Asymptotically almost all \lambda-terms are strongly normalizing

David, René; Grygiel, Katarzyna; Kozik, Jakub; Raffalli, Christophe; Theyssier, Guillaume; Zaionc, Marek

doi:10.2168/lmcs-9(1:2)2013

Cited by 24 publications

(57 citation statements)

References 22 publications

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“…The limit value (3.30), coinciding in the natural size notion with the corresponding mean for head abstractions (3.24), is close to 0.4196. This result stands, again, is sharp contrast to the canonical model of David et al [19] in which variables (in closed terms) tend to be arbitrarily far from their binding abstractions. Let us point out that such a disparity is a consequence of the different combinatorial models for λ-terms.…”

Section: De Bruijn Index Values In Plain Lambda Termscontrasting

confidence: 67%

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Statistical Properties of Lambda Terms

Bendkowski

Bodini²,

Dovgal³

2019

Electron. J. Combin.

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We present a quantitative, statistical analysis of random lambda terms in the de Bruijn notation. Following an analytic approach using multivariate generating functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the de Bruijn index value profile. Moreover, we conduct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical ingredient of our analysis is a novel method of dealing with combinatorial parameters inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms.

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Section: De Bruijn Index Values In Plain Lambda Termscontrasting

confidence: 67%

“…This central difference of both combinatorial models leads to remarkably contrasting asymptotic properties, including normalisation of large random λ-terms, cf. [19,7,5].…”

Section: De Bruijn Index Values In Plain Lambda Termsmentioning

confidence: 99%

Statistical Properties of Lambda Terms

Bendkowski

Bodini²,

Dovgal³

2019

Electron. J. Combin.

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“…The to our knowledge first enumerative investigation of lambda terms was performed in [10]. Later, particular classes of lambda terms like linear and affine terms have been enumerated [4,25].…”

Section: Introductionmentioning

confidence: 99%

Enumerating lambda terms by weighted length of their De Bruijn representation

Bodini

Gittenberger

Gołębiewski

2018

Discrete Applied Mathematics

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John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of 0-1-strings using the de Bruijn representation along with a weighting scheme. 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 and according to Tromp's weights) is o n −3/2 τ −n for τ ≈ 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 Θ n −3/2 ρ −n with some class dependent constant ρ, which in particular disproves the above mentioned conjecture.The methodology used is setting up the generating functions for the classes of lambda terms. These are infinitely nested radicals which are investigated then by a singularity analysis.We show further how some properties of random lambda terms can be analyzed and present a way to sample lambda terms uniformly at random in a very efficient way. This allows to generate terms of size more than one million within a reasonable time, which is significantly better than the samplers presented in the literature so far.

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“…Representing a rather functional approach to logic and computations, lambda calculus was first studied by David et al (see [10]). Assuming a canonical representation of closed λ-terms, David et al showed that asymptotically almost all λ-terms are strongly normalising.…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of $\lambda$-terms: a natural approach

Bendkowski

Grygiel

Lescanne

et al. 2017

Journal of Logic and Computation

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We consider combinatorial aspects of λ-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for λ-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral λ-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size sampler for Motzkin trees of Bodini et alli. Using the powerful theory of analytic combinatorics, we state several results concerning the asymptotic growth rate of λ-terms in neutral, normal, and head normal forms. Finally, we investigate the asymptotic density of λ-terms containing arbitrary fixed subterms showing that, inter alia, strongly normalising or typeable terms are asymptotically negligible in the set of all λ-terms.

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Asymptotically almost all \lambda-terms are strongly normalizing

Cited by 24 publications

References 22 publications

Statistical Properties of Lambda Terms

Statistical Properties of Lambda Terms

Enumerating lambda terms by weighted length of their De Bruijn representation

Combinatorics of $\lambda$-terms: a natural approach

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