On counting untyped lambda terms

Despite λ-calculus is now three quarters of a century old, no formula counting λ-terms has been proposed yet, and the combinatorics of λ-calculus is considered a hard problem. The difficulty lies in the fact that the recursive expression of the numbers of terms of size n with at most m free variable...

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Veröffentlicht in:Theoretical computer science Jg. 474; S. 80 - 97
1. Verfasser: Lescanne, Pierre
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier B.V 25.02.2013
Elsevier
Schlagworte:
Catalan numbers
Combinatorics
Computer Science
Data Structures and Algorithms
Discrete Mathematics
Functional programming
Lambda calculus
Logic in Computer Science
Mathematics
Randomization
Randomization
Catalan numbers
Combinatorics
Functional programming
Lambda calculus
Enumeration
Lambda calculus complexity enumeration expected complexity
Expected complexity
Complexity
ISSN:0304-3975, 1879-2294
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Beschreibung
Zusammenfassung:Despite λ-calculus is now three quarters of a century old, no formula counting λ-terms has been proposed yet, and the combinatorics of λ-calculus is considered a hard problem. The difficulty lies in the fact that the recursive expression of the numbers of terms of size n with at most m free variables contains the number of terms of size n−1 with at most m+1 variables. This leads to complex recurrences that cannot be handled by classical analytic methods. Here based on de Bruijn indices (another presentation of λ-calculus) we propose several results on counting untyped lambda terms, i.e., on telling how many terms belong to such or such class, according to the size of the terms and/or to the number of free variables. We extend the results to normal forms.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2012.11.019