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On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms
Uniquely closable skeletons of lambda terms are Motzkin-trees that
predetermine the unique closed lambda term that can be obtained by labeling
their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of
closed lambda terms predetermine the unique simply-typed lambda term that can
be obtained by labeling their leaves with de Bruijn indices.
We derive, through a sequence of logic program transformations, efficient
code for their combinatorial generation and study their statistical properties.
As a result, we obtain context-free grammars describing closable and uniquely
closable skeletons of lambda terms, opening the door for their in-depth study
with tools from analytic combinatorics.
Our empirical study of the more difficult case of (uniquely) typable terms
reveals some interesting open problems about their density and asymptotic
behavior.
As a connection between the two classes of terms, we also show that uniquely
typable closed lambda term skeletons of size are in a bijection with
binary trees of size .Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
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