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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.  ...  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.  ...  We are deeply grateful to Professor Harry Mairson and Professor Mitsuhiro Okada for useful comments and stimulating discussions.  ... 
doi:10.1007/s00153-007-0042-6 fatcat:krmcs76flfe6thyjhkddc57toa

Light types for polynomial time computation in lambda calculus

Patrick Baillot, Kazushige Terui
2009 Information and Computation  
We present a polymorphic 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).  ...  LAL and reduction It was shown in [39] that light affine lambda calculus admits polynomial step strong normalization: the bound of Theorem 1 holds on the length of any reduction sequence of light affine  ... 
doi:10.1016/j.ic.2008.08.005 fatcat:vtr4yo6kz5dsfp6qzoqe3szxte

Light types for polynomial time computation in lambda-calculus [article]

Patrick Baillot, Kazushige Terui
2004 arXiv   pre-print
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 are grateful to Paolo Coppola, Simone Martini and Ugo Dal Lago for their accurate reading and important suggestions.  ... 
arXiv:cs/0402059v2 fatcat:q2ricscan5giboyeygjk3j5pba

Light types for polynomial time computation in lambda-calculus

P. Baillot, K. Terui
2004 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.  
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 are grateful to Paolo Coppola, Simone Martini and Ugo Dal Lago for their accurate reading and important suggestions.  ... 
doi:10.1109/lics.2004.1319621 dblp:conf/lics/BaillotT04 fatcat:vgv6qbozbffv7axhycotxzsjpa

An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus [article]

Patrick Baillot, Marco Pedicini
2006 arXiv   pre-print
Given a fixed ring structure K we define an extension of Terui's light affine lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K.  ...  We show that this calculus captures the polynomial time function class FP(K): every typed term can be evaluated in polynomial time and conversely every polynomial time BSS machine over K can be simulated  ...  In section 2 we recall Light affine logic and light affine lambda-calculus and in section 3 the BSS model.  ... 
arXiv:cs/0608040v1 fatcat:zpyjpm7iafd6zfpvbytnuz3xqu

Soft lambda-Calculus: A Language for Polynomial Time Computation [chapter]

Patrick Baillot, Virgile Mogbil
2004 Lecture Notes in Computer Science  
We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time.  ...  It has been later simplified by Asperti into Light affine logic ([3]) which allows Work partially supported by Action Spécifique CNRS Méthodes formelles pour la Mobilité and ACI Sécurité Informatique CRISS  ...  We wish to thank Marcel Masseron for the stimulating discussions we had together on Soft linear logic and which led to the present paper.  ... 
doi:10.1007/978-3-540-24727-2_4 fatcat:xi3sw7bcd5dnrgqrm2x7pwkdda

Intersection Types for Light Affine Lambda Calculus

Daniel de Carvalho
2005 Electronical Notes in Theoretical Computer Science  
Light Affine Lambda Calculus is a term calculus for polynomial time computation ([12]). Some of the terms of Light Affine Lambda Calculus must however be regarded as errors.  ...  Intuitionistic Light Affine Logic (ILAL) types only terms without errors, but not all of them.  ...  Acknowledgement We have to thank Patrick Baillot for all his comments, remarks and suggestions ; we do it here warmly.  ... 
doi:10.1016/j.entcs.2005.06.011 fatcat:6gzxljsqcfd7rcnad4eyhsf5uu

Soft lambda-calculus: a language for polynomial time computation [article]

Patrick Baillot, Virgile Mogbil
2003 arXiv   pre-print
We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time.  ...  Using these datatypes we examine the concrete expressivity of Soft lambda-calculus with the example of the insertion sort algorithm.  ...  Our Soft lambda-calculus is inspired from Terui's Light affine lambda-calculus ([Ter01]), which is a calculus with a polynomial bound on reduction sequences that can be typed in Light affine logic.  ... 
arXiv:cs/0312015v1 fatcat:kyrtagxsybfe3gy27u4jayh7sm

Algebras and coalgebras in the light affine Lambda calculus

Marco Gaboardi, Romain Péchoux
2015 SIGPLAN notices  
Algebra and coalgebra are widely used to model data types in functional programming languages and proof assistants.  ...  Their use permits to better structure the computations and also to enhance the expressivity of a language or of a proof system.  ...  The Light Affine Lambda Calculus The Light Affine Lambda Calculus is the affine version of the Light Linear Lambda Calculus [39] and provide a concrete syntax for Intuitionistic Light Affine Logic [  ... 
doi:10.1145/2858949.2784759 fatcat:ezlynnzdxrepfpxnj3qtddwj2q

Soft Linear Set Theory

Richard McKinley
2008 The Journal of Logic and Algebraic Programming  
A formulation of naive set theory is given in Lafont's Soft Linear Logic, a logic with polynomial time cut-elimination.  ...  A novelty of this approach is the representation of the unary/binary natural numbers by two distinct sets (the safe naturals and the soft naturals).  ...  Acknowledgements I should like to thank Yves Lafont for some initial help, and Thomas Strahm for discussions on the work of Bellantoni and Cook.  ... 
doi:10.1016/j.jlap.2008.02.010 fatcat:jkg26rx7b5b5pixne4xi5m7cfq

Light Logics and the Call-by-Value Lambda Calculus

Paolo Coppola, Ugo Dal Lago, Simona Ronchi Della Rocca, Harry Mairson
2008 Logical Methods in Computer Science  
In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic.  ...  The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties.  ...  This phenomenon has catastrophic consequences in the context of Light Affine Logic, where polynomial time bounds cannot be transferred from the logic to pure lambda-calculus [6] .  ... 
doi:10.2168/lmcs-4(4:5)2008 fatcat:li2b2uivfrc23lxtselbo6eszi

Type inference for light affine logic via constraints on words

Patrick Baillot
2004 Theoretical Computer Science  
LAL provides a typing for lambda-calculus which guarantees that a well-typed program is executable in polynomial time on any input.  ...  Light Affine Logic (LAL) is a system due to Girard and Asperti capturing the complexity class P in a proof-theoretical approach based on Linear Logic.  ...  bound (using proof-nets or light affine lambda-calculus as intermediate language).  ... 
doi:10.1016/j.tcs.2004.08.014 fatcat:hewps5vkmvhtpcy42bmfgaol7q

Elementary Affine Logic and the Call-by-Value Lambda Calculus [chapter]

Paolo Coppola, Ugo Dal Lago, Simona Ronchi Della Rocca
2005 Lecture Notes in Computer Science  
Light and elementary linear logics have been introduced as logical systems enjoying quite remarkable normalization properties.  ...  In this paper, we show that shifting from usual call-by-name to call-by-value lambda calculus allows to regain strong connections with the underlying logic.  ...  This phenomenon has catastrophic consequences in the context of light affine logic, where polynomial time bounds cannot be transferred from the logic to pure lambda-calculus [1] .  ... 
doi:10.1007/11417170_11 fatcat:yj7f7zel6jcenoiamfhxga7po4

From light logics to type assignments: a case study

M. Gaboardi, S. R. D. Rocca
2009 Logic Journal of the IGPL  
Namely the typing assures the strong normalization in a number of steps polynomial in the size of the term, and moreover all polynomial functions can be computed by λ-terms that can be typed in the system  ...  Proofs of LLL and SLL characterize polynomial time computations, while ELL characterizes elementary time computations.  ...  In Section 2, SLL is introduced, its strong normalization in polynomial time is proved and its PTIME completeness is recalled.  ... 
doi:10.1093/jigpal/jzp019 fatcat:hrdmcpi7pbed3gsxd56tf7x4vy

Combining linear logic and size types for implicit complexity

Patrick Baillot, Alexis Ghyselen
2019 Theoretical Computer Science  
a way that typed programs admit polynomial time complexity.  ...  very strong...  ...  Elementary Affine Logic and Sizes We work on an elementary affine lambda calculus based on [24] without multithreading and side-effects, that we present here.  ... 
doi:10.1016/j.tcs.2019.09.032 fatcat:wbzm7ll2dbe55lenvkvzv2bnqu
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