Enumerating lambda terms by weighted length of their De Bruijn representation

Abstract

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.

Statistical Properties of Lambda Terms

“…Such a result should not be surprising given the exponential convergence speed at which h-shallow λ-terms of size N tend to closed λ-terms of size N as h → ∞, see Section 5, cf. [15]. Consequently, virtually all presented histograms for closed λ-terms are also correct histograms for h-shallow terms.…”

“…Using Theorem 5.9 it is possible to prove that a m and b m converge to their respective limits exponentially fast. Consequently, the approximation procedure proposed in [15] convergences exponentially fast, as well.…”

“…Nonetheless, unlike the limit distribution of longest runs, the discovered distribution does not suggest a limit concentration due to the significant values of the observed variances. [15] and efficient techniques allowing to obtain numerical estimates for the relative asymptotic density of m-open terms in the set of all plain λ-terms [27] it is possible to approximate the limit distribution of the m-openness parameter in plain terms. However, in order to avoid the laborious computations involved in this approach, we offer a simple Monte Carlo approximation scheme.…”

“…In [15,Lemma 8] the authors prove that for each m 0 the generating functions for m-open λ-terms L m (z) admit Puiseux expansions in form of…”

How to generate random lambda terms?

“…Given these concerns, we have to decide on a representation in which we do not differentiate bound variable names. We can therefore pick one canonical representative for each α-equivalence class of terms, in effect sampling terms up to α-equivalence, see [54,55,12], or use a representation in which there exists just one such representative, for instance the De Bruijn notation [18], see [37,14,8]. Recall that in the De Bruijn notation there are no named variables.…”