Luca Roversi

Intuitionistic Light Affine Logic

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.

A P-Time Completeness Proof for Light Logics

We explain why the original proofs of P-Time completeness forLigh t Affine Logic and Light LinearLogic can not work, and we fully develop a working one.

Lambda Calculus and Intuitionistic Linear Logic

The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.This paper introduces a typed functional language Λ! and a categorical model for it.The terms of Λ! encode a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of Λ! are built out of two disjoint sets of variables. Moreover, the λ-abstractions of Λ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the one-step operational semantics of Λ!, unlike all other extensions of Curry-Howard Isomorphism to Intuitionistic Linear Logic. The language Λ! is Church-Rosser and enjoys both Strong Normalizability and Subject Reduction.The categorical model induces operational equivalences like, for example, a set of extensional equivalences.The paper presents also an untyped version of Λ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from Λ! all the good computational properties and enjoys also the Principal-Type Property.

The call-by-value λ-calculus: a semantic investigation

This paper is about a categorical approach for modelling the pure (i.e., without constants) call-by-value λ-calculus, defined by Plotkin as a restriction of the call-by-name λ-calculus. In particular, we give the properties that a category Cbv must enjoy to describe a model of call-by-value λ-calculus. The category Cbv is general enough to catch models in Scott Domains and Coherence Spaces.

LIGHT AFFINE LOGIC AS A PROGRAMMING LANGUAGE: A FIRST CONTRIBUTION

This work is about an experimental paradigmatic functional language for programming with P-TIME functions. The language is designed from Intuitionistic Light Affine Logic. It can be typed automatically by a type inference algorithm that deduces polymorphic types a la ML.

Higher-Order Linear Ramified Recurrence

Higher-Order Linear Ramified Recurrence (HOLRR) is a linear (affine) λ-calculus — every variable occurs at most once — extended with a recursive scheme on free algebras. Two simple conditions on type derivations enforce both polytime completeness and a strong notion of polytime soundness on typeable terms. Completeness for PTIME holds by embedding Leivant’s ramified recurrence on words into HOLRR. Soundness is established at all types — and not only for first order terms. Type connectives are limited to tensor and linear implication. Moreover, typing rules are given as a simple deductive system.

A Polymorphic Language Which Is Typable and Poly-step

A functional language ΛLA is given. A sub-set ΛLA T of ΛLA is automatically typable. The types are formulas of Intuitionistic Light Affine Logic with polymorphism a la ML. Every term of ΛLA T can reduce to its normal form in, at most, poly-steps. ΛLA T can be used as a prototype of programming language for P-TIME algorithms

Intersection Types from a Proof-theoretic Perspective

In this work we present a proof-theoretical justification for the intersection type assignment system IT by means of the logical system Intersection Synchronous Logic ISL. ISL builds classes of equivalent deductions of the implicative and conjunctive fragment of the intuitionistic logic NJ. ISL results from decomposing intuitionistic conjunction into two connectives: a synchronous conjunction, that can be used only among equivalent deductions of NJ, and an asynchronous one, that can be applied among any sets of deductions of NJ. A term decoration of ISL exists so that it matches both: the IT assignment system, when only the synchronous conjunction is used, and the simple types assignment with pairs and projections, when the asynchronous conjunction is used. Moreover, the proof of strong normalization property for ISL is a simple consequence of the same property in NJ and hence strong normalization for IT comes for free.

Linear lambda calculus and deep inference

We introduce a deep inference logical system SBVr which extends SBV [6] with Rename, a self-dual atom-renaming operator. We prove that the cut free subsystem BVr of SBVr exists. We embed the terms of linear λ-calculus with explicit substitutions into formulas of SBVr. Our embedding recalls the one of full λ-calculus into π-calculus. The proof-search inside SBVr and BVr is complete with respect to the evaluation of linear λ-calculus with explicit substitutions. Instead, only soundness of proof-search in SBVr holds. Rename is crucial to let proof-search simulate the substitution of a linear λ-term for a variable in the course of linear β;-reduction. Despite SBVr is a minimal extension of SBV its proof-search can compute all boolean functions, exactly like linear λ-calculus with explicit substitutions can do.

Safe Recursion on Notation into a Light Logic by Levels

This volume contains the proceedings of the International Workshop on Developments in Implicit Computational complExity (DICE 2010), which took place on March 27-28 2010 in Paphos, Cyprus, as a satellite event of the Joint European Conference on Theory and Practice of Software, ETAPS 2010. Implicit Computational Complexity aims at studying computational complexity without referring to external measuring conditions or particular machine models, but instead by considering restrictions on programming languages or logical principles implying complexity properties. The aim of this workshop was to bring together researchers working on implicit computational complexity, from its logical and semantical aspects to those related to the static analysis of programs, so as to foster their interaction and to give newcomers an overview of the current trends in this area.We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels ( LALL), derived from the logic ML 4 . LALL is an intuitionistic deductive system, with a polynomial ti me cut elimination strategy. The embedding allows to represent every term t of SRN as a family of nets hdte l il2N in LALL. Every net dte l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the “fuzzy” borders of paragraph boxes that LALL inherits from ML 4 to “freely” duplicate the arguments, especially the safe ones, of t. Finally, the type of dte l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of dte l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principle s SRN relies on, and the proof theoretic one, called stratification, at the base of Light Linear Logic.