Paths , Computations and Labelsin the - calculus

We provide a new characterization of L evy's redex-families in the-calculus 12] as suitable paths in the initial term of the derivation. The idea is that redexes in a same family are created by \contraction" (via-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 12] and labeling) as in practice (our characterization underlies all recent works on optimal graph reduction techniques for the-calculus 10, 7, 8, 1], 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 L evy's labeling.