λ-Terms as total or partial functions on normal forms

Abstract

In this paper the set of λ -terras is split into 2ω+1 disjoint classes P _h (−ω≤h≤ω). This classification takes into account the meaning of a λ-term F as function on normal forms, and more precisely:

• 1 iff when auccessively applied to any number of normal forms it gives a λ-term without normal form

• 2 (0<h<ω) iff when successively applied to h-1 arbitrary normal forms it gives a λ-term without normal form, but there exist h normal forms X₁,...,X_h such that FX₁...X_h possesses normal form

• 3 (0≤h>ω) iff when successively applied to h arbitrary normal forms it gives a λ-term which possesses normal form, but there exist h+1 normal forms X₁,...,X_h+1 such that FX₁...X_h+1 possesses no normal form

• 4 iff when successively applied to any number of normal forms it gives a λ-term which possesses normal form.

This classification can be effectively determined only for λ-terms in normal form.