Abstract
A redex R in a lambda-term M is called needed if in every reduction of M to normal form (some residual of) R is contracted. Among others the following results are proved: 1. R is needed in M iff R is contracted in the leftmost reduction path of M. 2. Let R: M0 →M1 → M2 → … reduce redexes Ri: Mi → Mi+1, and have the property that ∀i.∃j≥i.Rj is needed in Mj. Then R is normalising, i.e., if M0 has a normal form, then R is finite and terminates at that normal form. 3. Neededness is an undecidable property, but has several efficiently decidable approximations, various versions of the so-called spine redexes.
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