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On the Number of Lambda Terms With Prescribed Size of Their De Bruijn Representation

Authors Bernhard Gittenberger, Zbigniew Golebiewski

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  • lambda calculus
  • terms enumeration
  • analytic combinatorics

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Abstract

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of 0-1-strings. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n) is o(n^{-3/2} tau^{-n}) for tau ~ 1.963448... . We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is Theta(n^{-3/2} * rho^{-n}) with some class dependent constant rho, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.

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Bernhard Gittenberger and Zbigniew Golebiewski. On the Number of Lambda Terms With Prescribed Size of Their De Bruijn Representation. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.STACS.2016.40

Author Details

Bernhard Gittenberger
Zbigniew Golebiewski

References

  1. Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc. A natural counting of lambda terms. CoRR, abs/1506.02367, 2015. URL: http://arxiv.org/abs/1506.02367.
  2. Olivier Bodini, Danièle Gardy, and Bernhard Gittenberger. Lambda-terms of bounded unary height. In Philippe Flajolet and Daniel Panario, editors, Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2011, San Francisco, California, USA, January 22, 2011, pages 23-32. SIAM, 2011. URL: http://dx.doi.org/10.1137/1.9781611973013.3.
  3. N. G. de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math., 34:381-392, 1972. Google Scholar
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