Paths, computations and labels in the λ-calculus
We provide a new characterization of Lévy's redex-families in the λ-calculus  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. From this point of view, our characterization underlies all recent works on optimal graph reduction techniques for the λ-calculus [9,6,7,1], providing an original and intuitive understanding of optimal implementations.
As an easy by-product, we prove that neither overlining nor underlining are required in Lévy's labelling.
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