Almost Every Simply Typed \(\lambda \)-Term Has a Long \(\beta \)-Reduction Sequence

  • Ryoma Sin’yaEmail author
  • Kazuyuki Asada
  • Naoki Kobayashi
  • Takeshi Tsukada
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)


It is well known that the length of a \(\beta \)-reduction sequence of a simply typed \(\lambda \)-term of order \(k\) can be huge; it is as large as \(k\)-fold exponential in the size of the \(\lambda \)-term in the worst case. We consider the following relevant question about quantitative properties, instead of the worst case: how many simply typed \(\lambda \)-terms have very long reduction sequences? We provide a partial answer to this question, by showing that asymptotically almost every simply typed \(\lambda \)-term of order \(k\) has a reduction sequence as long as \((k-2)\)-fold exponential in the term size, under the assumption that the arity of functions and the number of variables that may occur in every subterm are bounded above by a constant. The work has been motivated by quantitative analysis of the complexity of higher-order model checking.



We would like to thank anonymous referees for useful comments. This work was supported by JSPS KAKENHI Grant Number JP15H05706.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Ryoma Sin’ya
    • 1
    Email author
  • Kazuyuki Asada
    • 1
  • Naoki Kobayashi
    • 1
  • Takeshi Tsukada
    • 1
  1. 1.The University of TokyoTokyoJapan

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