Light types for polynomial time computation in lambda-calculus

We propose a new type system for lambda-calculus ensuring that well-typed programs can be executed in polynomial time: Dual light affine logic (DLAL). DLAL has a simple type language with a linear and an intuitionistic type arrow, and one modality. It corresponds to a fragment of Light affine logic (LAL). We show that contrarily to LAL, DLAL ensures good properties on lambda-terms: subject reduction is satisfied and a well-typed term admits a polynomial bound on the reduction by any strategy. We establish that as LAL, DLAL allows to represent all polytime functions. Finally we give a type inference procedure for propositional DLAL.Comment: 20 pages (including 10 pages of appendix). (revised version; in particular section 5 has been modified). A short version is to appear in the proceedings of the conference LICS 2004 (IEEE Computer Society Press

MacNeille Completion and Buchholz\u27 Omega Rule for Parameter-Free Second Order Logics

Buchholz\u27 Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada\u27s algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the (Dedekind-)MacNeille completion together with Girard\u27s reducibility candidates. Interestingly, it turns out that the Omega-rule, formulated as a rule of logical inference, finds its algebraic foundation in the MacNeille completion. In this paper, we consider a family of sequent calculi LIP = cup_{n >= -1} LIP_n for the parameter-free fragments of second order intuitionistic logic, that corresponds to the family ID_{<omega} = cup_{n <omega} ID_n of arithmetical theories of inductive definitions up to omega. In this setting, we observe a formal connection between the Omega-rule and the MacNeille completion, that leads to a way of interpreting second order quantifiers in a first order way in Heyting-valued semantics, called the Omega-interpretation. Based on this, we give a (partly) algebraic proof of cut elimination for LIP_n, in which quantification over reducibility candidates, that are genuinely second order, is replaced by the Omega-interpretation, that is essentially first order. As a consequence, our proof is locally formalizable in ID-theories

Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.

Following Aehlig, we consider a hierarchy F^p= { F^p_n }_{n in Nat} of parameter-free subsystems of System F, where each F^p_n corresponds to ID_n, the theory of n-times iterated inductive definitions (thus our F^p_n corresponds to the n+1th system of Aehlig). We here present two proofs of strong normalization for F^p_n, which are directly formalizable with inductive definitions. The first one, based on the Joachimski-Matthes method, can be fully formalized in ID_n+1. This provides a tight upper bound on the complexity of the normalization theorem for System F^p_n. The second one, based on the Godel-Tait method, can be locally formalized in ID_n. This provides a direct proof to the known result that the representable functions in F^p_n are provably total in ID_n. In both cases, Buchholz\u27 Omega-rule plays a central role

Verification of Ptime reducibility for system F terms via Dual Light Affine Logic.

Proceedings of Computer Science Logic 2006 (CSL'06), volume 4207 of Lecture Notes in Computer Science, pp.150-166. � SpringerIn a previous work we introduced Dual Light Affine Logic (DLAL) ([BaillotTerui04]) 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 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, finds all possible ways to decorate it into a DLAL typed term. We show that our procedure can be run in time polynomial in the size of the original Church typed system F term

On the meaning of focalization

Abstract In this paper, we use Girard&apos;s Ludics to analyze focalization, a fundamental property of linear logic. In particular, we show how this can be realized interactively thanks to section-retraction pairs (u αβ , f αβ ) between behaviours α ˆ(β Y ), X and αβ Y, X

Sublogarithmic uniform Boolean proof nets

Using a proofs-as-programs correspondence, Terui was able to compare two models of parallel computation: Boolean circuits and proof nets for multiplicative linear logic. Mogbil et. al. gave a logspace translation allowing us to compare their computational power as uniform complexity classes. This paper presents a novel translation in AC0 and focuses on a simpler restricted notion of uniform Boolean proof nets. We can then encode constant-depth circuits and compare complexity classes below logspace, which were out of reach with the previous translations.Comment: In Proceedings DICE 2011, arXiv:1201.034

The syntax and semantics of quantitative type theory

We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories