Paths, Computations and Labels in the lambda-Calculus

Andrea Asperti, Cosimo Laneve

Abstract: We provide a new characterization of Lévy's redex-families in the λ-calculus (Lévy, 1978) as suitable paths in the initial term of the derivation. The idea is that redexes in a same family are created by “contraction” (via β-reduction) of a unique common path in the initial term. This fact gives new evidence about the “common nature” of redexes in a same family, and about the possibility of sharing their reduction. In general, paths seem to provide a very friendly and intuitive tool for reasoning about redex-families, as well in theory (using paths, we shall provide a remarkably simple proof of the equivalence between extraction (Lévy, 1978) and labeling) as in practice (our characterization underlies all recent works on optimal graph reduction techniques for the λ-calculus (Lamping, 1990; Gonthier et al., 1992, Asperti, to appear), providing an original and intuitive understanding of optimal implementations).Finally, as an easy by-product of the path-characterization, we prove that neither overlining nor underlining are required in Lévy's labeling.