On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Baillot, Patrick

doi:10.1016/j.ic.2014.10.005

Cited by 5 publications

(22 citation statements)

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“…modality one obtains systems corresponding to different complexity classes, like light linear logic (LLL) for the class FP [15] and elementary linear logic (ELL) for the classes k-FEXPTIME, for k ě 0. [15,3,14]. These logical systems can be seen as type systems for some variants of lambda-calculi.…”

Section: State Of the Artmentioning

confidence: 99%

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Combining linear logic and size types for implicit complexity

Ghyselen

2020

Theoretical Computer Science

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Several type systems have been proposed to statically control the time complexity of lambda-calculus programs and characterize complexity classes such as FPTIME or FEXPTIME. A first line of research stems from linear logic and defines type systems based on restricted versions of the "!" modality controlling duplication. An instance of this is light linear logic for polynomial time computation [Girard98]. A second perspective relies on the idea of tracking the size increase between input and output, and together with a restricted use of recursion, to deduce from that time complexity bounds. This second approach is illustrated for instance by non-size-increasing types [Hofmann99]. However both approaches suffer from limitations. The first one, that of linear logic, has a limited intensional expressivity, that is to say some natural polynomial time programs are not typable. As to the second approach it is essentially linear, more precisely it does not allow for a non-linear use of functional arguments. In the present work we adress the problem of incorporating both approaches into a common type system. The source language we consider is a lambda-calculus with data-types and iteration, that is to say a variant of Godel's system T. Our goal is to design a system for this language allowing both to handle non-linear functional arguments and to keep a good intensional expressivity. We illustrate our methodology by choosing the system of elementary linear logic (ELL) and combining it with a system of linear size types. We discuss the expressivity of this new type system and prove that it gives a characterization of the complexity classes FPTIME and 2k-FEXPTIME, for k ě 0.

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Section: State Of the Artmentioning

confidence: 99%

“…i W (where W is a type for binary words), the larger the integer i, the more computational power we get... This results in a system that can characterize the classes k-FEXPTIME, for k ě 0 [3]. The paper [20] gives a reformulation of the principles of ELL in an extended lambda-calculus with constructions for !.…”

Section: State Of the Artmentioning

confidence: 99%

Combining linear logic and size types for implicit complexity

Ghyselen

2020

Theoretical Computer Science

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“…(This is an instance of the "type-theoretic" or "Curry-Howard" approach to implicit computational complexity.) This was refined by Baillot [1] into a characterization of each level of the k-EXPTIME hierarchy, in an affine variant of ELL.…”

Section: Introductionmentioning

confidence: 99%

“…This can be replayed in the elementary affine λ -calculus without type fixpoints, which we shall denote by EAλ . (The detailed proof given in [1] is for Elementary Affine Logic; it can be directly transposed to EAλ .) However, the characterization of P by !Str ⊸ !…”

Section: Introductionmentioning

confidence: 99%

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On the Elementary Affine Lambda-Calculus with and Without Fixed Points

Nguyên

2019

Electron. Proc. Theor. Comput. Sci.

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The elementary affine λ -calculus was introduced as a polyvalent setting for implicit computational complexity, allowing for characterizations of polynomial time and hyperexponential time predicates. But these results rely on type fixpoints (a.k.a. recursive types), and it was unknown whether this feature of the type system was really necessary. We give a positive answer by showing that without type fixpoints, we get a characterization of regular languages instead of polynomial time. The proof uses the semantic evaluation method. We also propose an aesthetic improvement on the characterization of the function classes FP and k-FEXPTIME in the presence of recursive types. * Partially supported by the ANR project ELICA (ANR-14-CE25-0005).On the elementary affine λ -calculus with and without type fixpoints Type fixpoints and Scott encodings We wish to draw attention to a particular feature of this language: the presence of type fixpoints 1 , a.k.a. recursive types. An example is the type of Scott binary strings:In the elementary affine λ -calculus as defined in [2], this recursive equation can be turned into a valid type definition, by using a fixed point operator µ on types (this explains our name µEAλ ):

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Polynomial time in untyped elementary linear logic

Olivier

2020

Theoretical Computer Science

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On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Cited by 5 publications

References 24 publications

Combining linear logic and size types for implicit complexity

Combining linear logic and size types for implicit complexity

On the Elementary Affine Lambda-Calculus with and Without Fixed Points

Polynomial time in untyped elementary linear logic

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