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Light Affine Logic
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 1998
"... Much effort has been recently devoted to the study of polytime formal (and especially logical) systems [GSS92, LM93, Le94, Gi96]. The purpose of such systems is manyfold. On the theoretical side, they provide a better understanding of what is the logical essence of polytime reduction (and other comp ..."
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Cited by 56 (3 self)
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Much effort has been recently devoted to the study of polytime formal (and especially logical) systems [GSS92, LM93, Le94, Gi96]. The purpose of such systems is manyfold. On the theoretical side, they provide a better understanding of what is the logical essence of polytime reduction (and other complexity classes). On the practical side, via the well known CurryHoward correspondence, they yield sophisticated typing systems, where types provide (statically) an accurate upper bound on the complexity of the computation. Even more, the type annotations give essential information on the "efficient way" to reduce the term. The most promising of these logical systems is Girard 's Light Linear Logic [Gi96] (see the same paper for a comparison with other approaches). In this paper, we introduce a slight variation of LLL, by adding full weakening (for this reason, we call it Light Affine Logic). This modification does not alter the good complexity properties of LLL: cutelimination is still pol...
Adventures in time and space
 33th ACM Symposium on Principles of Programming Languages
, 2006
"... Abstract. This paper investigates what is essentially a callbyvalue version of PCF under a complexitytheoretically motivated type system. The programming formalism, ATR, has its firstorder programs characterize the polynomialtime computable functions, and its secondorder programs characterize ..."
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Cited by 3 (3 self)
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Abstract. This paper investigates what is essentially a callbyvalue version of PCF under a complexitytheoretically motivated type system. The programming formalism, ATR, has its firstorder programs characterize the polynomialtime computable functions, and its secondorder programs characterize the type2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATRtypes are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The sizerestricted part is motivated by Bellantoni and Cook’s and Leivant’s implicit characterizations of polynomialtime. The timerestricting part is an affine version of Barber and Plotkin’s DILL. Two semantics are constructed for ATR. The first is a pruning of the naïve denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR’s time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield secondorder polynomial time bounds. The timecomplexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine. 1.
On the Expressive Power of Existential Quantification in PolynomialTime Computability
"... this paper to study the expressive power of bounded existential quantification in polynomialtime computability. Our goal was to characterize nondeterministic polynomialtime computations in a machineindependent way. The following considerations are intended to make our idea clear. Let # be the fin ..."
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this paper to study the expressive power of bounded existential quantification in polynomialtime computability. Our goal was to characterize nondeterministic polynomialtime computations in a machineindependent way. The following considerations are intended to make our idea clear. Let # be the finite alphabet
Light Affine Logic: Proof Nets, Programming Notation, PTime Correctness And Completeness
"... This paper is the rst fully structured introduction to Light Ane Logic, and to its intuitionistic fragment. Light Ane Logic has a polynomially bound cut elimination complexity (PTime correctness), and encodes all PTime Turing machines (PTime completeness). PTime correctness is proved introducin ..."
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This paper is the rst fully structured introduction to Light Ane Logic, and to its intuitionistic fragment. Light Ane Logic has a polynomially bound cut elimination complexity (PTime correctness), and encodes all PTime Turing machines (PTime completeness). PTime correctness is proved introducing Proof nets. PTime completeness is based on a very compact program notation. Every of such proofs is completely selfcontained. On one side, PTime correctness proof describes how the complexity of cut elimination is controlled, thanks to a suitable cut elimination strategy that exploits structural properties of Proof nets. This allows to have a good catch on the meaning of the quite mysterious x modality. On the other side, PTime completeness proof, with a lot of examples gives a avor of the non trivial task of programming with resource limitations, using Intuitionistic Light Ane Logic derivations as programs.