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On the number of lambda terms with prescribed size of their De Bruijn representation | | Bernhard Gittenberger
; Zbigniew Gołębiewski
; | | Date: |
21 Sep 2015 | | Abstract: | John Tromp introduced the so-called ’binary lambda calculus’ as a way to
encode lambda terms in terms of binary words. 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 $o(n^{-3/2} au^{-n})$ for
$ au approx 1.963448ldots$. 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 $Theta(n^{-3/2}
ho^{-n})$ with some class dependent constant $
ho$, 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. | | Source: | arXiv, 1509.6139 | | Services: | Forum | Review | PDF | Favorites | |
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