Torino, Corso Svizzera 185 — Torino — Italy. †e-mail:roversi@di.unito.it. home page:http://www.di.unito.it/˜rover. 1 7 From QlSRN to WALT 29 8 Conclusions and further work 31 A Completeness 33 A.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.2 Quantitative part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.3 Qualitative part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A.3.1 The look-up table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A.3.2 The transition map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.4 Encoding of a poly-time Turing machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B Details about the proofs 41 1 Introduction Implicit computational complexity (ICC) explores machine-independent characterizations of complexity classes with- out any explicit reference to resource usage bounds, which, instead, result from restricting suitable computational struc- tures. ICC systems originate from recursion theory [Cob65, BC92, LM94, Lei95, Lei99, LM], structural proof-theory and linear logic [Gir98, Laf04], rewriting systems or functional programming [Hue80, Der82, Jon99, Lei93, Lei94], 5 type systems [Hof97, Hof99a, Hof99b, Hof00, BNS00, BS01] .... This work is mainly concerned with the theoretical aspects of ICC whose essential goal is to support the evidence that the notions of the known complexity classes are natural concepts. Classically, a complexity class is defined in terms of some specific computational model. ICC aims to show that such computational models have mathematical counterparts, independent from them. Here, we approach ICC from a type-theoretical point of view. 10 We start from generalizing the structural proof-theoretical design principles, used for Light linear logic (LLL) [Gir98] and Light affine logic (LAL) [Asp98, Rov99, AR02]. The reason is that, so far, such principles look quite restrictive. Indeed, we know that LAL is polynomially strongly normalizable: the normalization of every of its deriva- tions is polynomial under every rewriting strategy [Ter01,

…(Full text truncated)…

This content is AI-processed based on ArXiv data.