Let be associated to each element N of the set of the normal forms of the λ-κ-β calculus and to each integer r > 0 the semi but non - decidable domain D [N,r] $$ \subseteq $$ r onto which N, considered as partial map ping r→, is total (that is the computation starting from NX1 ... Xr where N ∈ and X1, ..., xr ∈ D [N,r] and evolging through a β -reduction algorithm terminates). The decidability of the relation D [N,r] = r has been proved in a previous paper. In the present paper, for any N and r, an infinite, decidable subdomain C [N,r] $$ \subseteq $$ D [N,r] is defined in a constructive way. The ensuing sufficient condition for the termination of a computation starting from N X1 ... Xr can be tested in a number of steps negligible with respect to those needed for reaching the n.f., if there is one.