this paper is to present a new paradox for Type Theory, which is a type-theoretic refinement of Reynolds' result [24] that there is no set-theoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and the axiom of description in impredicative Type Theories. 1 Minimal and Polymorphic Higher-Order Logic

We present an experimental evaluation of Heintze and McAllester's linear-time subtransitive CFA algorithm, in the context of SML/NJ v110.7. As described in [9], linear-time termination of the algorithm depends on the existence of a simple typing of the program such that the type tree at each node has a bounded size. We show that this condition is violated by many programs, especially heavily functorized ones, and so the type-directed version of the algorithm (which explores a program's entire type structure) is only practical on small programs. We also show that the demand-driven variant of the algorithm does not always terminate for programs with polymorphic type. Our main result is that a hybrid algorithm that combines both approaches avoids both the problems. Whereas Heintze and McAllester's original formulation relied on the existence of types to bound the runtime of the algorithm, our work shows that an effective implementation must actually use the types to control termination.