  We define two extensions of the typed linear lambda-calculus that yield
minimal Turing-complete systems. The extensions are based on unbounded
recursion in one case, and bounded recursion with minimisation in the other. We
show that both approaches are compatible with linearity and typeability
constraints. Both extensions of the typed linear lambda-calculus are minimal,
in the sense that taking out any of the components breaks the universality of
the system. We discuss implementation techniques that exploit the linearity of
the calculi. Finally, we apply the results to languages with fixpoint
operators: we give a compilation of the programming language PCF into a linear
lambda-calculus with linear unbounded recursion.
