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