**Abstract** |
This paper analyses the computational behaviour of lambda-term
applications. The properties we are interested in are weak
normalisation (i.e. there is a terminating reduction) and strong
normalisation (i.e. all reductions are terminating).
We can prove that the application of a lambda-term M to a fixed
number n of copies of the same arbitrary strongly normalising
lambda-term is strongly normalising if and only if the application of
M to n different arbitrary strongly normalising
lambda-terms is strongly normalising. I.e. we have that
MX ... X (n times) is strongly normalising for an arbitrary
strongly normalising X if and only if MX1 ... Xn is strongly
normalising for arbitrary strongly normalising X1, ..., Xn. The
analogous property holds when replacing strongly normalising by weakly
normalising.
As an application of the result on strong normalisation we show that
the lambda-terms whose interpretation is the top element (in the
environment which associates the top element to all variables) of the
Honsell-Lenisa model are exactly the lambda-terms which applied to an
arbitrary number of strongly normalising lambda-terms produce always
strongly normalising lambda-terms. This proof uses a finitary logical
description of the model by means of intersection types. Therefore we
solve an open question stated by Dezani, Honsell and Motohama. |