On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

Combinatorics of explicit substitutions

$\lambda\upsilon$ is an extension of the $\lambda$-calculus which internalises the calculus of substitutions. In the current paper, we investigate the combinatorial properties of $\lambda\upsilon$ focusing on the quantitative aspects of substitution resolution. We exhibit an unexpected correspondence between the counting sequence for $\lambda\upsilon$-terms and famous Catalan numbers. As a by-product, we establish effective sampling schemes for random $\lambda\upsilon$-terms. We show that typical $\lambda\upsilon$-terms represent, in a strong sense, non-strict computations in the classic $\lambda$-calculus. Moreover, typically almost all substitutions are in fact suspended, i.e. unevaluated, under closures. Consequently, we argue that $\lambda\upsilon$ is an intrinsically non-strict calculus of explicit substitutions. Finally, we investigate the distribution of various redexes governing the substitution resolution in $\lambda\upsilon$ and investigate the quantitative contribution of various substitution primitives

Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence

It is well known that the length of a beta-reduction sequence of a simply typed lambda-term of order k can be huge; it is as large as k-fold exponential in the size of the lambda-term in the worst case. We consider the following relevant question about quantitative properties, instead of the worst case: how many simply typed lambda-terms have very long reduction sequences? We provide a partial answer to this question, by showing that asymptotically almost every simply typed lambda-term of order k has a reduction sequence as long as (k-1)-fold exponential in the term size, under the assumption that the arity of functions and the number of variables that may occur in every subterm are bounded above by a constant. To prove it, we have extended the infinite monkey theorem for strings to a parametrized one for regular tree languages, which may be of independent interest. The work has been motivated by quantitative analysis of the complexity of higher-order model checking