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Lambda-Lifting in Quadratic Time

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Functional and Logic Programming (FLOPS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2441))

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Abstract

Lambda-lifting is a program transformation used in compilers and in partial evaluators and that operates in cubic time. In this article, we show how to reduce this complexity to quadratic time.

Lambda-lifting transforms a block-structured program into a set of recursive equations, one for each local function in the source program. Each equation carries extra parameters to account for the free variables of the corresponding local function and of all its callees. It is the search for these extra parameters that yields the cubic factor in the traditional formulation of lambda-lifting, which is due to Johnsson. This search is carried out by a transitive closure.

Instead, we partition the call graph of the source program into strongly connected components, based on the simple observation that all functions in each component need the same extra parameters and thus a transitive closure is not needed. We therefore simplify the search for extra parameters by treating each strongly connected component instead of each function as a unit, thereby reducing the time complexity of lambda-lifting from \( \mathcal{O}(n^3 log n) \) to \( \mathcal{O}(n^2 log n) \), where n is the size of the program.

Since a lambda-lifter can output programs of size \( \mathcal{O}(n^2 ) \), we believe that our algorithm is close to optimal.

An extended version of this article is available in the BRICS Research Series [14].

Basic Research in Computer Science (www.brics.dk), funded by the Danish National Research Foundation.

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Author information

Authors and Affiliations

  1. BRICS Department of Computer Science, University of Aarhus, Ny Munkegade, Building 540, DK-8000, Aarhus C, Denmark

    Olivier Danvy

  2. Center for Pervasive Computing, Department of Computer Science, University of Aarhus, IT-Byen, Aabogade 34, DK-8200, Aarhus C, Denmark

    Ulrik P. Schultz

Authors
  1. Olivier Danvy
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  2. Ulrik P. Schultz
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Editor information

Editors and Affiliations

  1. Department of Mathematical Informatics, University of Tokyo, Hongo 7-3-1, Bunkyo-Ku, Tokyo, 113-8656, Japan

    Zhenjiang Hu

  2. Department of Informatic Languages and Systems, Universidad Complutense de Madrid, Av. Complutense s/n, 28040, Madrid, Spain

    Mario Rodríguez-Artalejo

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Danvy, O., Schultz, U.P. (2002). Lambda-Lifting in Quadratic Time. In: Hu, Z., Rodríguez-Artalejo, M. (eds) Functional and Logic Programming. FLOPS 2002. Lecture Notes in Computer Science, vol 2441. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45788-7_8

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  • DOI: https://doi.org/10.1007/3-540-45788-7_8

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  • Print ISBN: 978-3-540-44233-2

  • Online ISBN: 978-3-540-45788-6

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