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Journal Articles Theoretical Computer Science Year : 2020

Combining Linear Logic and Size Types for Implicit Complexity

Abstract

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 in-tensional 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 >= 0.
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Dates and versions

hal-01687224 , version 1 (18-01-2018)

Identifiers

  • HAL Id : hal-01687224 , version 1

Cite

Patrick Baillot, Alexis Ghyselen. Combining Linear Logic and Size Types for Implicit Complexity. Theoretical Computer Science, 2020, 813, pp.70-99. ⟨hal-01687224⟩
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