Boltzmann samplers for random generation of lambda terms

Randomly generating structured objects is important in testing and optimizing functional programs, whereas generating random $'l$-terms is more specifically needed for testing and optimizing compilers. For that a tool called QuickCheck has been proposed, but in this tool the control of the random generation is left to the programmer. Ten years ago, a method called Boltzmann samplers has been proposed to generate combinatorial structures. In this paper, we show how Boltzmann samplers can be developed to generate lambda-terms, but also other data structures like trees. These samplers rely on a critical value which parameters the main random selector and which is exhibited here with explanations on how it is computed. Haskell programs are proposed to show how samplers are actually implemented

Counting and Generating Terms in the Binary Lambda Calculus (Extended version)

In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows roughly like 1.963447954. .. n. In a second part we use this approach to generate random lambda terms using Boltzmann samplers.Comment: extended version of arXiv:1401.037

Unary profile of lambda terms with restricted De Bruijn indices

In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms

Statistical properties of lambda terms

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.Comment: Major revision of section 5. In particular, proofs of Lemma 5.7 and Theorem 5.

Statistical properties of lambda terms

We present a quantitative, statistical analysis of random lambda terms in the De Bruijn notation. Following an analytic approach using multivariate generat-ing 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 con-duct 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 paramet-ers 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