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Combining Linear Logic and Size Types for Implicit Complexity

Authors Patrick Baillot, Alexis Ghyselen

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Patrick Baillot
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude-Bernard Lyon 1, LIP , F-69342, Lyon Cedex 07, France
Alexis Ghyselen
  • ENS Paris-Saclay, 94230 Cachan, France

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Patrick Baillot and Alexis Ghyselen. Combining Linear Logic and Size Types for Implicit Complexity. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 9:1-9:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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 restricted versions of its !-modality controlling duplication. A second approach relies on the idea of tracking the size increase between input and output, and together with a restricted recursion scheme, to deduce time complexity bounds. However both approaches suffer from limitations : either a limited intensional expressivity, or linearity restrictions. In the present work we incorporate both approaches into a common type system, in order to overcome their respective constraints. Our system is based on elementary linear logic combined with linear size types, called sEAL, and leads to characterizations of the complexity classes FPTIME and 2k-FEXPTIME, for k >= 0.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Linear logic
  • Theory of computation → Turing machines
  • Software and its engineering → Functional languages
  • Implicit computational complexity
  • lambda-calculus
  • linear logic
  • type systems
  • polynomial time complexity
  • size types


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