In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL. In this paper, we consider a naive set theory based on Intuitionistic Light Affine Logic (ILAL), a simplification of LLL introduced by [1], and call it Light Affine Set Theory (LAST). The simplicity of LAST allows us to rigorously verify its polytime character. In particular, we prove that a function over {0, 1} ∗ is computable in polynomial time if and only if it is provably total in LAST.

Abstract. We give a new type inference algorithm for typing lambda-terms in Elementary Affine Logic (EAL), which is motivated by applications to complexity and optimal reduction. Following previous references on this topic, the variant of EAL type system we consider (denoted EAL ⋆ ) is a variant where sharing is restricted to variables and without polymorphism. Our algorithm improves over the ones already known in that it offers a better complexity bound: if a simple type derivation for the term t is given our algorithm performs EAL ⋆ type inference in polynomial time in the size of the derivation. 1

We show that context semantics can be fruitfully applied to the quantitative analysis of proof normalization in linear logic. In particular, context semantics lets us define the weight of a proof-net as a measure of its inherent complexity: it is both an upper bound to normalization time (modulo a polynomial overhead, independently on the reduction strategy) and a lower bound to the number of steps to normal form (for certain reduction strategies). Weights are then exploited in proving strong soundness theorems for various subsystems of linear logic, namely elementary linear logic, soft linear logic and light linear logic. 1

We propose a type inference algorithm for lambda terms in Elementary Affine Logic (EAL). The algorithm decorates the syntax tree of a simple typed lambda term and collects a set of linear constraints. The result is a parametric elementary type that can be instantiated with any solution of the set of collected constraints. We point out that the typeability of lambda terms in EAL has a practical counterpart, since it is possible to reduce any EAL-typeable lambda terms with the Lamping’s abstract algorithm obtaining a substantial increasing of performances. We show how to apply the same techniques to obtain decorations of intuitionistic proofs into Linear Logic proofs.

A fragment of second-order lambda calculus (System F) is defined that characterizes the elementary recursive functions. Type quantification is restricted to be non-interleaved and stratified, i.e., the types are assigned levels, and a quantified variable can only be instantiated by a type of smaller level, with a slightly liberalized treatment of the level zero.

We address the question of injectivity of coherent semantics of linear logic proof-nets. Starting from Girard's denition of experiment, we introduce the key-notion of \injective obsessional experiment ", which allows to give a positive answer to our question for certain fragments of linear logic, and to build counter-examples to the injectivity of coherent semantics in the general case.

This paper fits in the area of implicit polytime computational systems [Girard et al. 1998; Leivant and Marion 1993; Leivant 1994; Girard 1998]. The purpose of such systems is manifold. On the theoretical side, they provide a better understanding about the logical essence of calculating with time restrictions. Those ones admitting a Curry-Howard correspondence

This article is a structured introduction to Intuitionistic Light Affine Logic (ILAL). ILAL has a polynomially costing normalization, and it is expressive enough to encode, and simulate, all PolyTime Turing machines. The bound on the normalization cost is proved by introducing the proof-nets for ILAL. The bound follows from a suitable normalization strategy that exploits structural properties of the proof-nets. This allows us to have a good understanding of the meaning of the § modality, which is a peculiarity of light logics. The expressive power of ILAL is demonstrated in full detail. Such a proof gives a hint of the nontrivial task of programming with resource limitations, using ILAL derivations as programs.

This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofs-asprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity bound on the cut-elimination procedure. Here, we propose a more flexible control principle, that of indexing, which allows us to extend Girard’s systems while keeping the same complexity properties. A consequence of the higher flexibility of indexing with respect to stratification is the absence of boxes for handling the § modality. We finally propose a variant of our polytime system in which the § modality is only allowed on atoms, and which may thus serve as a basis for developing λ-calculus type assignment systems with more efficient typing algorithms than existing ones.

Abstract. In a previous work we introduced Dual light affine logic (DLAL) ([BT04]) as a variant of Light linear logic suitable for guaranteeing complexity properties on lambda calculus terms: all typable terms can be evaluated in polynomial time by beta reduction and all Ptime functions can be represented. In the present work we address the problem of typing lambda-terms in second-order DLAL. For that we give a procedure which, starting with a term typed in system F, determines whether it is typable in DLAL and outputs a concrete typing if there exists any. We show that our procedure can be run in time polynomial in the size of the original Church typed system F term. 1.