A combinatory system (or equivalently the set of its
basic combinators) is called combinatorially complete for a
functional system, if any member of the latter can be defined
by an entity of the former system. In this paper the
decision problem of combinatory completeness for
finite sets of proper combinators is studied for three
subsystems of the pure lambda calculus.
Precise characterizations of proper combinator bases for
the linear and the affine λ-calculus
are given, and the respective decision problems are shown
to be decidable. Furthermore, it is
determined which extensions with proper combinators of bases
for the linear λ-calculus are
combinatorially complete for the λI-calculus.